X3Automorphism
Subgroups of the automorphism group of X3
The page displays a subgroup lattice of S6 as the automorphism group of the Segre cubic defined in P4 by
-y[1]^2*y[2] - y[1]*y[2]^2 - y[1]^2*y[3] - 2*y[1]*y[2]*y[3] - y[2]^2*y[3] - y[1]*y[3]^2 - y[2]*y[3]^2 - y[1]^2*y[4] - 2*y[1]*y[2]*y[4] -
y[2]^2*y[4] - 2*y[1]*y[3]*y[4] - 2*y[2]*y[3]*y[4] - y[3]^2*y[4] - y[1]*y[4]^2 - y[2]*y[4]^2 - y[3]*y[4]^2 - y[1]^2*y[5] - 2*y[1]*y[2]*y[5] -
y[2]^2*y[5] - 2*y[1]*y[3]*y[5] - 2*y[2]*y[3]*y[5] - y[3]^2*y[5] - 2*y[1]*y[4]*y[5] - 2*y[2]*y[4]*y[5] - 2*y[3]*y[4]*y[5] -
y[4]^2*y[5] - y[1]*y[5]^2 - y[2]*y[5]^2 - y[3]*y[5]^2 - y[4]*y[5]^2.
and generators of every subgroup (up to conjugacy) of it.
In particular, a subgroup of S6 is linearizable if and only if
- fixed any singular points and, or
- is contained in the nonstandard S5 and, or
- is isomorphic to C22 and leaves three invariant planes in X3.
There are 36 linearizable conjugacy classes, their indices in the subgroup lattice below are
[ 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 27, 28, 30, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 48, 52, 53 ]
There are 19 nonlinearizable conjugacy classes, their indices in the subgroup lattice below are
[ 8, 23, 24, 25, 26, 29, 31, 38, 39, 40, 41, 42, 43, 49, 50, 51, 54, 55, 56]
The source code of generators of the groups can be found here.
Partially ordered set of subgroup classes
-----------------------------------------
[56] Order 720 Length 1 Maximal Subgroups: 50 51 52 53 54 55
---
[55] Order 360 Length 1 Maximal Subgroups: 39 42 47 48 49
---
[54] Order 120 Length 6 Nonsolvable Maximal Subgroups: 32 37 40 49 ---standard S5
[53] Order 120 Length 6 Nonsolvable Maximal Subgroups: 33 37 44 48 ---nonstandard S5
[52] Order 72 Length 10 Maximal Subgroups: 27 45 46 47
[51] Order 48 Length 15 Maximal Subgroups: 32 38 40 41 42
[50] Order 48 Length 15 Maximal Subgroups: 33 38 39 43 44
---
[49] Order 60 Length 6 Maximal Subgroups: 17 22 31
[48] Order 60 Length 6 Maximal Subgroups: 19 22 30
[47] Order 36 Length 10 Maximal Subgroups: 11 34
[46] Order 36 Length 10 Maximal Subgroups: 33 34 35
[45] Order 36 Length 10 Maximal Subgroups: 32 34 36
[44] Order 24 Length 15 Maximal Subgroups: 16 28 30
[43] Order 24 Length 15 Maximal Subgroups: 18 24 30
[42] Order 24 Length 15 Maximal Subgroups: 17 26 31
[41] Order 24 Length 15 Maximal Subgroups: 20 23 31
[40] Order 24 Length 15 Maximal Subgroups: 15 29 31
[39] Order 24 Length 15 Maximal Subgroups: 19 26 30
[38] Order 16 Length 45 Maximal Subgroups: 23 24 25 26 27 28 29
---
[37] Order 20 Length 36 Maximal Subgroups: 13 22
[36] Order 18 Length 20 Maximal Subgroups: 15 20 21
[35] Order 18 Length 20 Maximal Subgroups: 16 18 21
