================================================= Cross 1 [ 1 .. 12 ] Group( () ) [ 1, 1 ] 2 [ 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6 ] Group( [ ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12) ] ) [ 2, 1 ] 3 [ 1, 2, 3, 4, 5, 6, -1, -2, -3, -4, -5, -6 ] Group( [ ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)(13,14) ] ) [ 2, 1 ] 4 [ 1, 2, 2, 1, 5, 6, 7, 8, 8, 7, 6, 5 ] Group( [ ( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9) ] ) [ 2, 1 ] [ 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 11, 11 ] Group( [ ( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,12) ] ) [ 2, 1 ] 5 [ 1, 2, -2, -1, 5, 6, 7, 8, -8, -7, -6, -5 ] Group( [ ( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9)(13,14) ] ) [ 2, 1 ] [ 1, 2, 3, 4, 5, -5, -4, -3, -2, -1, 11, -11 ] Group( [ ( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,12)(13,14) ] ) [ 2, 1 ] 6 [ 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2 ] Group( [ ( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8) ] ) [ 2, 1 ] [ 1, 2, 3, 4, 3, 2, 1, 8, 9, 10, 9, 8 ] Group( [ ( 1, 7)( 2, 6)( 3, 5)( 8,12)( 9,11) ] ) [ 2, 1 ] 7 [ 0, 2, 3, 4, 5, 6, 0, -6, -5, -4, -3, -2 ] Group( [ ( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)(13,14) ] ) [ 2, 1 ] [ 1, 2, 3, 0, -3, -2, -1, 8, 9, 0, -9, -8 ] Group( [ ( 1, 7)( 2, 6)( 3, 5)( 8,12)( 9,11)(13,14) ] ) [ 2, 1 ] 8 [ 1, 2, -2, -1, 5, -5, 1, 2, -2, -1, 5, -5 ] Group( [ ( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9)(13,14), ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12) ] ) [ 4, 2 ] 9 [ 0, 2, 3, 0, -3, -2, 0, 2, 3, 0, -3, -2 ] Group( [ ( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)(13,14), ( 1, 7)( 2, 6)( 3, 5)( 8,12)( 9,11)(13,14) ] ) [ 4, 2 ] 10 [ 1, 2, 3, 4, 3, 2, 1, 2, 3, 4, 3, 2 ] Group( [ ( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8), ( 1, 7)( 2, 6)( 3, 5)( 8,12)( 9,11) ] ) [ 4, 2 ] 11 [ 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3 ] Group( [ ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12), ( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12) ] ) [ 4, 1 ] 12 [ 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5 ] Group( [ ( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9), ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12) ] ) [ 4, 2 ] 13 [ 1, 2, 3, -1, -2, -3, 1, 2, 3, -1, -2, -3 ] Group( [ ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12), ( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)(13,14) ] ) [ 4, 1 ] 14 [ 1, 2, 2, 1, 5, -5, -1, -2, -2, -1, -5, 5 ] Group( [ ( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9), ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)(13,14) ] ) [ 4, 2 ] [ 1, 2, -2, -1, 5, 5, -1, -2, 2, 1, -5, -5 ] Group( [ ( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,12), ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)(13,14) ] ) [ 4, 2 ] 15 [ 1, 2, 3, 0, -3, -2, -1, -2, -3, 0, 3, 2 ] Group( [ ( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8), ( 1, 7)( 2, 6)( 3, 5)( 8,12)( 9,11)(13,14) ] ) [ 4, 2 ] [ 0, 2, 3, 4, 3, 2, 0, -2, -3, -4, -3, -2 ] Group( [ ( 1, 7)( 2, 6)( 3, 5)( 8,12)( 9,11), ( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)(13,14) ] ) [ 4, 2 ] 16 [ 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2 ] Group( [ ( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8), ( 1, 7)( 2, 6)( 3, 5)( 8,12)( 9,11), ( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9) ] ) [ 8, 3 ] 17 [ 1, 2, -2, -1, -2, 2, 1, 2, -2, -1, -2, 2 ] Group( [ ( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8), ( 1, 7)( 2, 6)( 3, 5)( 8,12)( 9,11), ( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9)(13,14) ] ) [ 8, 3 ] 18 [ 0, 2, 2, 0, -2, -2, 0, 2, 2, 0, -2, -2 ] Group( [ ( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)(13,14), ( 1, 7)( 2, 6)( 3, 5)( 8,12)( 9,11)(13,14), ( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9) ] ) [ 8, 3 ] 19 [ 0, 2, -2, 0, 2, -2, 0, 2, -2, 0, 2, -2 ] Group( [ ( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)(13,14), ( 1, 7)( 2, 6)( 3, 5)( 8,12)( 9,11)(13,14), ( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9)(13,14) ] ) [ 8, 3 ] 20 [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] Group( [ ( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12), ( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9), (13,14) ] ) [ 16, 11 ] Subgroup structure [ [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 ], [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ], [ 2, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1 ], [ 3, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1 ], [ 4, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1 ], [ 5, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1 ], [ 6, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1 ], [ 7, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1 ], [ 8, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1 ], [ 9, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1 ], [ 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1 ], [ 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1 ], [ 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1 ], [ 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1 ], [ 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1 ], [ 15, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1 ], [ 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1 ], [ 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1 ], [ 18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1 ], [ 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1 ], [ 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ] ] Symmetry type S1 Group( [ () ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] Symmetry type S2 Group( [ ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6 ] Symmetry type S3 Group( [ ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)(13,14) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6 ] Symmetry type S4 Group( [ ( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 4, 3, 2, 1, 12, 11, 10, 9, 8, 7, 6, 5 ] Group( [ ( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,12) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11 ] Symmetry type S5 Group( [ ( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9)(13,14) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 4, 3, 2, 1, 12, 11, 10, 9, 8, 7, 6, 5 ] Group( [ ( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,12)(13,14) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11 ] Symmetry type S6 Group( [ ( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2 ] Group( [ ( 1, 7)( 2, 6)( 3, 5)( 8,12)( 9,11) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8 ] Symmetry type S7 Group( [ ( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)(13,14) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2 ] Group( [ ( 1, 7)( 2, 6)( 3, 5)( 8,12)( 9,11)(13,14) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8 ] Symmetry type S8 Group( [ ( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9)(13,14), ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 4, 3, 2, 1, 12, 11, 10, 9, 8, 7, 6, 5 ] [ 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6 ] [ 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11 ] Symmetry type S9 Group( [ ( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)(13,14), ( 1, 7)( 2, 6)( 3, 5)( 8,12)( 9,11)(13,14) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2 ] [ 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8 ] [ 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6 ] Symmetry type S10 Group( [ ( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8), ( 1, 7)( 2, 6)( 3, 5)( 8,12)( 9,11) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2 ] [ 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8 ] [ 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6 ] Symmetry type S11 Group( [ ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12), ( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6 ] [ 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3 ] [ 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9 ] Symmetry type S12 Group( [ ( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9), ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 4, 3, 2, 1, 12, 11, 10, 9, 8, 7, 6, 5 ] [ 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6 ] [ 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11 ] Symmetry type S13 Group( [ ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12), ( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)(13,14) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6 ] [ 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3 ] [ 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9 ] Symmetry type S14 Group( [ ( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9), ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)(13,14) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 4, 3, 2, 1, 12, 11, 10, 9, 8, 7, 6, 5 ] [ 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6 ] [ 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11 ] Group( [ ( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,12), ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)(13,14) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11 ] [ 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6 ] [ 4, 3, 2, 1, 12, 11, 10, 9, 8, 7, 6, 5 ] Symmetry type S15 Group( [ ( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8), ( 1, 7)( 2, 6)( 3, 5)( 8,12)( 9,11)(13,14) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2 ] [ 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8 ] [ 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6 ] Group( [ ( 1, 7)( 2, 6)( 3, 5)( 8,12)( 9,11), ( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)(13,14) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8 ] [ 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2 ] [ 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6 ] Symmetry type S16 Group( [ ( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8), ( 1, 7)( 2, 6)( 3, 5)( 8,12)( 9,11), ( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2 ] [ 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8 ] [ 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6 ] [ 4, 3, 2, 1, 12, 11, 10, 9, 8, 7, 6, 5 ] [ 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9 ] [ 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3 ] [ 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11 ] Symmetry type S17 Group( [ ( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8), ( 1, 7)( 2, 6)( 3, 5)( 8,12)( 9,11), ( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9)(13,14) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2 ] [ 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8 ] [ 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6 ] [ 4, 3, 2, 1, 12, 11, 10, 9, 8, 7, 6, 5 ] [ 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9 ] [ 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3 ] [ 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11 ] Symmetry type S18 Group( [ ( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)(13,14), ( 1, 7)( 2, 6)( 3, 5)( 8,12)( 9,11)(13,14), ( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2 ] [ 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8 ] [ 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6 ] [ 4, 3, 2, 1, 12, 11, 10, 9, 8, 7, 6, 5 ] [ 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9 ] [ 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3 ] [ 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11 ] Symmetry type S19 Group( [ ( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)(13,14), ( 1, 7)( 2, 6)( 3, 5)( 8,12)( 9,11)(13,14), ( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9)(13,14) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2 ] [ 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8 ] [ 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6 ] [ 4, 3, 2, 1, 12, 11, 10, 9, 8, 7, 6, 5 ] [ 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9 ] [ 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3 ] [ 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11 ] Symmetry type S20 Group( [ ( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12), ( 1, 4)( 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9), (13,14) ] ) [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] [ 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3 ] [ 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3 ] [ 4, 3, 2, 1, 12, 11, 10, 9, 8, 7, 6, 5 ] [ 4, 3, 2, 1, 12, 11, 10, 9, 8, 7, 6, 5 ] [ 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6 ] [ 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6 ] [ 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2 ] [ 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2 ] [ 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11 ] [ 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11 ] [ 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9 ] [ 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9 ] [ 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8 ] [ 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8 ] Symmetry H=H1, Symmetry type S1, Id(H)=[ 1, 1 ] Representation 1, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 12, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1 ], maximals , factor group id(H/H')=[ 1, 1 ] Jumps from H1 to H[ ] Typejumps from S1 to S[ ] Symmetry H=H2, Symmetry type S2, Id(H)=[ 2, 1 ] Representation 1, dimension