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## G = C24⋊C6order 96 = 25·3

### 1st semidirect product of C24 and C6 acting faithfully

Aliases: C241C6, C231A4, C22≀C2⋊C3, C22⋊A41C2, C22.2(C2×A4), SmallGroup(96,70)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C24⋊C6
 Chief series C1 — C22 — C24 — C22⋊A4 — C24⋊C6
 Lower central C24 — C24⋊C6
 Upper central C1

Generators and relations for C24⋊C6
G = < a,b,c,d,e | a2=b2=c2=d2=e6=1, ab=ba, ac=ca, ad=da, eae-1=db=bd, bc=cb, ebe-1=abcd, ede-1=cd=dc, ece-1=d >

3C2
4C2
6C2
6C2
16C3
2C22
2C22
3C22
3C22
6C22
6C22
6C22
6C22
6C22
6C4
16C6
3C23
6C23
6D4
6C23
6D4
4A4
8A4
8A4

Character table of C24⋊C6

 class 1 2A 2B 2C 2D 3A 3B 4 6A 6B size 1 3 4 6 6 16 16 12 16 16 ρ1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 1 1 1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 ζ3 ζ32 1 ζ32 ζ3 linear of order 3 ρ4 1 1 -1 1 1 ζ32 ζ3 -1 ζ65 ζ6 linear of order 6 ρ5 1 1 -1 1 1 ζ3 ζ32 -1 ζ6 ζ65 linear of order 6 ρ6 1 1 1 1 1 ζ32 ζ3 1 ζ3 ζ32 linear of order 3 ρ7 3 3 3 -1 -1 0 0 -1 0 0 orthogonal lifted from A4 ρ8 3 3 -3 -1 -1 0 0 1 0 0 orthogonal lifted from C2×A4 ρ9 6 -2 0 2 -2 0 0 0 0 0 orthogonal faithful ρ10 6 -2 0 -2 2 0 0 0 0 0 orthogonal faithful

Permutation representations of C24⋊C6
On 8 points - transitive group 8T33
Generators in S8
```(1 4)(2 5)(3 7)(6 8)
(1 6)(2 7)(3 5)(4 8)
(1 6)(2 3)(4 8)(5 7)
(1 8)(2 5)(3 7)(4 6)
(1 2)(3 4 5 6 7 8)```

`G:=sub<Sym(8)| (1,4)(2,5)(3,7)(6,8), (1,6)(2,7)(3,5)(4,8), (1,6)(2,3)(4,8)(5,7), (1,8)(2,5)(3,7)(4,6), (1,2)(3,4,5,6,7,8)>;`

`G:=Group( (1,4)(2,5)(3,7)(6,8), (1,6)(2,7)(3,5)(4,8), (1,6)(2,3)(4,8)(5,7), (1,8)(2,5)(3,7)(4,6), (1,2)(3,4,5,6,7,8) );`

`G=PermutationGroup([[(1,4),(2,5),(3,7),(6,8)], [(1,6),(2,7),(3,5),(4,8)], [(1,6),(2,3),(4,8),(5,7)], [(1,8),(2,5),(3,7),(4,6)], [(1,2),(3,4,5,6,7,8)]])`

`G:=TransitiveGroup(8,33);`

On 12 points - transitive group 12T58
Generators in S12
```(2 11)(6 9)
(2 11)(4 7)
(2 11)(3 12)(5 8)(6 9)
(1 10)(2 11)(4 7)(5 8)
(1 2 3 4 5 6)(7 8 9 10 11 12)```

`G:=sub<Sym(12)| (2,11)(6,9), (2,11)(4,7), (2,11)(3,12)(5,8)(6,9), (1,10)(2,11)(4,7)(5,8), (1,2,3,4,5,6)(7,8,9,10,11,12)>;`

`G:=Group( (2,11)(6,9), (2,11)(4,7), (2,11)(3,12)(5,8)(6,9), (1,10)(2,11)(4,7)(5,8), (1,2,3,4,5,6)(7,8,9,10,11,12) );`

