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## G = C24.6A4order 192 = 26·3

### 6th non-split extension by C24 of A4 acting faithfully

Aliases: C24.6A4, C41D4⋊C6, C42⋊(C2×C6), C42⋊C6⋊C2, C422C2⋊C6, C42⋊C32C22, C23.5(C2×A4), C23.A41C2, C22.54C24⋊C3, C22.5(C22×A4), SmallGroup(192,1008)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C24.6A4
 Chief series C1 — C22 — C42 — C42⋊C3 — C23.A4 — C24.6A4
 Lower central C42 — C24.6A4
 Upper central C1

Generators and relations for C24.6A4
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g3=1, e2=dc=gcg-1=cd, f2=gdg-1=c, ab=ba, faf-1=ac=ca, ad=da, eae-1=acd, ag=ga, ebe-1=bc=cb, fbf-1=bd=db, bg=gb, ce=ec, cf=fc, gfg-1=de=ed, df=fd, ef=fe, geg-1=cef >

Subgroups: 342 in 67 conjugacy classes, 16 normal (10 characteristic)
C1, C2 [×5], C3, C4 [×3], C22, C22 [×8], C6 [×3], C2×C4 [×4], D4 [×4], C23, C23 [×2], C23 [×2], A4, C2×C6, C42, C22⋊C4 [×4], C4⋊C4 [×2], C22×C4, C2×D4 [×4], C24, C2×A4 [×3], C22≀C2, C4⋊D4 [×2], C22.D4, C422C2 [×2], C41D4, C42⋊C3, C22×A4, C22.54C24, C42⋊C6 [×2], C23.A4, C24.6A4
Quotients: C1, C2 [×3], C3, C22, C6 [×3], A4, C2×C6, C2×A4 [×3], C22×A4, C24.6A4

Character table of C24.6A4

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 6A 6B 6C 6D 6E 6F size 1 3 4 4 4 12 16 16 12 12 12 16 16 16 16 16 16 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 -1 1 1 1 -1 1 1 -1 -1 -1 1 -1 linear of order 2 ρ3 1 1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ4 1 1 -1 1 -1 1 1 1 1 -1 -1 -1 1 -1 -1 -1 1 linear of order 2 ρ5 1 1 1 -1 -1 -1 ζ32 ζ3 1 1 -1 ζ65 ζ65 ζ32 ζ3 ζ6 ζ6 linear of order 6 ρ6 1 1 -1 -1 1 -1 ζ3 ζ32 1 -1 1 ζ32 ζ6 ζ65 ζ6 ζ3 ζ65 linear of order 6 ρ7 1 1 1 1 1 1 ζ3 ζ32 1 1 1 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 linear of order 3 ρ8 1 1 -1 1 -1 1 ζ3 ζ32 1 -1 -1 ζ6 ζ32 ζ65 ζ6 ζ65 ζ3 linear of order 6 ρ9 1 1 1 -1 -1 -1 ζ3 ζ32 1 1 -1 ζ6 ζ6 ζ3 ζ32 ζ65 ζ65 linear of order 6 ρ10 1 1 -1 -1 1 -1 ζ32 ζ3 1 -1 1 ζ3 ζ65 ζ6 ζ65 ζ32 ζ6 linear of order 6 ρ11 1 1 -1 1 -1 1 ζ32 ζ3 1 -1 -1 ζ65 ζ3 ζ6 ζ65 ζ6 ζ32 linear of order 6 ρ12 1 1 1 1 1 1 ζ32 ζ3 1 1 1 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 linear of order 3 ρ13 3 3 -3 -3 3 1 0 0 -1 1 -1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ14 3 3 -3 3 -3 -1 0 0 -1 1 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ15 3 3 3 -3 -3 1 0 0 -1 -1 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ16 3 3 3 3 3 -1 0 0 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ17 12 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C24.6A4
On 16 points - transitive group 16T420
Generators in S16
(2 4)(5 12)(6 11)(7 10)(8 9)(14 16)
(2 14)(4 16)(5 10)(6 8)(7 12)(9 11)
(1 13)(2 14)(3 15)(4 16)(5 12)(6 9)(7 10)(8 11)
(1 15)(2 16)(3 13)(4 14)(5 10)(6 11)(7 12)(8 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 12 13 5)(2 9 14 6)(3 10 15 7)(4 11 16 8)
(2 10 6)(3 13 15)(4 7 11)(5 8 14)(9 16 12)

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

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

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

G:=TransitiveGroup(16,420);

On 24 points - transitive group 24T368
Generators in S24
(1 3)(2 4)(10 12)(13 15)(17 19)(22 24)
(2 4)(5 7)(9 11)(17 19)(21 23)(22 24)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(17 19)(18 20)
(1 3)(2 4)(5 7)(6 8)(13 15)(14 16)(21 23)(22 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)
(1 7 3 5)(2 8 4 6)(9 17 11 19)(10 18 12 20)(13 24)(14 21)(15 22)(16 23)
(1 15 12)(2 24 19)(3 13 10)(4 22 17)(5 21 11)(6 16 18)(7 23 9)(8 14 20)