of irred subspace U_i = 1 K=H2, dim Fix_P(U) K = 6, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 2 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 6, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H2, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 2 ], maximals 1 , factor group id(H/H')=[ 2, 1 ] Jumps from H2 to H[ 1 ] Typejumps from S2 to S[ 1 ] Symmetry H=H3, Symmetry type S3, Id(H)=[ 2, 1 ] Representation 1, dimension of irred subspace U_i = 1 K=H3, dim Fix_P(U) K = 6, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 3 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 6, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H3, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 3 ], maximals 1 , factor group id(H/H')=[ 2, 1 ] Jumps from H3 to H[ 1 ] Typejumps from S3 to S[ 1 ] Symmetry H=H4, Symmetry type S4, Id(H)=[ 2, 1 ] Representation 1, dimension of irred subspace U_i = 1 K=H4, dim Fix_P(U) K = 6, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 4 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 6, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H4, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 4 ], maximals 1 , factor group id(H/H')=[ 2, 1 ] Jumps from H4 to H[ 1 ] Typejumps from S4 to S[ 1 ] Symmetry H=H5, Symmetry type S4, Id(H)=[ 2, 1 ] Representation 1, dimension of irred subspace U_i = 1 K=H5, dim Fix_P(U) K = 6, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 5 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 6, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H5, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 5 ], maximals 1 , factor group id(H/H')=[ 2, 1 ] Jumps from H5 to H[ 1 ] Typejumps from S4 to S[ 1 ] Symmetry H=H6, Symmetry type S5, Id(H)=[ 2, 1 ] Representation 1, dimension of irred subspace U_i = 1 K=H6, dim Fix_P(U) K = 6, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 6 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 6, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H6, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 6 ], maximals 1 , factor group id(H/H')=[ 2, 1 ] Jumps from H6 to H[ 1 ] Typejumps from S5 to S[ 1 ] Symmetry H=H7, Symmetry type S5, Id(H)=[ 2, 1 ] Representation 1, dimension of irred subspace U_i = 1 K=H7, dim Fix_P(U) K = 6, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 7 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 6, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H7, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 7 ], maximals 1 , factor group id(H/H')=[ 2, 1 ] Jumps from H7 to H[ 1 ] Typejumps from S5 to S[ 1 ] Symmetry H=H8, Symmetry type S6, Id(H)=[ 2, 1 ] Representation 1, dimension of irred subspace U_i = 1 K=H8, dim Fix_P(U) K = 7, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 8 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 5, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H8, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 8 ], maximals 1 , factor group id(H/H')=[ 2, 1 ] Jumps from H8 to H[ 1 ] Typejumps from S6 to S[ 1 ] Symmetry H=H9, Symmetry type S6, Id(H)=[ 2, 1 ] Representation 1, dimension of irred subspace U_i = 1 K=H9, dim Fix_P(U) K = 7, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 9 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 5, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H9, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 9 ], maximals 1 , factor group id(H/H')=[ 2, 1 ] Jumps from H9 to H[ 1 ] Typejumps from S6 to S[ 1 ] Symmetry H=H10, Symmetry type S7, Id(H)=[ 2, 1 ] Representation 1, dimension of irred subspace U_i = 1 K=H10, dim Fix_P(U) K = 5, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 10 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 7, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H10, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 10 ], maximals 1 , factor group id(H/H')=[ 2, 1 ] Jumps from H10 to H[ 1 ] Typejumps from S7 to S[ 1 ] Symmetry H=H11, Symmetry type S7, Id(H)=[ 2, 1 ] Representation 1, dimension of irred subspace U_i = 1 K=H11, dim Fix_P(U) K = 5, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 11 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H1, dim Fix_P(U) K = 7, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H11, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 11 ], maximals 1 , factor group id(H/H')=[ 2, 1 ] Jumps from H11 to H[ 1 ] Typejumps from S7 to S[ 1 ] Symmetry H=H12, Symmetry type S8, Id(H)=[ 4, 2 ] Representation 