Group([ (1,2,3,4,5), (2,5)(3,4) ])
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D5
[ (), (2,5)(3,4), (1,2)(3,5), (1,2,3,4,5), (1,3)(4,5), (1,3,5,2,4), (1,4)(2,3), (1,4,2,5,3), (1,5,4,3,2), (1,5)(2,4) ] D5 as permutation of 5 points
1
[ 1 .. 10 ] Group( () ) [ 1, 1 ]
2
[ 1, 1, 3, 4, 5, 6, 6, 5, 3, 4 ] Group( [ ( 1, 2)( 3, 9)( 4,10)( 5, 8)( 6, 7) ] ) [ 2, 1 ]
[ 1, 2, 3, 3, 1, 2, 7, 8, 7, 8 ] Group( [ ( 1, 5)( 2, 6)( 3, 4)( 7, 9)( 8,10) ] ) [ 2, 1 ]
[ 1, 2, 3, 4, 5, 5, 4, 3, 2, 1 ] Group( [ ( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6) ] ) [ 2, 1 ]
[ 1, 2, 1, 2, 5, 6, 7, 7, 5, 6 ] Group( [ ( 1, 3)( 2, 4)( 5, 9)( 6,10)( 7, 8) ] ) [ 2, 1 ]
[ 1, 2, 3, 4, 4, 3, 1, 2, 9, 9 ] Group( [ ( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,10) ] ) [ 2, 1 ]
3
[ 1, -1, 3, 4, 5, 6, -6, -5, -3, -4 ] Group( [ ( 1, 2)( 3, 9)( 4,10)( 5, 8)( 6, 7)(11,12) ] ) [ 2, 1 ]
[ 1, 2, 3, -3, -1, -2, 7, 8, -7, -8 ] Group( [ ( 1, 5)( 2, 6)( 3, 4)( 7, 9)( 8,10)(11,12) ] ) [ 2, 1 ]
[ 1, 2, 3, 4, 5, -5, -4, -3, -2, -1 ] Group( [ ( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,12) ] ) [ 2, 1 ]
[ 1, 2, -1, -2, 5, 6, 7, -7, -5, -6 ] Group( [ ( 1, 3)( 2, 4)( 5, 9)( 6,10)( 7, 8)(11,12) ] ) [ 2, 1 ]
[ 1, 2, 3, 4, -4, -3, -1, -2, 9, -9 ] Group( [ ( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,10)(11,12) ] ) [ 2, 1 ]
4
[ 1, 2, 2, 1, 2, 1, 2, 1, 1, 2 ] Group( [ ( 1, 4, 6, 8, 9)( 2, 3, 5, 7,10) ] ) [ 5, 1 ]
5
[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] Group( [ ( 1, 4, 6, 8, 9)( 2, 3, 5, 7,10), ( 1, 2)( 3, 9)( 4,10)( 5, 8)( 6, 7) ] ) [ 10, 1 ]
6
[ 1, -1, -1, 1, -1, 1, -1, 1, 1, -1 ] Group( [ ( 1, 4, 6, 8, 9)( 2, 3, 5, 7,10), ( 1, 2)( 3, 9)( 4,10)( 5, 8)( 6, 7)(11,12) ] ) [ 10, 1 ]
7
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] Group( [ ( 1, 4, 6, 8, 9)( 2, 3, 5, 7,10), ( 1, 2)( 3, 9)( 4,10)( 5, 8)( 6, 7), (11,12) ] ) [ 20, 4 ]
Subgroup structure
[ [ 0, 1, 2, 3, 4, 5, 6, 7 ],
[ 1, 1, 1, 1, 1, 1, 1, 1 ],
[ 2, 0, 1, 0, 0, 1, 0, 1 ],
[ 3, 0, 0, 1, 0, 0, 1, 1 ],
[ 4, 0, 0, 0, 1, 1, 1, 1 ],
[ 5, 0, 0, 0, 0, 1, 0, 1 ],
[ 6, 0, 0, 0, 0, 0, 1, 1 ],
[ 7, 0, 0, 0, 0, 0, 0, 1 ] ]
Symmetry type S1 [ 1, 1 ]
Representation 1, dimension of U_i = 1
1, dim Fix_P(U) K = 10, dim Fix_U_i K = 1, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1 ], maximals
Possible jumps from S1 [ ]
Symmetry type S2 [ 2, 1 ]
Representation 1, dimension of U_i = 1
2, dim Fix_P(U) K = 5, dim Fix_U_i K = 1, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 2 ], maximals
Representation 2, dimension of U_i = 1
1, dim Fix_P(U) K = 5, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
2, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 2 ], maximals 1
Possible jumps from S2 [ 1 ]
Symmetry type S3 [ 2, 1 ]
Representation 1, dimension of U_i = 1
3, dim Fix_P(U) K = 5, dim Fix_U_i K = 1, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 3 ], maximals
Representation 2, dimension of U_i = 1
1, dim Fix_P(U) K = 5, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
3, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 3 ], maximals 1
Possible jumps from S3 [ 1 ]
Symmetry type S4 [ 5, 1 ]
Representation 1, dimension of U_i = 1
4, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 4 ], maximals
Representation 2, dimension of U_i = 1
NON-REAL projection
1, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K [ 5, 1 ]
4, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 4 ], maximals 1
Representation 3, dimension of U_i = 1