[34] Order 18 Length 10 Maximal Subgroups: 17 19 21
[33] Order 12 Length 60 Maximal Subgroups: 14 16 18 19
[32] Order 12 Length 60 Maximal Subgroups: 12 15 17 20
[31] Order 12 Length 15 Maximal Subgroups: 5 8
[30] Order 12 Length 15 Maximal Subgroups: 6 9
[29] Order 8 Length 45 Maximal Subgroups: 8 12 13
[28] Order 8 Length 45 Maximal Subgroups: 9 13 14
[27] Order 8 Length 45 Maximal Subgroups: 11 12 14
[26] Order 8 Length 45 Maximal Subgroups: 8 9 11
[25] Order 8 Length 45 Maximal Subgroups: 10 11 13
[24] Order 8 Length 15 Maximal Subgroups: 9 10 12
[23] Order 8 Length 15 Maximal Subgroups: 8 10 14
---
[22] Order 10 Length 36 Maximal Subgroups: 4 7
[21] Order 9 Length 10 Maximal Subgroups: 5 6
[20] Order 6 Length 60 Maximal Subgroups: 2 5
[19] Order 6 Length 60 Maximal Subgroups: 4 6
[18] Order 6 Length 60 Maximal Subgroups: 3 6
[17] Order 6 Length 60 Maximal Subgroups: 4 5
[16] Order 6 Length 20 Maximal Subgroups: 3 6
[15] Order 6 Length 20 Maximal Subgroups: 2 5
[14] Order 4 Length 45 Maximal Subgroups: 3 4
[13] Order 4 Length 45 Maximal Subgroups: 4
[12] Order 4 Length 45 Maximal Subgroups: 2 4
[11] Order 4 Length 45 Maximal Subgroups: 4
[10] Order 4 Length 45 Maximal Subgroups: 2 3 4
[ 9] Order 4 Length 15 Maximal Subgroups: 4
[ 8] Order 4 Length 15 Maximal Subgroups: 4
---
[ 7] Order 5 Length 36 Maximal Subgroups: 1
[ 6] Order 3 Length 20 Maximal Subgroups: 1
[ 5] Order 3 Length 20 Maximal Subgroups: 1
[ 4] Order 2 Length 45 Maximal Subgroups: 1
[ 3] Order 2 Length 15 Maximal Subgroups: 1
[ 2] Order 2 Length 15 Maximal Subgroups: 1
---
[ 1] Order 1 Length 1 Maximal Subgroups:
Concretely, they are:
2: The group name is C2, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2
Generators:
[1 0 0 0 0]
[0 0 0 1 0]
[0 0 1 0 0]
[0 1 0 0 0]
[0 0 0 0 1]
---------------------------
3: The group name is C2, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2
Generators:
[ 0 0 0 -1 1]
[ 0 0 1 -1 0]
[ 0 1 0 -1 0]
[ 0 0 0 -1 0]
[ 1 0 0 -1 0]
---------------------------
4: The group name is C2, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2
Generators:
[ 1 0 -1 0 0]
[ 0 0 -1 1 0]
[ 0 0 -1 0 0]
[ 0 1 -1 0 0]
[ 0 0 -1 0 1]
---------------------------
5: The group name is C3, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 3
Generators:
[1 0 0 0 0]
[0 0 0 1 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 0 1]
---------------------------
6: The group name is C3, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 3
Generators:
[ 0 0 -1 0 0]
[ 0 0 -1 1 0]
[ 1 0 -1 0 0]
[ 0 0 -1 0 1]
[ 0 1 -1 0 0]
---------------------------
7: The group name is C5, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 5
Generators:
