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Q8
1
[ 1 .. 8 ] Group( () ) [ 1, 1 ]
2
[ 1, 2, 1, 2, 5, 6, 5, 6 ] Group( [ (1,3)(2,4)(5,7)(6,8) ] ) [ 2, 1 ]
3
[ 1, 2, -1, -2, 5, 6, -5, -6 ] Group( [ ( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,10) ] ) [ 2, 1 ]
4
[ 1, 1, 1, 1, 5, 5, 5, 5 ] Group( [ (1,3)(2,4)(5,7)(6,8), (1,2,3,4)(5,6,7,8) ] ) [ 4, 1 ]
5
[ 1, -1, 1, -1, 5, -5, 5, -5 ] Group( [ (1,3)(2,4)(5,7)(6,8), ( 1, 2, 3, 4)( 5, 6, 7, 8)( 9,10) ] ) [ 4, 1 ]
6
[ 1, 2, 1, 2, 1, 2, 1, 2 ] Group( [ (1,3)(2,4)(5,7)(6,8), (1,5,3,7)(2,8,4,6) ] ) [ 4, 1 ]
7
[ 1, 2, 1, 2, -1, -2, -1, -2 ] Group( [ (1,3)(2,4)(5,7)(6,8), ( 1, 5, 3, 7)( 2, 8, 4, 6)( 9,10) ] ) [ 4, 1 ]
8
[ 1, 2, 1, 2, 2, 1, 2, 1 ] Group( [ (1,3)(2,4)(5,7)(6,8), (1,6,3,8)(2,5,4,7) ] ) [ 4, 1 ]
9
[ 1, 2, 1, 2, -2, -1, -2, -1 ] Group( [ (1,3)(2,4)(5,7)(6,8), ( 1, 6, 3, 8)( 2, 5, 4, 7)( 9,10) ] ) [ 4, 1 ]
10
[ 1, 1, 1, 1, 1, 1, 1, 1 ] Group( [ (1,3)(2,4)(5,7)(6,8), (1,2,3,4)(5,6,7,8), (1,5,3,7)(2,8,4,6) ] ) [ 8, 4 ]
11
[ 1, 1, 1, 1, -1, -1, -1, -1 ] Group( [ (1,3)(2,4)(5,7)(6,8), (1,2,3,4)(5,6,7,8), ( 1, 5, 3, 7)( 2, 8, 4, 6)( 9,10) ] ) [ 8, 4 ]
12
[ 1, -1, 1, -1, 1, -1, 1, -1 ] Group( [ (1,3)(2,4)(5,7)(6,8), ( 1, 2, 3, 4)( 5, 6, 7, 8)( 9,10), (1,5,3,7)(2,8,4,6) ] ) [ 8, 4 ]
13
[ 1, -1, 1, -1, -1, 1, -1, 1 ] Group( [ (1,3)(2,4)(5,7)(6,8), ( 1, 2, 3, 4)( 5, 6, 7, 8)( 9,10), ( 1, 5, 3, 7)( 2, 8, 4, 6)( 9,10) ] ) [ 8, 4 ]
14
[ 0, 0, 0, 0, 0, 0, 0, 0 ] Group( [ (1,2,3,4)(5,6,7,8), (1,5,3,7)(2,8,4,6), ( 9,10) ] ) [ 16, 12 ]
Subgroup structure
[ [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 ],
[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ],
[ 2, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ],
[ 3, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ],
[ 4, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1 ],
[ 5, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1 ],
[ 6, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1 ],
[ 7, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1 ],
[ 8, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1 ],
[ 9, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1 ],
[ 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1 ],
[ 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1 ],
[ 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1 ],
[ 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1 ],
[ 14, 0, 0, 0, 0, 0, 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 = 8, 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 = 4, 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 = 4, 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 = 4, 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 = 4, 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, 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
2, dim Fix_P(U) K = 2, 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 [ 2, 4 ], maximals 2
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 [ 4, 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 [ 4, 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, 2 ]
Symmetry type S5 [ 4, 1 ]
Representation 1, dimension of U_i = 1
5, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 5 ], maximals
Representation 2, dimension of U_i = 1
2, 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 [ 2, 5 ], maximals 2
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 [ 4, 1 ]
5, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 5 ], 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 [ 4, 1 ]
5, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 5 ], maximals 1
Possible jumps from S5 [ 1, 1, 2 ]
Symmetry type S6 [ 4, 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 2, dimension of U_i = 1
2, dim Fix_P(U) K = 2, 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 [ 2, 6 ], maximals 2
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 [ 4, 1 ]
6, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 6 ], 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 [ 4, 1 ]
6, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 6 ], maximals 1
Possible jumps from S6 [ 1, 1, 2 ]
Symmetry type S7 [ 4, 1 ]
Representation 1, dimension of U_i = 1
7, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 7 ], maximals
Representation 2, dimension of U_i = 1
2, 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 [ 2, 7 ], maximals 2
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 [ 4, 1 ]
7, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 7 ], 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 [ 4, 1 ]
7, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 7 ], maximals 1
Possible jumps from S7 [ 1, 1, 2 ]
Symmetry type S8 [ 4, 1 ]
Representation 1, dimension of U_i = 1
8, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 8 ], maximals
Representation 2, dimension of U_i = 1
2, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
8, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 2, 8 ], maximals 2
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 [ 4, 1 ]
8, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 8 ], 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 [ 4, 1 ]
8, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 8 ], maximals 1
Possible jumps from S8 [ 1, 1, 2 ]
Symmetry type S9 [ 4, 1 ]
Representation 1, dimension of U_i = 1