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

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

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

G:=TransitiveGroup(24,368);

On 24 points - transitive group 24T373
Generators in S24
(1 6)(2 5)(3 7)(4 8)(9 10)(11 12)(13 17)(14 20)(15 19)(16 18)(21 24)(22 23)
(1 3)(6 7)(13 15)(17 19)(21 23)(22 24)
(1 3)(2 4)(5 8)(6 7)(13 15)(14 16)(17 19)(18 20)
(1 3)(2 4)(5 8)(6 7)(9 11)(10 12)(21 23)(22 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)
(1 8 3 5)(2 7 4 6)(9 22)(10 23)(11 24)(12 21)(13 17 15 19)(14 18 16 20)
(1 22 19)(2 11 16)(3 24 17)(4 9 14)(5 12 18)(6 23 15)(7 21 13)(8 10 20)

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

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

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

G:=TransitiveGroup(24,373);

On 24 points - transitive group 24T377
Generators in S24
(3 4)(7 8)(9 11)(13 15)(18 20)(21 23)
(1 5)(2 6)(3 8)(4 7)(9 23)(10 22)(11 21)(12 24)(13 20)(14 17)(15 18)(16 19)
(1 2)(3 4)(5 6)(7 8)(9 11)(10 12)(21 23)(22 24)
(1 2)(3 4)(5 6)(7 8)(13 15)(14 16)(17 19)(18 20)
(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4 2 3)(5 8 6 7)(9 12 11 10)(13 15)(14 16)(21 22 23 24)
(1 14 22)(2 16 24)(3 15 21)(4 13 23)(5 17 10)(6 19 12)(7 20 9)(8 18 11)

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

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

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

G:=TransitiveGroup(24,377);

On 24 points - transitive group 24T380
Generators in S24
(1 3)(2 4)(5 8)(6 7)(9 10)(11 12)(13 14)(15 16)(17 18)(19 20)(21 22)(23 24)
(1 8)(2 7)(3 5)(4 6)(9 21)(10 22)(11 23)(12 24)(13 20)(14 19)(15 18)(16 17)
(1 2)(3 4)(5 6)(7 8)(13 15)(14 16)(17 19)(18 20)
(1 2)(3 4)(5 6)(7 8)(9 11)(10 12)(21 23)(22 24)
(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4 2 3)(5 7 6 8)(9 11)(10 12)(13 14 15 16)(17 20 19 18)
(1 9 16)(2 11 14)(3 10 15)(4 12 13)(5 22 18)(6 24 20)(7 23 19)(8 21 17)

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

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

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

G:=TransitiveGroup(24,380);

On 24 points - transitive group 24T382
Generators in S24
(1 6)(2 5)(3 4)(7 8)(9 17)(10 20)(11 19)(12 18)(13 23)(14 22)(15 21)(16 24)
(1 7)(2 4)(3 5)(6 8)(9 20)(10 17)(11 18)(12 19)(13 16)(14 15)(21 22)(23 24)
(1 5)(2 6)(3 7)(4 8)(13 15)(14 16)(21 23)(22 24)
(1 5)(2 6)(3 7)(4 8)(9 11)(10 12)(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)
(1 7 5 3)(2 8 6 4)(9 17)(10 18)(11 19)(12 20)(13 22 15 24)(14 23 16 21)
(1 17 21)(2 11 13)(3 12 24)(4 18 16)(5 19 23)(6 9 15)(7 10 22)(8 20 14)

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

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

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

G:=TransitiveGroup(24,382);

Matrix representation of C24.6A4 in GL12(ℤ)

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

G:=sub<GL(12,Integers())| [1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,1,0],[0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0],[0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1],[0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0],[0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0],[1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0] >;

C24.6A4 in GAP, Magma, Sage, TeX

C_2^4._6A_4
% in TeX

G:=Group("C2^4.6A4");
// GroupNames label

G:=SmallGroup(192,1008);
// by ID

G=gap.SmallGroup(192,1008);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-2,2,4371,850,185,360,2524,2111,1173,102,1027,1784]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=g^3=1,e^2=d*c=g*c*g^-1=c*d,f^2=g*d*g^-1=c,a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,e*a*e^-1=a*c*d,a*g=g*a,e*b*e^-1=b*c=c*b,f*b*f^-1=b*d=d*b,b*g=g*b,c*e=e*c,c*f=f*c,g*f*g^-1=d*e=e*d,d*f=f*d,e*f=f*e,g*e*g^-1=c*e*f>;
// generators/relations

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