[0 1 0 0 0]
[0 0 0 1 0]
[1 0 0 0 0]
[0 0 0 0 1]
[0 0 1 0 0]
---------------------------
8: The group name is C2^2, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^2
Generators:
[ 1 0 -1 0 0]
[ 0 0 -1 1 0]
[ 0 0 -1 0 0]
[ 0 1 -1 0 0]
[ 0 0 -1 0 1]
[ 1 -1 0 0 0]
[ 0 -1 0 0 0]
[ 0 -1 0 1 0]
[ 0 -1 1 0 0]
[ 0 -1 0 0 1]
---------------------------
9: The group name is C2^2, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^2
Generators:
[ 0 -1 0 0 1]
[ 0 -1 0 0 0]
[ 0 -1 1 0 0]
[ 0 -1 0 1 0]
[ 1 -1 0 0 0]
[ 1 -1 0 0 0]
[ 0 -1 0 0 0]
[ 0 -1 0 1 0]
[ 0 -1 1 0 0]
[ 0 -1 0 0 1]
---------------------------
10: The group name is C2^2, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^2
Generators:
[ 1 0 -1 0 0]
[ 0 0 -1 1 0]
[ 0 0 -1 0 0]
[ 0 1 -1 0 0]
[ 0 0 -1 0 1]
[0 0 0 0 1]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[1 0 0 0 0]
---------------------------
11: The group name is C4, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^2
Generators:
[ 0 0 1 0 -1]
[ 0 0 0 0 -1]
[ 1 0 0 0 -1]
[ 0 1 0 0 -1]
[ 0 0 0 1 -1]
[ 1 0 0 -1 0]
[ 0 0 0 -1 1]
[ 0 0 1 -1 0]
[ 0 0 0 -1 0]
[ 0 1 0 -1 0]
---------------------------
12: The group name is C2^2, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^2
Generators:
[0 0 0 0 1]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[1 0 0 0 0]
[1 0 0 0 0]
[0 0 0 1 0]
[0 0 1 0 0]
[0 1 0 0 0]
[0 0 0 0 1]
---------------------------
13: The group name is C4, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^2
Generators:
[0 0 0 0 1]
[0 0 0 1 0]
[0 0 1 0 0]
[0 1 0 0 0]
[1 0 0 0 0]
[0 0 0 1 0]
[1 0 0 0 0]
[0 0 1 0 0]
[0 0 0 0 1]
[0 1 0 0 0]
---------------------------
14: The group name is C2^2, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^2
Generators:
[ 1 0 -1 0 0]
[ 0 0 -1 1 0]
[ 0 0 -1 0 0]
[ 0 1 -1 0 0]
[ 0 0 -1 0 1]
[ 0 -1 0 0 1]
[ 0 -1 0 0 0]
[ 0 -1 0 1 0]
[ 0 -1 1 0 0]
[ 1 -1 0 0 0]
---------------------------
15: The group name is S3, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2 * 3
Generators:
[1 0 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 1 0 0 0]
[0 0 0 0 1]
[1 0 0 0 0]
[0 0 0 1 0]
[0 0 1 0 0]
[0 1 0 0 0]
[0 0 0 0 1]
---------------------------
16: The group name is S3, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2 * 3
Generators:
[-1 0 1 0 0]
[-1 0 0 0 1]
[-1 0 0 0 0]
[-1 1 0 0 0]
[-1 0 0 1 0]
[ 0 0 0 -1 1]
[ 0 0 1 -1 0]
[ 0 1 0 -1 0]
[ 0 0 0 -1 0]
[ 1 0 0 -1 0]
---------------------------
17: The group name is S3, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2 * 3
Generators:
[1 0 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 1 0 0 0]
[0 0 0 0 1]
[0 0 0 0 1]
[0 0 0 1 0]
[0 0 1 0 0]
[0 1 0 0 0]
[1 0 0 0 0]
---------------------------
