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

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

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

G=PermutationGroup([(1,14),(2,5),(3,11),(4,8),(6,13),(7,15),(9,10),(12,16)], [(1,16),(2,13),(3,10),(4,7),(5,6),(8,15),(9,11),(12,14)], [(1,3),(2,4),(5,8),(6,15),(7,13),(9,12),(10,16),(11,14)], [(1,2),(3,4),(5,14),(6,12),(7,10),(8,11),(9,15),(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 8)(2 13)(3 14)(4 11)(5 24)(6 19)(7 17)(9 21)(10 22)(12 16)(15 23)(18 20)
(1 20)(2 21)(3 10)(4 15)(5 16)(6 7)(8 18)(9 13)(11 23)(12 24)(14 22)(17 19)
(2 9)(3 10)(5 12)(6 7)(13 21)(14 22)(16 24)(17 19)
(1 8)(2 9)(4 11)(5 12)(13 21)(15 23)(16 24)(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)| (1,8)(2,13)(3,14)(4,11)(5,24)(6,19)(7,17)(9,21)(10,22)(12,16)(15,23)(18,20), (1,20)(2,21)(3,10)(4,15)(5,16)(6,7)(8,18)(9,13)(11,23)(12,24)(14,22)(17,19), (2,9)(3,10)(5,12)(6,7)(13,21)(14,22)(16,24)(17,19), (1,8)(2,9)(4,11)(5,12)(13,21)(15,23)(16,24)(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( (1,8)(2,13)(3,14)(4,11)(5,24)(6,19)(7,17)(9,21)(10,22)(12,16)(15,23)(18,20), (1,20)(2,21)(3,10)(4,15)(5,16)(6,7)(8,18)(9,13)(11,23)(12,24)(14,22)(17,19), (2,9)(3,10)(5,12)(6,7)(13,21)(14,22)(16,24)(17,19), (1,8)(2,9)(4,11)(5,12)(13,21)(15,23)(16,24)(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([(1,8),(2,13),(3,14),(4,11),(5,24),(6,19),(7,17),(9,21),(10,22),(12,16),(15,23),(18,20)], [(1,20),(2,21),(3,10),(4,15),(5,16),(6,7),(8,18),(9,13),(11,23),(12,24),(14,22),(17,19)], [(2,9),(3,10),(5,12),(6,7),(13,21),(14,22),(16,24),(17,19)], [(1,8),(2,9),(4,11),(5,12),(13,21),(15,23),(16,24),(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(24,181);

On 24 points - transitive group 24T182
Generators in S24
(2 9)(4 21)(6 17)(7 23)(11 15)(13 19)
(2 13)(4 11)(6 23)(7 17)(9 19)(15 21)
(1 24)(2 9)(3 14)(4 21)(5 12)(6 17)(7 23)(8 18)(10 20)(11 15)(13 19)(16 22)
(1 8)(2 13)(3 20)(4 11)(5 16)(6 23)(7 17)(9 19)(10 14)(12 22)(15 21)(18 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)| (2,9)(4,21)(6,17)(7,23)(11,15)(13,19), (2,13)(4,11)(6,23)(7,17)(9,19)(15,21), (1,24)(2,9)(3,14)(4,21)(5,12)(6,17)(7,23)(8,18)(10,20)(11,15)(13,19)(16,22), (1,8)(2,13)(3,20)(4,11)(5,16)(6,23)(7,17)(9,19)(10,14)(12,22)(15,21)(18,24), (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( (2,9)(4,21)(6,17)(7,23)(11,15)(13,19), (2,13)(4,11)(6,23)(7,17)(9,19)(15,21), (1,24)(2,9)(3,14)(4,21)(5,12)(6,17)(7,23)(8,18)(10,20)(11,15)(13,19)(16,22), (1,8)(2,13)(3,20)(4,11)(5,16)(6,23)(7,17)(9,19)(10,14)(12,22)(15,21)(18,24), (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([(2,9),(4,21),(6,17),(7,23),(11,15),(13,19)], [(2,13),(4,11),(6,23),(7,17),(9,19),(15,21)], [(1,24),(2,9),(3,14),(4,21),(5,12),(6,17),(7,23),(8,18),(10,20),(11,15),(13,19),(16,22)], [(1,8),(2,13),(3,20),(4,11),(5,16),(6,23),(7,17),(9,19),(10,14),(12,22),(15,21),(18,24)], [(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(24,182);