1, dimension of irred subspace U_i = 1 K=H12, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 12 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H6, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H12, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 6, 12 ], maximals 6 , factor group id(H/H')=[ 2, 1 ] Representation 3, dimension of irred subspace U_i = 1 K=H2, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H12, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 2, 12 ], maximals 2 , factor group id(H/H')=[ 2, 1 ] Representation 4, dimension of irred subspace U_i = 1 K=H7, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H12, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 7, 12 ], maximals 7 , factor group id(H/H')=[ 2, 1 ] Jumps from H12 to H[ 2, 6, 7 ] Typejumps from S8 to S[ 2, 5, 5 ] Symmetry H=H13, Symmetry type S9, Id(H)=[ 4, 2 ] Representation 1, dimension of irred subspace U_i = 1 K=H13, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 13 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H10, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H13, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 10, 13 ], maximals 10 , factor group id(H/H')=[ 2, 1 ] Representation 3, dimension of irred subspace U_i = 1 K=H11, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H13, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 11, 13 ], maximals 11 , factor group id(H/H')=[ 2, 1 ] Representation 4, dimension of irred subspace U_i = 1 K=H2, dim Fix_P(U) K = 4, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H13, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 2, 13 ], maximals 2 , factor group id(H/H')=[ 2, 1 ] Jumps from H13 to H[ 2, 10, 11 ] Typejumps from S9 to S[ 2, 7, 7 ] Symmetry H=H14, Symmetry type S10, Id(H)=[ 4, 2 ] Representation 1, dimension of irred subspace U_i = 1 K=H14, dim Fix_P(U) K = 4, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 14 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H8, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H14, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 8, 14 ], maximals 8 , factor group id(H/H')=[ 2, 1 ] Representation 3, dimension of irred subspace U_i = 1 K=H9, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H14, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 9, 14 ], maximals 9 , factor group id(H/H')=[ 2, 1 ] Representation 4, dimension of irred subspace U_i = 1 K=H2, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H14, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 2, 14 ], maximals 2 , factor group id(H/H')=[ 2, 1 ] Jumps from H14 to H[ 2, 8, 9 ] Typejumps from S10 to S[ 2, 6, 6 ] Symmetry H=H15, Symmetry type S11, Id(H)=[ 4, 1 ] Representation 1, dimension of irred subspace U_i = 1 K=H15, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 15 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H2, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H15, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 2, 15 ], maximals 2 , factor group id(H/H')=[ 2, 1 ] Representation 3, dimension of irred subspace U_i = 1 NON-REAL projection K=H1, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 4, 1 ] K=H15, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 15 ], maximals 1 , factor group id(H/H')=[ 4, 1 ] Representation 4, dimension of irred subspace U_i = 1 NON-REAL projection K=H1, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 4, 1 ] K=H15, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 15 ], maximals 1 , factor group id(H/H')=[ 4, 1 ] Jumps from H15 to H[ 1, 1, 2 ] Typejumps from S11 to S[ 1, 1, 2 ] Symmetry H=H16, Symmetry type S12, Id(H)=[ 4, 2 ] Representation 1, dimension of irred subspace U_i = 1 K=H16, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 16 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H4, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H16, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 4, 16 ], maximals 4 , factor group id(H/H')=[ 2, 1 ] Representation 3, dimension of irred subspace U_i = 1 K=H2, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H16, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 2, 16 ], maximals 2 , factor group id(H/H')=[ 2, 1 ] Representation 4, dimension of irred subspace U_i = 1 K=H5, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H16, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 5, 16 ], maximals 5 , factor group id(H/H')=[ 2, 1 ] Jumps from H16 to H[ 2, 4, 5 ] Typejumps from