1 Group( () ) [ 1 .. 12 ] 2 Group( [ ( 1, 4)( 2, 5)( 3, 6)( 7,10)( 8,11)( 9,12) ] ) [ 1, 2, 3, 1, 2, 3, 7, 8, 9, 7, 8, 9 ] 3 Group( [ ( 1, 4)( 2, 5)( 3, 6)( 7,10)( 8,11)( 9,12)(13,14) ] ) [ 1, 2, 3, -1, -2, -3, 7, 8, 9, -7, -8, -9 ] 4 Group( [ ( 1, 7)( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)(13,14) ] ) [ 1, 2, 3, 4, 5, 6, -1, -6, -5, -4, -3, -2 ] 5 Group( [ ( 1, 7)( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8) ] ) [ 1, 2, 3, 4, 5, 6, 1, 6, 5, 4, 3, 2 ] 6 Group( [ ( 1, 8)( 2, 7)( 3,12)( 4,11)( 5,10)( 6, 9) ] ) [ 1, 2, 3, 4, 5, 6, 2, 1, 6, 5, 4, 3 ] 7 Group( [ ( 1, 8)( 2, 7)( 3,12)( 4,11)( 5,10)( 6, 9)(13,14) ] ) [ 1, 2, 3, 4, 5, 6, -2, -1, -6, -5, -4, -3 ] 8 Group( [ ( 1, 3, 5)( 2, 4, 6)( 7, 9,11)( 8,10,12) ] ) [ 1, 2, 1, 2, 1, 2, 7, 8, 7, 8, 7, 8 ] 9 Group( [ ( 1, 4)( 2, 5)( 3, 6)( 7,10)( 8,11)( 9,12)(13,14), ( 1, 7)( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8) ] ) [ 1, 2, 3, -1, -2, -3, 1, -3, -2, -1, 3, 2 ] 10 Group( [ ( 1, 4)( 2, 5)( 3, 6)( 7,10)( 8,11)( 9,12)(13,14), ( 1, 7)( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)(13,14) ] ) [ 1, 2, 3, -1, -2, -3, -1, 3, 2, 1, -3, -2 ] 11 Group( [ ( 1, 4)( 2, 5)( 3, 6)( 7,10)( 8,11)( 9,12), ( 1, 7)( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8) ] ) [ 1, 2, 3, 1, 2, 3, 1, 3, 2, 1, 3, 2 ] 12 Group( [ ( 1, 4)( 2, 5)( 3, 6)( 7,10)( 8,11)( 9,12), ( 1, 7)( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)(13,14) ] ) [ 1, 2, 3, 1, 2, 3, -1, -3, -2, -1, -3, -2 ] 13 Group( [ ( 1, 3, 5)( 2, 4, 6)( 7, 9,11)( 8,10,12), ( 1, 4)( 2, 5)( 3, 6)( 7,10)( 8,11)( 9,12) ] ) [ 1, 1, 1, 1, 1, 1, 7, 7, 7, 7, 7, 7 ] 14 Group( [ ( 1, 3, 5)( 2, 4, 6)( 7, 9,11)( 8,10,12), ( 1, 4)( 2, 5)( 3, 6)( 7,10)( 8,11)( 9,12)(13,14) ] ) [ 1, -1, 1, -1, 1, -1, 7, -7, 7, -7, 7, -7 ] 15 Group( [ ( 1, 3, 5)( 2, 4, 6)( 7, 9,11)( 8,10,12), ( 1, 7)( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8) ] ) [ 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2 ] 16 Group( [ ( 1, 3, 5)( 2, 4, 6)( 7, 9,11)( 8,10,12), ( 1, 7)( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)(13,14) ] ) [ 1, 2, 1, 2, 1, 2, -1, -2, -1, -2, -1, -2 ] 17 Group( [ ( 1, 3, 5)( 2, 4, 6)( 7, 9,11)( 8,10,12), ( 1, 8)( 2, 7)( 3,12)( 4,11)( 5,10)( 6, 9) ] ) [ 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1 ] 18 Group( [ ( 1, 3, 5)( 2, 4, 6)( 7, 9,11)( 8,10,12), ( 1, 8)( 2, 7)( 3,12)( 4,11)( 5,10)( 6, 9)(13,14) ] ) [ 1, 2, 1, 2, 1, 2, -2, -1, -2, -1, -2, -1 ] 19 Group( [ ( 1, 3, 5)( 2, 4, 6)( 7, 9,11)( 8,10,12), ( 1, 4)( 2, 5)( 3, 6)( 7,10)( 8,11)( 9,12), ( 1, 7)( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8) ] ) [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] 20 Group( [ ( 1, 3, 5)( 2, 4, 6)( 7, 9,11)( 8,10,12), ( 1, 4)( 2, 5)( 3, 6)( 7,10)( 8,11)( 9,12), ( 1, 7)( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)(13,14) ] ) [ 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1 ] 21 Group( [ ( 1, 3, 5)( 2, 4, 6)( 7, 9,11)( 8,10,12), ( 1, 4)( 2, 5)( 3, 6)( 7,10)( 8,11)( 9,12)(13,14), ( 1, 7)( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8) ] ) [ 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1 ] 22 Group( [ ( 1, 3, 5)( 2, 4, 6)( 7, 9,11)( 8,10,12), ( 1, 4)( 2, 5)( 3, 6)( 7,10)( 8,11)( 9,12)(13,14), ( 1, 7)( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)(13,14) ] ) [ 1, -1, 1, -1, 1, -1, -1, 1, -1, 1, -1, 1 ] 23 Group( [ ( 1, 2, 3, 4, 5, 6)( 7, 8, 9,10,11,12), ( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7), (13,14) ] ) [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] Subgroup structure [ [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23 ], [ 1, 1, 1, 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, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1 ], [ 3, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1 ], [ 4, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1 ], [ 5, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1 ], [ 6, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1 ], [ 7, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1 ], [ 8, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ], [ 9, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1 ], [ 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1 ], [ 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1 ], [ 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1 ], [ 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1 ], [ 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1 ], [ 15, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1 ], [ 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1 ], [ 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1 ], [ 18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1 ], [ 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1 ], [ 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1 ], [ 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1 ], [ 22, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1 ], [ 23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ] ]