`G=PermutationGroup([[(2,11),(6,9)], [(2,11),(4,7)], [(2,11),(3,12),(5,8),(6,9)], [(1,10),(2,11),(4,7),(5,8)], [(1,2,3,4,5,6),(7,8,9,10,11,12)]])`

`G:=TransitiveGroup(12,58);`

On 12 points - transitive group 12T59
Generators in S12
```(2 8)(3 12)(4 9)(6 11)
(1 10)(2 8)(5 7)(6 11)
(2 6)(3 4)(8 11)(9 12)
(1 5)(2 6)(7 10)(8 11)
(1 2 3)(4 5 6)(7 8 9 10 11 12)```

`G:=sub<Sym(12)| (2,8)(3,12)(4,9)(6,11), (1,10)(2,8)(5,7)(6,11), (2,6)(3,4)(8,11)(9,12), (1,5)(2,6)(7,10)(8,11), (1,2,3)(4,5,6)(7,8,9,10,11,12)>;`

`G:=Group( (2,8)(3,12)(4,9)(6,11), (1,10)(2,8)(5,7)(6,11), (2,6)(3,4)(8,11)(9,12), (1,5)(2,6)(7,10)(8,11), (1,2,3)(4,5,6)(7,8,9,10,11,12) );`

`G=PermutationGroup([[(2,8),(3,12),(4,9),(6,11)], [(1,10),(2,8),(5,7),(6,11)], [(2,6),(3,4),(8,11),(9,12)], [(1,5),(2,6),(7,10),(8,11)], [(1,2,3),(4,5,6),(7,8,9,10,11,12)]])`

`G:=TransitiveGroup(12,59);`

On 16 points - transitive group 16T183
Generators in S16
```(1 6)(2 11)(3 9)(4 14)(5 12)(7 13)(8 10)(15 16)
(1 8)(2 5)(3 16)(4 13)(6 10)(7 14)(9 15)(11 12)
(1 3)(2 4)(5 13)(6 9)(7 12)(8 16)(10 15)(11 14)
(1 2)(3 4)(5 8)(6 11)(7 15)(9 14)(10 12)(13 16)
(2 3 4)(5 6 7 8 9 10)(11 12 13 14 15 16)```

`G:=sub<Sym(16)| (1,6)(2,11)(3,9)(4,14)(5,12)(7,13)(8,10)(15,16), (1,8)(2,5)(3,16)(4,13)(6,10)(7,14)(9,15)(11,12), (1,3)(2,4)(5,13)(6,9)(7,12)(8,16)(10,15)(11,14), (1,2)(3,4)(5,8)(6,11)(7,15)(9,14)(10,12)(13,16), (2,3,4)(5,6,7,8,9,10)(11,12,13,14,15,16)>;`

`G:=Group( (1,6)(2,11)(3,9)(4,14)(5,12)(7,13)(8,10)(15,16), (1,8)(2,5)(3,16)(4,13)(6,10)(7,14)(9,15)(11,12), (1,3)(2,4)(5,13)(6,9)(7,12)(8,16)(10,15)(11,14), (1,2)(3,4)(5,8)(6,11)(7,15)(9,14)(10,12)(13,16), (2,3,4)(5,6,7,8,9,10)(11,12,13,14,15,16) );`

`G=PermutationGroup([[(1,6),(2,11),(3,9),(4,14),(5,12),(7,13),(8,10),(15,16)], [(1,8),(2,5),(3,16),(4,13),(6,10),(7,14),(9,15),(11,12)], [(1,3),(2,4),(5,13),(6,9),(7,12),(8,16),(10,15),(11,14)], [(1,2),(3,4),(5,8),(6,11),(7,15),(9,14),(10,12),(13,16)], [(2,3,4),(5,6,7,8,9,10),(11,12,13,14,15,16)]])`