NON-REAL projection
1, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K [ 5, 1 ]
4, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 4 ], maximals 1
Representation 4, dimension of U_i = 1
NON-REAL projection
1, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K [ 5, 1 ]
4, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 4 ], maximals 1
Representation 5, dimension of U_i = 1
NON-REAL projection
1, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K [ 5, 1 ]
4, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 4 ], maximals 1
Possible jumps from S4 [ 1, 1, 1, 1 ]
Symmetry type S5 [ 10, 1 ]
Representation 1, dimension of U_i = 1
5, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 5 ], maximals
Representation 2, dimension of U_i = 1
4, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
5, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 4, 5 ], maximals 4
Representation 3, dimension of U_i = 2
1, dim Fix_P(U) K = 4, dim Fix_U_i K = 2, N_H(K)/K [ 10, 1 ]
2, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K [ 1, 1 ]
5, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 2, 5 ], maximals 2
Representation 4, dimension of U_i = 2
1, dim Fix_P(U) K = 4, dim Fix_U_i K = 2, N_H(K)/K [ 10, 1 ]
2, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K [ 1, 1 ]
5, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 2, 5 ], maximals 2
Possible jumps from S5 [ 2, 2, 4 ]
Symmetry type S6 [ 10, 1 ]
Representation 1, dimension of U_i = 1
6, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 6 ], maximals
Representation 2, dimension of U_i = 1
4, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
6, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 4, 6 ], maximals 4
Representation 3, dimension of U_i = 2
1, dim Fix_P(U) K = 4, dim Fix_U_i K = 2, N_H(K)/K [ 10, 1 ]
3, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K [ 1, 1 ]
6, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 3, 6 ], maximals 3
Representation 4, dimension of U_i = 2
1, dim Fix_P(U) K = 4, dim Fix_U_i K = 2, N_H(K)/K [ 10, 1 ]
3, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K [ 1, 1 ]
6, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 3, 6 ], maximals 3
Possible jumps from S6 [ 3, 3, 4 ]
Symmetry type S7 [ 20, 4 ]
Representation 3, dimension of U_i = 1
5, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
7, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 5, 7 ], maximals 5
Representation 4, dimension of U_i = 1
6, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
7, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 6, 7 ], maximals 6
Representation 6, dimension of U_i = 2
1, dim Fix_P(U) K = 4, dim Fix_U_i K = 2, N_H(K)/K [ 20, 4 ]
2, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
3, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
7, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 2, 3, 7 ], maximals 2 3
Representation 8, dimension of U_i = 2
1, dim Fix_P(U) K = 4, dim Fix_U_i K = 2, N_H(K)/K [ 20, 4 ]
2, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
3, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
7, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 2, 3, 7 ], maximals 2 3
Possible jumps from S7 [ 2, 2, 3, 3, 5, 6 ]
Possible jumps from S1 [ ]
Possible jumps from S2 [ 1 ]
Possible jumps from S3 [ 1 ]
Possible jumps from S4 [ 1, 1, 1, 1 ]
Possible jumps from S5 [ 2, 2, 4 ]
Possible jumps from S6 [ 3, 3, 4 ]
Possible jumps from S7 [ 2, 2, 3, 3, 5, 6 ]