18: The group name is C6, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2 * 3
Generators:
[-1 0 1 0 0]
[-1 0 0 0 1]
[-1 0 0 0 0]
[-1 1 0 0 0]
[-1 0 0 1 0]
[ 0 -1 0 0 1]
[ 0 -1 0 0 0]
[ 0 -1 0 1 0]
[ 0 -1 1 0 0]
[ 1 -1 0 0 0]
---------------------------
19: The group name is S3, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2 * 3
Generators:
[ 1 0 -1 0 0]
[ 0 0 -1 1 0]
[ 0 0 -1 0 0]
[ 0 1 -1 0 0]
[ 0 0 -1 0 1]
[-1 0 1 0 0]
[-1 0 0 0 1]
[-1 0 0 0 0]
[-1 1 0 0 0]
[-1 0 0 1 0]
---------------------------
20: The group name is C6, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2 * 3
Generators:
[1 0 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 1 0 0 0]
[0 0 0 0 1]
[0 0 0 0 1]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[1 0 0 0 0]
---------------------------
21: The group name is C3^2, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 3^2
Generators:
[ 0 0 -1 0 1]
[ 1 0 -1 0 0]
[ 0 0 -1 1 0]
[ 0 0 -1 0 0]
[ 0 1 -1 0 0]
[0 0 0 0 1]
[1 0 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 1 0 0 0]
---------------------------
22: The group name is D5, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2 * 5
Generators:
[0 0 0 0 1]
[0 0 1 0 0]
[0 0 0 1 0]
[1 0 0 0 0]
[0 1 0 0 0]
[0 0 0 0 1]
[0 0 0 1 0]
[0 0 1 0 0]
[0 1 0 0 0]
[1 0 0 0 0]
---------------------------
23: The group name is C2^3, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^3
Generators:
[ 1 0 -1 0 0]
[ 0 0 -1 1 0]
[ 0 0 -1 0 0]
[ 0 1 -1 0 0]
[ 0 0 -1 0 1]
[0 0 0 0 1]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[1 0 0 0 0]
[ 1 -1 0 0 0]
[ 0 -1 0 0 0]
[ 0 -1 0 1 0]
[ 0 -1 1 0 0]
[ 0 -1 0 0 1]
---------------------------
24: The group name is C2^3, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^3
Generators:
[ 0 -1 0 0 1]
[ 0 -1 0 0 0]
[ 0 -1 1 0 0]
[ 0 -1 0 1 0]
[ 1 -1 0 0 0]
[0 0 0 0 1]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[1 0 0 0 0]
[ 1 -1 0 0 0]
[ 0 -1 0 0 0]
[ 0 -1 0 1 0]
[ 0 -1 1 0 0]
[ 0 -1 0 0 1]
---------------------------
25: The group name is C2*C4, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^3
Generators:
[0 0 0 0 1]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[1 0 0 0 0]
[ 0 0 -1 0 1]
[ 0 0 -1 1 0]
[ 0 1 -1 0 0]
[ 0 0 -1 0 0]
[ 1 0 -1 0 0]
[ 1 -1 0 0 0]
[ 0 -1 0 0 0]
[ 0 -1 0 1 0]
[ 0 -1 1 0 0]
[ 0 -1 0 0 1]
---------------------------
26: The group name is D4, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^3
Generators:
[ 1 0 -1 0 0]
[ 0 0 -1 1 0]
[ 0 0 -1 0 0]
[ 0 1 -1 0 0]
[ 0 0 -1 0 1]
[ 0 -1 0 0 1]
[ 0 -1 0 0 0]
[ 0 -1 1 0 0]
[ 0 -1 0 1 0]
[ 1 -1 0 0 0]
[ 1 -1 0 0 0]
[ 0 -1 0 0 0]
[ 0 -1 0 1 0]
[ 0 -1 1 0 0]
[ 0 -1 0 0 1]
---------------------------