`G:=TransitiveGroup(16,183);`

On 24 points - transitive group 24T181
Generators in S24
```(1 20)(2 13)(3 14)(4 23)(5 12)(6 7)(8 18)(9 21)(10 22)(11 15)(16 24)(17 19)
(1 8)(2 9)(3 22)(4 15)(5 16)(6 19)(7 17)(10 14)(11 23)(12 24)(13 21)(18 20)
(2 21)(3 22)(5 24)(6 19)(7 17)(9 13)(10 14)(12 16)
(1 20)(2 21)(4 23)(5 24)(8 18)(9 13)(11 15)(12 16)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)```

`G:=sub<Sym(24)| (1,20)(2,13)(3,14)(4,23)(5,12)(6,7)(8,18)(9,21)(10,22)(11,15)(16,24)(17,19), (1,8)(2,9)(3,22)(4,15)(5,16)(6,19)(7,17)(10,14)(11,23)(12,24)(13,21)(18,20), (2,21)(3,22)(5,24)(6,19)(7,17)(9,13)(10,14)(12,16), (1,20)(2,21)(4,23)(5,24)(8,18)(9,13)(11,15)(12,16), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)>;`

`G:=Group( (1,20)(2,13)(3,14)(4,23)(5,12)(6,7)(8,18)(9,21)(10,22)(11,15)(16,24)(17,19), (1,8)(2,9)(3,22)(4,15)(5,16)(6,19)(7,17)(10,14)(11,23)(12,24)(13,21)(18,20), (2,21)(3,22)(5,24)(6,19)(7,17)(9,13)(10,14)(12,16), (1,20)(2,21)(4,23)(5,24)(8,18)(9,13)(11,15)(12,16), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24) );`

`G=PermutationGroup([[(1,20),(2,13),(3,14),(4,23),(5,12),(6,7),(8,18),(9,21),(10,22),(11,15),(16,24),(17,19)], [(1,8),(2,9),(3,22),(4,15),(5,16),(6,19),(7,17),(10,14),(11,23),(12,24),(13,21),(18,20)], [(2,21),(3,22),(5,24),(6,19),(7,17),(9,13),(10,14),(12,16)], [(1,20),(2,21),(4,23),(5,24),(8,18),(9,13),(11,15),(12,16)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)]])`

`G:=TransitiveGroup(24,181);`

On 24 points - transitive group 24T182
Generators in S24
```(2 21)(4 11)(6 17)(7 19)(9 13)(15 23)
(2 13)(4 23)(6 7)(9 21)(11 15)(17 19)
(1 8)(2 21)(3 14)(4 11)(5 24)(6 17)(7 19)(9 13)(10 22)(12 16)(15 23)(18 20)
(1 20)(2 13)(3 10)(4 23)(5 16)(6 7)(8 18)(9 21)(11 15)(12 24)(14 22)(17 19)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)```

`G:=sub<Sym(24)| (2,21)(4,11)(6,17)(7,19)(9,13)(15,23), (2,13)(4,23)(6,7)(9,21)(11,15)(17,19), (1,8)(2,21)(3,14)(4,11)(5,24)(6,17)(7,19)(9,13)(10,22)(12,16)(15,23)(18,20), (1,20)(2,13)(3,10)(4,23)(5,16)(6,7)(8,18)(9,21)(11,15)(12,24)(14,22)(17,19), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)>;`

`G:=Group( (2,21)(4,11)(6,17)(7,19)(9,13)(15,23), (2,13)(4,23)(6,7)(9,21)(11,15)(17,19), (1,8)(2,21)(3,14)(4,11)(5,24)(6,17)(7,19)(9,13)(10,22)(12,16)(15,23)(18,20), (1,20)(2,13)(3,10)(4,23)(5,16)(6,7)(8,18)(9,21)(11,15)(12,24)(14,22)(17,19), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24) );`