9, dim Fix_P(U) K = 2, dim Fix_U_i K = 1, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 9 ], maximals
Representation 2, dimension of U_i = 1
2, dim Fix_P(U) K = 2, 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 [ 2, 9 ], maximals 2
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 [ 4, 1 ]
9, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 9 ], 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 [ 4, 1 ]
9, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 9 ], maximals 1
Possible jumps from S9 [ 1, 1, 2 ]
Symmetry type S10 [ 8, 4 ]
Representation 1, dimension of U_i = 1
10, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 10 ], maximals
Representation 2, dimension of U_i = 1
6, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
10, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 6, 10 ], maximals 6
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 ]
10, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 4, 10 ], maximals 4
Representation 4, dimension of U_i = 1
8, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
10, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 8, 10 ], maximals 8
Representation 5, dimension of U_i = 2
1, dim Fix_P(U) K = 4, dim Fix_U_i K = 2, N_H(K)/K [ 8, 4 ]
10, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 10 ], maximals 1
Possible jumps from S10 [ 1, 4, 6, 8 ]
Symmetry type S11 [ 8, 4 ]
Representation 1, dimension of U_i = 1
11, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 11 ], maximals
Representation 2, dimension of U_i = 1
7, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
11, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 7, 11 ], maximals 7
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 ]
11, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 4, 11 ], maximals 4
Representation 4, dimension of U_i = 1
9, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
11, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 9, 11 ], maximals 9
Representation 5, dimension of U_i = 2
1, dim Fix_P(U) K = 4, dim Fix_U_i K = 2, N_H(K)/K [ 8, 4 ]
11, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 11 ], maximals 1
Possible jumps from S11 [ 1, 4, 7, 9 ]
Symmetry type S12 [ 8, 4 ]
Representation 1, dimension of U_i = 1
12, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 12 ], maximals
Representation 2, dimension of U_i = 1
6, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
12, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 6, 12 ], maximals 6
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 ]
12, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 5, 12 ], maximals 5
Representation 4, dimension of U_i = 1
9, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
12, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 9, 12 ], maximals 9
Representation 5, dimension of U_i = 2
1, dim Fix_P(U) K = 4, dim Fix_U_i K = 2, N_H(K)/K [ 8, 4 ]
12, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 12 ], maximals 1
Possible jumps from S12 [ 1, 5, 6, 9 ]
Symmetry type S13 [ 8, 4 ]
Representation 1, dimension of U_i = 1
13, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 13 ], maximals
Representation 2, dimension of U_i = 1
7, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
13, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 7, 13 ], maximals 7
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 ]
13, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 5, 13 ], maximals 5
Representation 4, dimension of U_i = 1
8, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
13, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 8, 13 ], maximals 8
Representation 5, dimension of U_i = 2
1, dim Fix_P(U) K = 4, dim Fix_U_i K = 2, N_H(K)/K [ 8, 4 ]
13, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 1, 13 ], maximals 1
Possible jumps from S13 [ 1, 5, 7, 8 ]
Symmetry type S14 [ 16, 12 ]
Representation 5, dimension of U_i = 1
10, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
14, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 10, 14 ], maximals 10
Representation 6, dimension of U_i = 1
12, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
14, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 12, 14 ], maximals 12
Representation 7, dimension of U_i = 1
11, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
14, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 11, 14 ], maximals 11
Representation 8, dimension of U_i = 1
13, dim Fix_P(U) K = 1, dim Fix_U_i K = 1, N_H(K)/K [ 2, 1 ]
14, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 13, 14 ], maximals 13
Representation 9, dimension of U_i = 2
3, dim Fix_P(U) K = 4, dim Fix_U_i K = 2, N_H(K)/K [ 8, 4 ]
14, dim Fix_P(U) K = 0, dim Fix_U_i K = 0, N_H(K)/K [ 1, 1 ]
isotropy subgroups [ 3, 14 ], maximals 3
Possible jumps from S14 [ 3, 10, 11, 12, 13 ]
Possible jumps from S1 [ ]
Possible jumps from S2 [ 1 ]
Possible jumps from S3 [ 1 ]
Possible jumps from S4 [ 1, 1, 2 ]
Possible jumps from S5 [ 1, 1, 2 ]
Possible jumps from S6 [ 1, 1, 2 ]
Possible jumps from S7 [ 1, 1, 2 ]
Possible jumps from S8 [ 1, 1, 2 ]
Possible jumps from S9 [ 1, 1, 2 ]
Possible jumps from S10 [ 1, 4, 6, 8 ]
Possible jumps from S11 [ 1, 4, 7, 9 ]
Possible jumps from S12 [ 1, 5, 6, 9 ]
Possible jumps from S13 [ 1, 5, 7, 8 ]
Possible jumps from S14 [ 3, 10, 11, 12, 13 ]