27: The group name is D4, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^3
Generators:
[ 0 0 1 0 -1]
[ 0 0 0 1 -1]
[ 1 0 0 0 -1]
[ 0 1 0 0 -1]
[ 0 0 0 0 -1]
[1 0 0 0 0]
[0 0 0 0 1]
[0 0 1 0 0]
[0 0 0 1 0]
[0 1 0 0 0]
[ 1 0 0 -1 0]
[ 0 0 0 -1 1]
[ 0 0 1 -1 0]
[ 0 0 0 -1 0]
[ 0 1 0 -1 0]
---------------------------
28: The group name is D4, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^3
Generators:
[ 0 0 -1 0 1]
[ 0 0 -1 1 0]
[ 0 0 -1 0 0]
[ 0 1 -1 0 0]
[ 1 0 -1 0 0]
[ 0 -1 0 0 1]
[ 0 -1 0 0 0]
[ 0 -1 1 0 0]
[ 0 -1 0 1 0]
[ 1 -1 0 0 0]
[ 1 -1 0 0 0]
[ 0 -1 0 0 0]
[ 0 -1 0 1 0]
[ 0 -1 1 0 0]
[ 0 -1 0 0 1]
---------------------------
29: The group name is D4, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^3
Generators:
[ 1 0 -1 0 0]
[ 0 0 -1 1 0]
[ 0 0 -1 0 0]
[ 0 1 -1 0 0]
[ 0 0 -1 0 1]
[ 1 -1 0 0 0]
[ 0 -1 0 0 0]
[ 0 -1 1 0 0]
[ 0 -1 0 1 0]
[ 0 -1 0 0 1]
[ 1 -1 0 0 0]
[ 0 -1 0 0 0]
[ 0 -1 0 1 0]
[ 0 -1 1 0 0]
[ 0 -1 0 0 1]
---------------------------
30: The group name is A4, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^2 * 3
Generators:
[-1 0 1 0 0]
[-1 0 0 0 1]
[-1 0 0 0 0]
[-1 1 0 0 0]
[-1 0 0 1 0]
[ 0 -1 0 0 1]
[ 0 -1 0 0 0]
[ 0 -1 1 0 0]
[ 0 -1 0 1 0]
[ 1 -1 0 0 0]
[0 0 0 0 1]
[0 1 0 0 0]
[0 0 0 1 0]
[0 0 1 0 0]
[1 0 0 0 0]
---------------------------
31: The group name is A4, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^2 * 3
Generators:
[ 1 0 -1 0 0]
[ 0 0 -1 1 0]
[ 0 0 -1 0 0]
[ 0 1 -1 0 0]
[ 0 0 -1 0 1]
[1 0 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 1 0 0 0]
[0 0 0 0 1]
[ 1 0 0 -1 0]
[ 0 0 1 -1 0]
[ 0 1 0 -1 0]
[ 0 0 0 -1 0]
[ 0 0 0 -1 1]
---------------------------
32: The group name is D6, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^2 * 3
Generators:
[0 0 0 0 1]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[1 0 0 0 0]
[1 0 0 0 0]
[0 0 0 1 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 0 1]
[1 0 0 0 0]
[0 0 0 1 0]
[0 0 1 0 0]
[0 1 0 0 0]
[0 0 0 0 1]
---------------------------
33: The group name is D6, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^2 * 3
Generators:
[ 1 0 -1 0 0]
[ 0 0 -1 1 0]
[ 0 0 -1 0 0]
[ 0 1 -1 0 0]
[ 0 0 -1 0 1]
[ 0 0 -1 0 0]
[ 0 0 -1 1 0]
[ 1 0 -1 0 0]
[ 0 0 -1 0 1]
[ 0 1 -1 0 0]
[ 0 -1 0 0 1]
[ 0 -1 0 0 0]
[ 0 -1 0 1 0]
[ 0 -1 1 0 0]
[ 1 -1 0 0 0]
---------------------------
34: The group name is C3:S3, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2 * 3^2
Generators:
[ 0 1 0 -1 0]
[ 0 0 0 -1 1]
[ 0 0 0 -1 0]
[ 0 0 1 -1 0]
[ 1 0 0 -1 0]
[ 0 1 -1 0 0]
[ 1 0 -1 0 0]
[ 0 0 -1 0 0]
[ 0 0 -1 1 0]
[ 0 0 -1 0 1]
[0 1 0 0 0]
[0 0 0 0 1]
[0 0 1 0 0]
[0 0 0 1 0]
[1 0 0 0 0]
---------------------------