`G=PermutationGroup([[(2,21),(4,11),(6,17),(7,19),(9,13),(15,23)], [(2,13),(4,23),(6,7),(9,21),(11,15),(17,19)], [(1,8),(2,21),(3,14),(4,11),(5,24),(6,17),(7,19),(9,13),(10,22),(12,16),(15,23),(18,20)], [(1,20),(2,13),(3,10),(4,23),(5,16),(6,7),(8,18),(9,21),(11,15),(12,24),(14,22),(17,19)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)]])`

`G:=TransitiveGroup(24,182);`

On 24 points - transitive group 24T183
Generators in S24
```(1 8)(2 21)(3 10)(6 19)(7 17)(9 13)(14 22)(18 20)
(2 21)(3 10)(4 23)(5 12)(9 13)(11 15)(14 22)(16 24)
(1 8)(2 21)(3 14)(4 11)(5 24)(6 17)(7 19)(9 13)(10 22)(12 16)(15 23)(18 20)
(1 20)(2 13)(3 10)(4 23)(5 16)(6 7)(8 18)(9 21)(11 15)(12 24)(14 22)(17 19)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)```

`G:=sub<Sym(24)| (1,8)(2,21)(3,10)(6,19)(7,17)(9,13)(14,22)(18,20), (2,21)(3,10)(4,23)(5,12)(9,13)(11,15)(14,22)(16,24), (1,8)(2,21)(3,14)(4,11)(5,24)(6,17)(7,19)(9,13)(10,22)(12,16)(15,23)(18,20), (1,20)(2,13)(3,10)(4,23)(5,16)(6,7)(8,18)(9,21)(11,15)(12,24)(14,22)(17,19), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)>;`

`G:=Group( (1,8)(2,21)(3,10)(6,19)(7,17)(9,13)(14,22)(18,20), (2,21)(3,10)(4,23)(5,12)(9,13)(11,15)(14,22)(16,24), (1,8)(2,21)(3,14)(4,11)(5,24)(6,17)(7,19)(9,13)(10,22)(12,16)(15,23)(18,20), (1,20)(2,13)(3,10)(4,23)(5,16)(6,7)(8,18)(9,21)(11,15)(12,24)(14,22)(17,19), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24) );`

`G=PermutationGroup([[(1,8),(2,21),(3,10),(6,19),(7,17),(9,13),(14,22),(18,20)], [(2,21),(3,10),(4,23),(5,12),(9,13),(11,15),(14,22),(16,24)], [(1,8),(2,21),(3,14),(4,11),(5,24),(6,17),(7,19),(9,13),(10,22),(12,16),(15,23),(18,20)], [(1,20),(2,13),(3,10),(4,23),(5,16),(6,7),(8,18),(9,21),(11,15),(12,24),(14,22),(17,19)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)]])`

`G:=TransitiveGroup(24,183);`

On 24 points - transitive group 24T184
Generators in S24
```(2 13)(3 22)(4 11)(5 24)(6 19)(7 17)(9 21)(10 14)(12 16)(15 23)
(1 20)(2 21)(4 15)(5 24)(6 7)(8 18)(9 13)(11 23)(12 16)(17 19)
(1 8)(2 9)(4 11)(5 12)(13 21)(15 23)(16 24)(18 20)
(1 8)(3 10)(4 11)(6 7)(14 22)(15 23)(17 19)(18 20)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)```

`G:=sub<Sym(24)| (2,13)(3,22)(4,11)(5,24)(6,19)(7,17)(9,21)(10,14)(12,16)(15,23), (1,20)(2,21)(4,15)(5,24)(6,7)(8,18)(9,13)(11,23)(12,16)(17,19), (1,8)(2,9)(4,11)(5,12)(13,21)(15,23)(16,24)(18,20), (1,8)(3,10)(4,11)(6,7)(14,22)(15,23)(17,19)(18,20), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)>;`

`G:=Group( (2,13)(3,22)(4,11)(5,24)(6,19)(7,17)(9,21)(10,14)(12,16)(15,23), (1,20)(2,21)(4,15)(5,24)(6,7)(8,18)(9,13)(11,23)(12,16)(17,19), (1,8)(2,9)(4,11)(5,12)(13,21)(15,23)(16,24)(18,20), (1,8)(3,10)(4,11)(6,7)(14,22)(15,23)(17,19)(18,20), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24) );`

