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K4
1
[ 1 .. 4 ] Group( () ) [ 1, 1 ]
2
[ 1, -1, 3, -3 ] Group( [ (1,2)(3,4)(5,6) ] ) [ 2, 1 ]
[ 1, 2, -1, -2 ] Group( [ (1,3)(2,4)(5,6) ] ) [ 2, 1 ]
[ 1, 2, -2, -1 ] Group( [ (1,4)(2,3)(5,6) ] ) [ 2, 1 ]
3
[ 1, 2, 3, 3 ] Group( [ (3,4) ] ) [ 2, 1 ]
[ 1, 2, 3, 2 ] Group( [ (2,4) ] ) [ 2, 1 ]
[ 1, 2, 2, 4 ] Group( [ (2,3) ] ) [ 2, 1 ]
[ 1, 2, 1, 4 ] Group( [ (1,3) ] ) [ 2, 1 ]
[ 1, 2, 3, 1 ] Group( [ (1,4) ] ) [ 2, 1 ]
[ 1, 1, 3, 4 ] Group( [ (1,2) ] ) [ 2, 1 ]
4
[ 1, 1, 3, 3 ] Group( [ (3,4), (1,2) ] ) [ 4, 2 ]
[ 1, 2, 1, 2 ] Group( [ (2,4), (1,3) ] ) [ 4, 2 ]
[ 1, 2, 2, 1 ] Group( [ (2,3), (1,4) ] ) [ 4, 2 ]
5
[ 1, -1, 0, 0 ] Group( [ (3,4), (1,2)(5,6) ] ) [ 4, 2 ]
[ 1, 0, -1, 0 ] Group( [ (2,4), (1,3)(5,6) ] ) [ 4, 2 ]
[ 1, 0, 0, -1 ] Group( [ (2,3), (1,4)(5,6) ] ) [ 4, 2 ]
[ 0, 2, 0, -2 ] Group( [ (1,3), (2,4)(5,6) ] ) [ 4, 2 ]
[ 0, 2, -2, 0 ] Group( [ (1,4), (2,3)(5,6) ] ) [ 4, 2 ]
[ 0, 0, 3, -3 ] Group( [ (1,2), (3,4)(5,6) ] ) [ 4, 2 ]
6
[ 1, 2, 2, 2 ] Group( [ (2,3,4), (3,4) ] ) [ 6, 1 ]
[ 1, 2, 1, 1 ] Group( [ (1,4,3), (3,4) ] ) [ 6, 1 ]
[ 1, 1, 3, 1 ] Group( [ (1,2,4), (1,2) ] ) [ 6, 1 ]
[ 1, 1, 1, 4 ] Group( [ (1,3,2), (1,2) ] ) [ 6, 1 ]
7
[ 1, 1, -1, -1 ] Group( [ (3,4), (1,2), (1,3)(2,4)(5,6) ] ) [ 8, 3 ]
[ 1, -1, 1, -1 ] Group( [ (2,4), (1,3), (1,4)(2,3)(5,6) ] ) [ 8, 3 ]
[ 1, -1, -1, 1 ] Group( [ (2,3), (1,4), (1,2)(3,4)(5,6) ] ) [ 8, 3 ]
8
[ 1, 1, 1, 1 ] Group( [ (1,2)(3,4), (1,3)(2,4), (2,3,4), (3,4) ] ) [ 24, 12 ]
9
[ 0, 0, 0, 0 ] Group( [ (1,2,3,4), (1,2), (5,6) ] ) [ 48, 48 ]
Subgroup structure
[ [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ],
[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ],
[ 2, 0, 1, 0, 0, 1, 0, 1, 0, 1 ],
[ 3, 0, 0, 1, 1, 1, 1, 1, 1, 1 ],
[ 4, 0, 0, 0, 1, 0, 0, 1, 1, 1 ],
[ 5, 0, 0, 0, 0, 1, 0, 0, 0, 1 ],
[ 6, 0, 0, 0, 0, 0, 1, 0, 1, 1 ],
[ 7, 0, 0, 0, 0, 0, 0, 1, 0, 1 ],
[ 8, 0, 0, 0, 0, 0, 0, 0, 1, 1 ],
[ 9, 0, 0, 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 = 4, 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 = 2, 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 = 2, 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 = 3, 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 = 1, 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 [ 4, 2 ]
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
3, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
4, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 3, 4 ], maximals 3
Representation 3, dimension of U_i = 1
3, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
4, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 3, 4 ], maximals 3
Possible jumps from S4 [ 3, 3 ]
Symmetry type S5 [ 4, 2 ]
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
3, dim Fix_P(U) K = 2, 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 [ 3, 5 ], maximals 3
Representation 4, dimension of U_i = 1
2, 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 [ 2, 5 ], maximals 2
Possible jumps from S5 [ 2, 3 ]
Symmetry type S6 [ 6, 1 ]
Representation 1, dimension of U_i = 1
6, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 6 ], maximals
Representation 3, dimension of U_i = 2
1, dim Fix_P(U) K = 2, dim Fix_U_i K = 2, N_H(K)/K [ 6, 1 ]
3, dim Fix_P(U) K = 1, 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 ]
Symmetry type S7 [ 8, 3 ]
Representation 1, dimension of U_i = 1
7, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 7 ], maximals
Representation 3, dimension of U_i = 1
4, 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 [ 4, 7 ], maximals 4
Representation 5, dimension of U_i = 2
1, dim Fix_P(U) K = 2, dim Fix_U_i K = 2, N_H(K)/K [ 8, 3 ]
3, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
2, 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 [ 1, 2, 3, 7 ], maximals 2 3
Possible jumps from S7 [ 2, 3, 4 ]
Symmetry type S8 [ 24, 12 ]
Representation 1, dimension of U_i = 1
8, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 8 ], maximals
Representation 5, dimension of U_i = 3
1, dim Fix_P(U) K = 3, dim Fix_U_i K = 3, N_H(K)/K [ 24, 12 ]
3, dim Fix_P(U) K = 2, dim Fix_U_i K = 2, N_H(K)/K [ 2, 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 = 1, dim Fix_U_i K = 1, N_H(K)/K [ 1, 1 ]
8, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 3, 4, 6, 8 ], maximals 4 6
Possible jumps from S8 [ 4, 6 ]
Symmetry type S9 [ 48, 48 ]
Representation 3, dimension of U_i = 1
8, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
9, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 8, 9 ], maximals 8
Representation 8, dimension of U_i = 3
1, dim Fix_P(U) K = 3, dim Fix_U_i K = 3, N_H(K)/K [ 48, 48 ]
3, dim Fix_P(U) K = 2, dim Fix_U_i K = 2, N_H(K)/K [ 4, 2 ]
2, dim Fix_P(U) K = 2, dim Fix_U_i K = 2, N_H(K)/K [ 8, 3 ]
5, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 2, 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 = 1, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
9, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 2, 3, 5, 6, 7, 9 ], maximals 5 6 7
Possible jumps from S9 [ 5, 6, 7, 8 ]
Possible jumps from S1 [ ]
Possible jumps from S2 [ 1 ]
Possible jumps from S3 [ 1 ]
Possible jumps from S4 [ 3, 3 ]
Possible jumps from S5 [ 2, 3 ]
Possible jumps from S6 [ 3 ]
Possible jumps from S7 [ 2, 3, 4 ]
Possible jumps from S8 [ 4, 6 ]
Possible jumps from S9 [ 5, 6, 7, 8 ]