35: The group name is C3*S3, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2 * 3^2
Generators:
[ 0 0 1 0 -1]
[ 0 0 0 1 -1]
[ 1 0 0 0 -1]
[ 0 1 0 0 -1]
[ 0 0 0 0 -1]
[ 0 1 0 -1 0]
[ 0 0 0 -1 1]
[ 0 0 0 -1 0]
[ 0 0 1 -1 0]
[ 1 0 0 -1 0]
[0 1 0 0 0]
[0 0 0 0 1]
[0 0 1 0 0]
[0 0 0 1 0]
[1 0 0 0 0]
---------------------------
36: The group name is C3*S3, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2 * 3^2
Generators:
[ 1 0 0 -1 0]
[ 0 1 0 -1 0]
[ 0 0 0 -1 0]
[ 0 0 1 -1 0]
[ 0 0 0 -1 1]
[0 1 0 0 0]
[0 0 0 0 1]
[0 0 1 0 0]
[0 0 0 1 0]
[1 0 0 0 0]
[1 0 0 0 0]
[0 0 0 0 1]
[0 0 1 0 0]
[0 0 0 1 0]
[0 1 0 0 0]
---------------------------
37: The group name is F5, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^2 * 5
Generators:
[0 0 0 0 1]
[0 0 0 1 0]
[0 0 1 0 0]
[0 1 0 0 0]
[1 0 0 0 0]
[0 1 0 0 0]
[0 0 0 1 0]
[1 0 0 0 0]
[0 0 0 0 1]
[0 0 1 0 0]
[0 0 0 1 0]
[1 0 0 0 0]
[0 0 1 0 0]
[0 0 0 0 1]
[0 1 0 0 0]
---------------------------
38: The group name is C2*D4, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^4
Generators:
[ 1 0 -1 0 0]
[ 0 0 -1 1 0]
[ 0 0 -1 0 0]
[ 0 1 -1 0 0]
[ 0 0 -1 0 1]
[ 0 -1 0 0 1]
[ 0 -1 0 0 0]
[ 0 -1 1 0 0]
[ 0 -1 0 1 0]
[ 1 -1 0 0 0]
[0 0 0 0 1]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[1 0 0 0 0]
[ 1 -1 0 0 0]
[ 0 -1 0 0 0]
[ 0 -1 0 1 0]
[ 0 -1 1 0 0]
[ 0 -1 0 0 1]
---------------------------
39: The group name is S4, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^3 * 3
Generators:
[ 1 0 -1 0 0]
[ 0 0 -1 1 0]
[ 0 0 -1 0 0]
[ 0 1 -1 0 0]
[ 0 0 -1 0 1]
[ 0 0 -1 0 0]
[ 0 0 -1 1 0]
[ 1 0 -1 0 0]
[ 0 0 -1 0 1]
[ 0 1 -1 0 0]
[ 0 -1 0 0 1]
[ 0 -1 0 0 0]
[ 0 -1 1 0 0]
[ 0 -1 0 1 0]
[ 1 -1 0 0 0]
[ 1 -1 0 0 0]
[ 0 -1 0 0 0]
[ 0 -1 0 1 0]
[ 0 -1 1 0 0]
[ 0 -1 0 0 1]
---------------------------
40: The group name is S4, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^3 * 3
Generators:
[ 1 0 -1 0 0]
[ 0 0 -1 1 0]
[ 0 0 -1 0 0]
[ 0 1 -1 0 0]
[ 0 0 -1 0 1]
[1 0 0 0 0]
[0 0 0 1 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 0 1]
[1 0 0 0 0]
[0 0 0 1 0]
[0 0 1 0 0]
[0 1 0 0 0]
[0 0 0 0 1]
[ 1 -1 0 0 0]
[ 0 -1 0 0 0]
[ 0 -1 0 1 0]
[ 0 -1 1 0 0]
[ 0 -1 0 0 1]
---------------------------
41: The group name is C2*A4, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^3 * 3
Generators:
[ 1 0 -1 0 0]
[ 0 0 -1 1 0]
[ 0 0 -1 0 0]
[ 0 1 -1 0 0]
[ 0 0 -1 0 1]
[0 0 0 0 1]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[1 0 0 0 0]
[1 0 0 0 0]
[0 0 0 1 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 0 1]
[ 1 -1 0 0 0]
[ 0 -1 0 0 0]
[ 0 -1 0 1 0]
[ 0 -1 1 0 0]
[ 0 -1 0 0 1]
---------------------------
42: The group name is S4, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^3 * 3
Generators:
[ 1 0 -1 0 0]
[ 0 0 -1 1 0]