`G=PermutationGroup([[(2,13),(3,22),(4,11),(5,24),(6,19),(7,17),(9,21),(10,14),(12,16),(15,23)], [(1,20),(2,21),(4,15),(5,24),(6,7),(8,18),(9,13),(11,23),(12,16),(17,19)], [(1,8),(2,9),(4,11),(5,12),(13,21),(15,23),(16,24),(18,20)], [(1,8),(3,10),(4,11),(6,7),(14,22),(15,23),(17,19),(18,20)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)]])`

`G:=TransitiveGroup(24,184);`

On 24 points - transitive group 24T185
Generators in S24
```(1 16)(2 10)(3 19)(4 20)(5 8)(6 15)(7 13)(9 22)(11 18)(12 23)(14 21)(17 24)
(1 12)(2 21)(3 18)(4 7)(5 17)(6 22)(8 24)(9 15)(10 14)(11 19)(13 20)(16 23)
(1 7)(3 9)(4 12)(6 11)(13 16)(15 18)(19 22)(20 23)
(2 8)(3 9)(5 10)(6 11)(14 17)(15 18)(19 22)(21 24)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)```

`G:=sub<Sym(24)| (1,16)(2,10)(3,19)(4,20)(5,8)(6,15)(7,13)(9,22)(11,18)(12,23)(14,21)(17,24), (1,12)(2,21)(3,18)(4,7)(5,17)(6,22)(8,24)(9,15)(10,14)(11,19)(13,20)(16,23), (1,7)(3,9)(4,12)(6,11)(13,16)(15,18)(19,22)(20,23), (2,8)(3,9)(5,10)(6,11)(14,17)(15,18)(19,22)(21,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)>;`

`G:=Group( (1,16)(2,10)(3,19)(4,20)(5,8)(6,15)(7,13)(9,22)(11,18)(12,23)(14,21)(17,24), (1,12)(2,21)(3,18)(4,7)(5,17)(6,22)(8,24)(9,15)(10,14)(11,19)(13,20)(16,23), (1,7)(3,9)(4,12)(6,11)(13,16)(15,18)(19,22)(20,23), (2,8)(3,9)(5,10)(6,11)(14,17)(15,18)(19,22)(21,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24) );`

`G=PermutationGroup([[(1,16),(2,10),(3,19),(4,20),(5,8),(6,15),(7,13),(9,22),(11,18),(12,23),(14,21),(17,24)], [(1,12),(2,21),(3,18),(4,7),(5,17),(6,22),(8,24),(9,15),(10,14),(11,19),(13,20),(16,23)], [(1,7),(3,9),(4,12),(6,11),(13,16),(15,18),(19,22),(20,23)], [(2,8),(3,9),(5,10),(6,11),(14,17),(15,18),(19,22),(21,24)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)]])`

`G:=TransitiveGroup(24,185);`

On 24 points - transitive group 24T186
Generators in S24
```(1 4)(2 24)(3 22)(5 18)(6 16)(7 10)(8 15)(9 13)(11 21)(12 19)(14 20)(17 23)
(1 14)(2 18)(3 6)(4 20)(5 24)(7 23)(8 21)(9 12)(10 17)(11 15)(13 19)(16 22)
(2 8)(3 9)(5 11)(6 12)(13 22)(15 24)(16 19)(18 21)
(1 7)(2 8)(4 10)(5 11)(14 23)(15 24)(17 20)(18 21)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)```

`G:=sub<Sym(24)| (1,4)(2,24)(3,22)(5,18)(6,16)(7,10)(8,15)(9,13)(11,21)(12,19)(14,20)(17,23), (1,14)(2,18)(3,6)(4,20)(5,24)(7,23)(8,21)(9,12)(10,17)(11,15)(13,19)(16,22), (2,8)(3,9)(5,11)(6,12)(13,22)(15,24)(16,19)(18,21), (1,7)(2,8)(4,10)(5,11)(14,23)(15,24)(17,20)(18,21), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)>;`