On 24 points - transitive group 24T183
Generators in S24
(1 24)(2 9)(3 20)(6 7)(8 18)(10 14)(13 19)(17 23)
(2 9)(3 20)(4 11)(5 22)(10 14)(12 16)(13 19)(15 21)
(1 24)(2 9)(3 14)(4 21)(5 12)(6 17)(7 23)(8 18)(10 20)(11 15)(13 19)(16 22)
(1 8)(2 13)(3 20)(4 11)(5 16)(6 23)(7 17)(9 19)(10 14)(12 22)(15 21)(18 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)| (1,24)(2,9)(3,20)(6,7)(8,18)(10,14)(13,19)(17,23), (2,9)(3,20)(4,11)(5,22)(10,14)(12,16)(13,19)(15,21), (1,24)(2,9)(3,14)(4,21)(5,12)(6,17)(7,23)(8,18)(10,20)(11,15)(13,19)(16,22), (1,8)(2,13)(3,20)(4,11)(5,16)(6,23)(7,17)(9,19)(10,14)(12,22)(15,21)(18,24), (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( (1,24)(2,9)(3,20)(6,7)(8,18)(10,14)(13,19)(17,23), (2,9)(3,20)(4,11)(5,22)(10,14)(12,16)(13,19)(15,21), (1,24)(2,9)(3,14)(4,21)(5,12)(6,17)(7,23)(8,18)(10,20)(11,15)(13,19)(16,22), (1,8)(2,13)(3,20)(4,11)(5,16)(6,23)(7,17)(9,19)(10,14)(12,22)(15,21)(18,24), (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([(1,24),(2,9),(3,20),(6,7),(8,18),(10,14),(13,19),(17,23)], [(2,9),(3,20),(4,11),(5,22),(10,14),(12,16),(13,19),(15,21)], [(1,24),(2,9),(3,14),(4,21),(5,12),(6,17),(7,23),(8,18),(10,20),(11,15),(13,19),(16,22)], [(1,8),(2,13),(3,20),(4,11),(5,16),(6,23),(7,17),(9,19),(10,14),(12,22),(15,21),(18,24)], [(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(24,183);

On 24 points - transitive group 24T184
Generators in S24
(2 13)(3 20)(4 11)(5 22)(6 23)(7 17)(9 19)(10 14)(12 16)(15 21)
(1 24)(2 19)(4 15)(5 22)(6 7)(8 18)(9 13)(11 21)(12 16)(17 23)
(1 8)(2 9)(4 11)(5 12)(13 19)(15 21)(16 22)(18 24)
(1 8)(3 10)(4 11)(6 7)(14 20)(15 21)(17 23)(18 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)| (2,13)(3,20)(4,11)(5,22)(6,23)(7,17)(9,19)(10,14)(12,16)(15,21), (1,24)(2,19)(4,15)(5,22)(6,7)(8,18)(9,13)(11,21)(12,16)(17,23), (1,8)(2,9)(4,11)(5,12)(13,19)(15,21)(16,22)(18,24), (1,8)(3,10)(4,11)(6,7)(14,20)(15,21)(17,23)(18,24), (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( (2,13)(3,20)(4,11)(5,22)(6,23)(7,17)(9,19)(10,14)(12,16)(15,21), (1,24)(2,19)(4,15)(5,22)(6,7)(8,18)(9,13)(11,21)(12,16)(17,23), (1,8)(2,9)(4,11)(5,12)(13,19)(15,21)(16,22)(18,24), (1,8)(3,10)(4,11)(6,7)(14,20)(15,21)(17,23)(18,24), (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([(2,13),(3,20),(4,11),(5,22),(6,23),(7,17),(9,19),(10,14),(12,16),(15,21)], [(1,24),(2,19),(4,15),(5,22),(6,7),(8,18),(9,13),(11,21),(12,16),(17,23)], [(1,8),(2,9),(4,11),(5,12),(13,19),(15,21),(16,22),(18,24)], [(1,8),(3,10),(4,11),(6,7),(14,20),(15,21),(17,23),(18,24)], [(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(24,184);

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