[ 0 0 -1 0 0]
[ 0 1 -1 0 0]
[ 0 0 -1 0 1]
[0 0 0 0 1]
[0 0 0 1 0]
[0 0 1 0 0]
[0 1 0 0 0]
[1 0 0 0 0]
[1 0 0 0 0]
[0 0 0 1 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 0 1]
[ 1 -1 0 0 0]
[ 0 -1 0 0 0]
[ 0 -1 0 1 0]
[ 0 -1 1 0 0]
[ 0 -1 0 0 1]
---------------------------
43: The group name is C2*A4, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^3 * 3
Generators:
[ 0 0 -1 0 0]
[ 0 0 -1 1 0]
[ 1 0 -1 0 0]
[ 0 0 -1 0 1]
[ 0 1 -1 0 0]
[ 0 -1 0 0 1]
[ 0 -1 0 0 0]
[ 0 -1 1 0 0]
[ 0 -1 0 1 0]
[ 1 -1 0 0 0]
[0 0 0 0 1]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[1 0 0 0 0]
[ 1 -1 0 0 0]
[ 0 -1 0 0 0]
[ 0 -1 0 1 0]
[ 0 -1 1 0 0]
[ 0 -1 0 0 1]
---------------------------
44: The group name is S4, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^3 * 3
Generators:
[ 0 0 -1 0 1]
[ 0 0 -1 1 0]
[ 0 0 -1 0 0]
[ 0 1 -1 0 0]
[ 1 0 -1 0 0]
[ 0 0 -1 0 0]
[ 0 0 -1 1 0]
[ 1 0 -1 0 0]
[ 0 0 -1 0 1]
[ 0 1 -1 0 0]
[ 0 -1 0 0 1]
[ 0 -1 0 0 0]
[ 0 -1 1 0 0]
[ 0 -1 0 1 0]
[ 1 -1 0 0 0]
[ 1 -1 0 0 0]
[ 0 -1 0 0 0]
[ 0 -1 0 1 0]
[ 0 -1 1 0 0]
[ 0 -1 0 0 1]
---------------------------
45: The group name is S3^2, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^2 * 3^2
Generators:
[0 0 0 0 1]
[1 0 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 1 0 0 0]
[ 1 0 -1 0 0]
[ 0 1 -1 0 0]
[ 0 0 -1 1 0]
[ 0 0 -1 0 0]
[ 0 0 -1 0 1]
[ 1 0 -1 0 0]
[ 0 0 -1 0 1]
[ 0 0 -1 0 0]
[ 0 0 -1 1 0]
[ 0 1 -1 0 0]
[1 0 0 0 0]
[0 0 0 0 1]
[0 0 1 0 0]
[0 0 0 1 0]
[0 1 0 0 0]
---------------------------
46: The group name is S3^2, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^2 * 3^2
Generators:
[ 0 0 1 0 -1]
[ 0 0 0 1 -1]
[ 1 0 0 0 -1]
[ 0 1 0 0 -1]
[ 0 0 0 0 -1]
[0 0 0 0 1]
[1 0 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 1 0 0 0]
[ 1 0 -1 0 0]
[ 0 1 -1 0 0]
[ 0 0 -1 1 0]
[ 0 0 -1 0 0]
[ 0 0 -1 0 1]
[ 1 0 -1 0 0]
[ 0 0 -1 0 1]
[ 0 0 -1 0 0]
[ 0 0 -1 1 0]
[ 0 1 -1 0 0]
---------------------------
47: The group name is C3:S3.C2, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^2 * 3^2
Generators:
[0 0 0 0 1]
[1 0 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 1 0 0 0]
[ 1 0 -1 0 0]
[ 0 1 -1 0 0]
[ 0 0 -1 1 0]
[ 0 0 -1 0 0]
[ 0 0 -1 0 1]
[ 1 0 -1 0 0]
[ 0 0 -1 0 1]
[ 0 0 -1 0 0]
[ 0 0 -1 1 0]
[ 0 1 -1 0 0]
[ 0 0 1 0 -1]
[ 0 0 0 0 -1]
[ 1 0 0 0 -1]
[ 0 1 0 0 -1]
[ 0 0 0 1 -1]
---------------------------
48: The group name is A5, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^2 * 3 * 5
Generators:
[ 0 0 -1 0 1]
[ 1 0 -1 0 0]
[ 0 0 -1 1 0]
[ 0 0 -1 0 0]
[ 0 1 -1 0 0]
[-1 0 0 0 0]
[-1 1 0 0 0]
[-1 0 0 1 0]
[-1 0 1 0 0]
[-1 0 0 0 1]
---------------------------
49: The group name is A5, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^2 * 3 * 5