`G:=Group( (1,4)(2,24)(3,22)(5,18)(6,16)(7,10)(8,15)(9,13)(11,21)(12,19)(14,20)(17,23), (1,14)(2,18)(3,6)(4,20)(5,24)(7,23)(8,21)(9,12)(10,17)(11,15)(13,19)(16,22), (2,8)(3,9)(5,11)(6,12)(13,22)(15,24)(16,19)(18,21), (1,7)(2,8)(4,10)(5,11)(14,23)(15,24)(17,20)(18,21), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24) );`

`G=PermutationGroup([[(1,4),(2,24),(3,22),(5,18),(6,16),(7,10),(8,15),(9,13),(11,21),(12,19),(14,20),(17,23)], [(1,14),(2,18),(3,6),(4,20),(5,24),(7,23),(8,21),(9,12),(10,17),(11,15),(13,19),(16,22)], [(2,8),(3,9),(5,11),(6,12),(13,22),(15,24),(16,19),(18,21)], [(1,7),(2,8),(4,10),(5,11),(14,23),(15,24),(17,20),(18,21)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)]])`

`G:=TransitiveGroup(24,186);`

C24⋊C6 is a maximal subgroup of
C24⋊Dic3  C24⋊D6  C24⋊A4  (C22×S3)⋊A4  A4≀C2  (C22×D5)⋊A4  F16⋊C2
C24⋊C6 is a maximal quotient of
C24⋊C12  C24.A4  (C22×C4).A4  C2≀A4  2+ 1+4.C6  C24⋊C18  (C22×S3)⋊A4  (C22×D5)⋊A4

Polynomial with Galois group C24⋊C6 over ℚ
actionf(x)Disc(f)
8T33x8-x7-15x6+15x5+62x4-72x3-53x2+78x-1922·36·56·74·2512
12T58x12-23x10+158x8-378x6+372x4-160x2+25212·36·510·76·138
12T59x12-4x11-9x10+48x9+5x8-164x7+69x6+188x5-89x4-82x3+23x2+14x+1215·138·474·20292

Matrix representation of C24⋊C6 in GL6(ℤ)

 0 1 0 0 0 0 1 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 1 0 0 0 0 1 0
,
 -1 -1 -1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 -1 -1 -1 0 0 0 1 0 0
,
 0 0 1 0 0 0 -1 -1 -1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1 -1 -1 0 0 0 1 0 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 -1 -1 -1
,
 0 0 0 1 0 0 0 0 0 -1 -1 -1 0 0 0 0 1 0 1 0 0 0 0 0 -1 -1 -1 0 0 0 0 1 0 0 0 0

`G:=sub<GL(6,Integers())| [0,1,-1,0,0,0,1,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,1,0,0,0,-1,1,0],[-1,0,0,0,0,0,-1,0,1,0,0,0,-1,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,1,-1,0],[0,-1,1,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,1,-1,0],[0,1,-1,0,0,0,1,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,-1,0,0,0,1,0,-1,0,0,0,0,0,-1],[0,0,0,1,-1,0,0,0,0,0,-1,1,0,0,0,0,-1,0,1,-1,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0] >;`

C24⋊C6 in GAP, Magma, Sage, TeX

`C_2^4\rtimes C_6`
`% in TeX`

`G:=Group("C2^4:C6");`
`// GroupNames label`

`G:=SmallGroup(96,70);`
`// by ID`

`G=gap.SmallGroup(96,70);`
`# by ID`

`G:=PCGroup([6,-2,-3,-2,2,-2,2,542,116,1443,225,730,1307]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=e^6=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=d*b=b*d,b*c=c*b,e*b*e^-1=a*b*c*d,e*d*e^-1=c*d=d*c,e*c*e^-1=d>;`
`// generators/relations`

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