S12 to S[ 2, 4, 4 ] Symmetry H=H17, Symmetry type S13, Id(H)=[ 4, 1 ] Representation 1, dimension of irred subspace U_i = 1 K=H17, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 17 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H2, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H17, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 2, 17 ], maximals 2 , factor group id(H/H')=[ 2, 1 ] Representation 3, dimension of irred subspace U_i = 1 NON-REAL projection K=H1, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 4, 1 ] K=H17, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 17 ], maximals 1 , factor group id(H/H')=[ 4, 1 ] Representation 4, dimension of irred subspace U_i = 1 NON-REAL projection K=H1, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 4, 1 ] K=H17, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 17 ], maximals 1 , factor group id(H/H')=[ 4, 1 ] Jumps from H17 to H[ 1, 1, 2 ] Typejumps from S13 to S[ 1, 1, 2 ] Symmetry H=H18, Symmetry type S14, Id(H)=[ 4, 2 ] Representation 1, dimension of irred subspace U_i = 1 K=H18, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 18 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H4, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H18, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 4, 18 ], maximals 4 , factor group id(H/H')=[ 2, 1 ] Representation 3, dimension of irred subspace U_i = 1 K=H3, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H18, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 3, 18 ], maximals 3 , factor group id(H/H')=[ 2, 1 ] Representation 4, dimension of irred subspace U_i = 1 K=H7, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H18, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 7, 18 ], maximals 7 , factor group id(H/H')=[ 2, 1 ] Jumps from H18 to H[ 3, 4, 7 ] Typejumps from S14 to S[ 3, 4, 5 ] Symmetry H=H19, Symmetry type S14, Id(H)=[ 4, 2 ] Representation 1, dimension of irred subspace U_i = 1 K=H19, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 19 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H5, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H19, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 5, 19 ], maximals 5 , factor group id(H/H')=[ 2, 1 ] Representation 3, dimension of irred subspace U_i = 1 K=H3, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H19, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 3, 19 ], maximals 3 , factor group id(H/H')=[ 2, 1 ] Representation 4, dimension of irred subspace U_i = 1 K=H6, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H19, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 6, 19 ], maximals 6 , factor group id(H/H')=[ 2, 1 ] Jumps from H19 to H[ 3, 5, 6 ] Typejumps from S14 to S[ 3, 4, 5 ] Symmetry H=H20, Symmetry type S15, Id(H)=[ 4, 2 ] Representation 1, dimension of irred subspace U_i = 1 K=H20, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 20 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H8, dim Fix_P(U) K = 4, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H20, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 8, 20 ], maximals 8 , factor group id(H/H')=[ 2, 1 ] Representation 3, dimension of irred subspace U_i = 1 K=H11, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H20, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 11, 20 ], maximals 11 , factor group id(H/H')=[ 2, 1 ] Representation 4, dimension of irred subspace U_i = 1 K=H3, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H20, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 3, 20 ], maximals 3 , factor group id(H/H')=[ 2, 1 ] Jumps from H20 to H[ 3, 8, 11 ] Typejumps from S15 to S[ 3, 6, 7 ] Symmetry H=H21, Symmetry type S15, Id(H)=[ 4, 2 ] Representation 1, dimension of irred subspace U_i = 1 K=H21, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 21 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H9, dim Fix_P(U) K = 4, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H21, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 9, 21 ], maximals 9 , factor group id(H/H')=[ 2, 1 ] Representation 3, dimension of irred subspace U_i = 1 K=H10, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H21, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 10, 21 ], maximals 10 , factor group id(H/H')=[ 2, 1 ] Representation 4, dimension of irred subspace U_i = 1 K=H3, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H21, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 3, 21 ], maximals 3 , factor group id(H/H')=[ 2, 1 ] Jumps from H21 to H[ 3, 9, 10 ] Typejumps from S15 to S[ 3, 6, 7 ] Symmetry H=H22, Symmetry type S16, Id(H)=[ 8, 3 ] Representation 1, dimension of irred subspace U_i = 1 K=H22, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 22 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H16, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H22, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 16, 22 ], maximals 16 , factor group id(H/H')=[ 2, 1 ] Representation 3, dimension of irred subspace U_i = 1 K=H14, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H22, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 14, 22 ], maximals 14 , factor group id(H/H')=[ 2, 1 ] Representation 4, dimension of irred subspace U_i = 1 K=H15, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H22, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 15, 22 ], maximals 15 , factor group id(H/H')=[ 2, 1 ] Representation 5, dimension of irred subspace U_i = 2 K=H1, dim Fix_P(U) K = 6, dim Fix_U_i K = 2, id(N_H(K)/K)=[ 8, 3 ] K=H8, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H4, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H22, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 4, 8, 22 ], maximals 4 8 , factor group id(H/H')=[ 8, 3 ] Jumps from H22 to H[ 4, 8, 14, 15, 16 ] Typejumps from S16 to S[ 4, 6, 10, 11, 12 ] Symmetry H=H23, Symmetry type S17, Id(H)=[ 8, 3 ] Representation 1, dimension of irred subspace U_i = 1 K=H23, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 23 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H12, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H23, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 12, 23 ], maximals 12 , factor group id(H/H')=[ 2, 1 ] Representation 3, dimension of irred subspace U_i = 1 K=H14, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H23, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 14, 23 ], maximals 14 , factor group id(H/H')=[ 2, 1 ] Representation 4, dimension of irred subspace U_i = 1 K=H17, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H23, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 17, 23 ], maximals 17 , factor group id(H/H')=[ 2, 1 ] Representation 5, dimension of irred subspace U_i = 2 K=H1, dim Fix_P(U) K = 6, dim Fix_U_i K = 2, id(N_H(K)/K)=[ 8, 3 ] K=H8, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H6, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H23, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 6, 8, 23 ], maximals 6 8 , factor group id(H/H')=[ 8, 3 ] Jumps from H23 to H[ 6, 8, 12, 14, 17 ] Typejumps from S17 to S[ 5, 6, 8, 10, 13 ] Symmetry H=H24, Symmetry type S18, Id(H)=[ 8, 3 ] Representation 1, dimension of irred subspace U_i = 1 K=H24, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 24 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H16, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H24, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 16, 24 ], maximals 16 , factor group id(H/H')=[ 2, 1 ] Representation 3, dimension of irred subspace U_i = 1 K=H13, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H24, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 13, 24 ], maximals 13 , factor group id(H/H')=[ 2, 1 ] Representation 4, dimension of irred subspace U_i = 1 K=H17, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H24, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 17, 24 ], maximals 17 , factor group id(H/H')=[ 2, 1 ] Representation 5, dimension of irred subspace U_i = 2 K=H1, dim Fix_P(U) K = 6, dim Fix_U_i K = 2, id(N_H(K)/K)=[ 8, 3 ] K=H10, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H4, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H24, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 4, 10, 24 ], maximals 4 10 , factor group id(H/H')=[ 8, 3 ] Jumps from H24 to H[ 4, 10, 13, 16, 17 ] Typejumps from S18 to S[ 4, 7, 9, 12, 13 ] Symmetry H=H25, Symmetry type S19, Id(H)=[ 8, 3 ] Representation 1, dimension of irred subspace U_i = 1 K=H25, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 25 ], maximals , factor group id(H/H')=[ 1, 1 ] Representation 2, dimension of irred subspace U_i = 1 K=H12, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H25, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 12, 25 ], maximals 12 , factor group id(H/H')=[ 2, 1 ] Representation 3, dimension of irred subspace U_i = 1 K=H13, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H25, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 