Generators:
[0 0 1 0 0]
[1 0 0 0 0]
[0 1 0 0 0]
[0 0 0 1 0]
[0 0 0 0 1]
[0 0 0 0 1]
[0 1 0 0 0]
[0 0 0 1 0]
[0 0 1 0 0]
[1 0 0 0 0]
---------------------------
50: The group name is C2*S4, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^4 * 3
Generators:
[1 0 0 0 0]
[0 1 0 0 0]
[0 0 0 1 0]
[0 0 1 0 0]
[0 0 0 0 1]
[-1 0 1 0 0]
[-1 0 0 0 1]
[-1 0 0 0 0]
[-1 1 0 0 0]
[-1 0 0 1 0]
[0 0 0 0 1]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[1 0 0 0 0]
[0 0 1 0 0]
[0 1 0 0 0]
[1 0 0 0 0]
[0 0 0 0 1]
[0 0 0 1 0]
[ 1 -1 0 0 0]
[ 0 -1 0 0 0]
[ 0 -1 1 0 0]
[ 0 -1 0 1 0]
[ 0 -1 0 0 1]
---------------------------
51: The group name is C2*S4, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^4 * 3
Generators:
[1 0 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 1 0 0 0]
[0 0 0 0 1]
[ 1 0 0 -1 0]
[ 0 0 1 -1 0]
[ 0 1 0 -1 0]
[ 0 0 0 -1 0]
[ 0 0 0 -1 1]
[0 0 0 0 1]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[1 0 0 0 0]
[1 0 0 0 0]
[0 0 1 0 0]
[0 1 0 0 0]
[0 0 0 1 0]
[0 0 0 0 1]
[ 1 -1 0 0 0]
[ 0 -1 0 0 0]
[ 0 -1 0 1 0]
[ 0 -1 1 0 0]
[ 0 -1 0 0 1]
---------------------------
52: The group name is S3wrC2, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^3 * 3^2
Generators:
[ 0 0 1 0 -1]
[ 0 0 0 1 -1]
[ 1 0 0 0 -1]
[ 0 1 0 0 -1]
[ 0 0 0 0 -1]
[ 1 0 0 -1 0]
[ 0 1 0 -1 0]
[ 0 0 0 -1 0]
[ 0 0 1 -1 0]
[ 0 0 0 -1 1]
[ 0 1 0 -1 0]
[ 1 0 0 -1 0]
[ 0 0 1 -1 0]
[ 0 0 0 -1 0]
[ 0 0 0 -1 1]
[0 1 0 0 0]
[0 0 0 0 1]
[0 0 1 0 0]
[0 0 0 1 0]
[1 0 0 0 0]
[ 1 0 0 -1 0]
[ 0 1 0 -1 0]
[ 0 0 1 -1 0]
[ 0 0 0 -1 0]
[ 0 0 0 -1 1]
---------------------------
53: The group name is S5, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^3 * 3 * 5
Generators:
[ 0 0 -1 0 1]
[ 0 0 -1 1 0]
[ 0 1 -1 0 0]
[ 1 0 -1 0 0]
[ 0 0 -1 0 0]
[ 0 -1 0 0 1]
[ 0 -1 0 0 0]
[ 0 -1 0 1 0]
[ 0 -1 1 0 0]
[ 1 -1 0 0 0]
---------------------------
54: The group name is S5, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^3 * 3 * 5
Generators:
[0 0 0 0 1]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[1 0 0 0 0]
[0 1 0 0 0]
[1 0 0 0 0]
[0 0 0 1 0]
[0 0 0 0 1]
[0 0 1 0 0]
---------------------------
55: The group name is A6, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^3 * 3^2 * 5
Generators:
[ 0 0 1 0 -1]
[ 1 0 0 0 -1]
[ 0 0 0 1 -1]
[ 0 1 0 0 -1]
[ 0 0 0 0 -1]
[0 0 0 0 1]
[0 1 0 0 0]
[0 0 0 1 0]
[0 0 1 0 0]
[1 0 0 0 0]
---------------------------
56: The group name is S6, the generators are:
MatrixGroup(5, Cyclotomic Field of order 3 and degree 2) of order 2^4 * 3^2 * 5
Generators:
[0 1 0 0 0]
[1 0 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]
[-1 1 0 0 0]
[-1 0 1 0 0]
[-1 0 0 1 0]
[-1 0 0 0 1]
[-1 0 0 0 0]
---------------------------