13, 25 ], maximals 13 , factor group id(H/H')=[ 2, 1 ] Representation 4, dimension of irred subspace U_i = 1 K=H15, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H25, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 15, 25 ], maximals 15 , factor group id(H/H')=[ 2, 1 ] Representation 5, dimension of irred subspace U_i = 2 K=H1, dim Fix_P(U) K = 6, dim Fix_U_i K = 2, id(N_H(K)/K)=[ 8, 3 ] K=H10, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H6, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H25, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 1, 6, 10, 25 ], maximals 6 10 , factor group id(H/H')=[ 8, 3 ] Jumps from H25 to H[ 6, 10, 12, 13, 15 ] Typejumps from S19 to S[ 5, 7, 8, 9, 11 ] Symmetry H=H26, Symmetry type S20, Id(H)=[ 16, 11 ] Representation 5, dimension of irred subspace U_i = 1 K=H22, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H26, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 22, 26 ], maximals 22 , factor group id(H/H')=[ 2, 1 ] Representation 6, dimension of irred subspace U_i = 1 K=H23, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H26, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 23, 26 ], maximals 23 , factor group id(H/H')=[ 2, 1 ] Representation 7, dimension of irred subspace U_i = 1 K=H25, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H26, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 25, 26 ], maximals 25 , factor group id(H/H')=[ 2, 1 ] Representation 8, dimension of irred subspace U_i = 1 K=H24, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H26, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 24, 26 ], maximals 24 , factor group id(H/H')=[ 2, 1 ] Representation 9, dimension of irred subspace U_i = 2 K=H3, dim Fix_P(U) K = 6, dim Fix_U_i K = 2, id(N_H(K)/K)=[ 8, 3 ] K=H21, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H19, dim Fix_P(U) K = 3, dim Fix_U_i K = 1, id(N_H(K)/K)=[ 2, 1 ] K=H26, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, id(N_H(K)/K)=[ 1, 1 ] isotropy subgroups H[ 3, 19, 21, 26 ], maximals 19 21 , factor group id(H/H')=[ 8, 3 ] Jumps from H26 to H[ 19, 21, 22, 23, 24, 25 ] Typejumps from S20 to S[ 14, 15, 16, 17, 18, 19 ] fromH : fromS : toK size(N_H(K)/K) ... 1 : 1 : 2 : 2 : 1 2 3 : 3 : 1 2 4 : 4 : 1 2 5 : 4 : 1 2 6 : 5 : 1 2 7 : 5 : 1 2 8 : 6 : 1 2 9 : 6 : 1 2 10 : 7 : 1 2 11 : 7 : 1 2 12 : 8 : 2 2 6 2 7 2 13 : 9 : 2 2 10 2 11 2 14 : 10 : 2 2 8 2 9 2 15 : 11 : 1 4 2 2 16 : 12 : 2 2 4 2 5 2 17 : 13 : 1 4 2 2 18 : 14 : 3 2 4 2 7 2 19 : 14 : 3 2 5 2 6 2 20 : 15 : 3 2 8 2 11 2 21 : 15 : 3 2 9 2 10 2 22 : 16 : 4 2 8 2 14 2 15 2 16 2 23 : 17 : 6 2 8 2 12 2 14 2 17 2 24 : 18 : 4 2 10 2 13 2 16 2 17 2 25 : 19 : 6 2 10 2 12 2 13 2 15 2 26 : 20 : 19 2 21 2 22 2 23 2 24 2 25 2 fromH toK size(N_H(K)/K) ... one from each type 1 2 1 2 3 1 2 4 1 2 5 1 2 6 1 2 7 1 2 8 1 2 9 1 2 10 1 2 11 1 2 12 2 2 6 2 13 2 2 10 2 14 2 2 8 2 15 1 4 2 2 16 2 2 4 2 17 1 4 2 2 18 3 2 4 2 7 2 19 3 2 5 2 6 2 20 3 2 8 2 11 2 21 3 2 9 2 10 2 22 4 2 8 2 14 2 15 2 16 2 23 6 2 8 2 12 2 14 2 17 2 24 4 2 10 2 13 2 16 2 17 2 25 6 2 10 2 12 2 13 2 15 2 26 19 2 21 2 22 2 23 2 24 2 25 2 fromS : toS size(N_H(K)/K) id(H/H')... 1 : 2 : 1 2 Z2 3 : 1 2 Z2 4 : 1 2 Z2 5 : 1 2 Z2 6 : 1 2 Z2 7 : 1 2 Z2 8 : 2 2 Z2 5 2 Z2 9 : 2 2 Z2 7 2 Z2 10 : 2 2 Z2 6 2 Z2 11 : 1 4 Z4 2 2 Z2 12 : 2 2 Z2 4 2 Z2 13 : 1 4 Z4 2 2 Z2 14 : 3 2 Z2 4 2 Z2 5 2 Z2 15 : 3 2 Z2 6 2 Z2 7 2 Z2 16 : 4 2 D4 6 2 D4 10 2 Z2 11 2 Z2 12 2 Z2 17 : 5 2 D4 6 2 D4 8 2 Z2 10 2 Z2 13 2 Z2 18 : 4 2 D4 7 2 D4 9 2 Z2 12 2 Z2 13 2 Z2 19 : 5 2 D4 7 2 D4 8 2 Z2 9 2 Z2 11 2 Z2 20 : 14 2 D4 15 2 D4 16 2 Z2 17 2 Z2 18 2 Z2 19 2 Z2 Typejumps from S1 to S[ ] Typejumps from S2 to S[ 1 ] Typejumps from S3 to S[ 1 ] Typejumps from S4 to S[ 1 ] Typejumps from S4 to S[ 1 ] Typejumps from S5 to S[ 1 ] Typejumps from S5 to S[ 1 ] Typejumps from S6 to S[ 1 ] Typejumps from S6 to S[ 1 ] Typejumps from S7 to S[ 1 ] Typejumps from S7 to S[ 1 ] Typejumps from S8 to S[ 2, 5, 5 ] Typejumps from S9 to S[ 2, 7, 7 ] Typejumps from S10 to S[ 2, 6, 6 ] Typejumps from S11 to S[ 1, 1, 2 ] Typejumps from S12 to S[ 2, 4, 4 ] Typejumps from S13 to S[ 1, 1, 2 ] Typejumps from S14 to S[ 3, 4, 5 ] Typejumps from S14 to S[ 3, 4, 5 ] Typejumps from S15 to S[ 3, 6, 7 ] Typejumps from S15 to S[ 3, 6, 7 ] Typejumps from S16 to S[ 4, 6, 10, 11, 12 ] Typejumps from S17 to S[ 5, 6, 8, 10, 13 ] Typejumps from S18 to S[ 4, 7, 9, 12, 13 ] Typejumps from S19 to S[ 5, 7, 8, 9, 11 ] Typejumps from S20 to S[ 14, 15, 16, 17, 18, 19 ]