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## G = C24⋊A4order 192 = 26·3

### 3rd semidirect product of C24 and A4 acting faithfully

Aliases: C243A4, C24⋊(C2×C6), C22≀C2⋊C6, C23.6(C2×A4), C24⋊C61C2, C22⋊A41C22, C24⋊C221C3, C22.6(C22×A4), SmallGroup(192,1009)

Series: Derived Chief Lower central Upper central

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

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

Subgroups: 374 in 70 conjugacy classes, 16 normal (7 characteristic)
C1, C2 [×5], C3, C4 [×3], C22, C22 [×10], C6 [×3], C2×C4 [×3], D4 [×3], Q8, C23 [×3], C23 [×3], A4 [×2], C2×C6, C42, C22⋊C4 [×6], C2×D4 [×3], C2×Q8, C24 [×2], C2×A4 [×3], C22≀C2 [×3], C22≀C2, C4.4D4 [×3], C22×A4, C22⋊A4, C24⋊C22, C24⋊C6 [×3], C24⋊A4
Quotients: C1, C2 [×3], C3, C22, C6 [×3], A4, C2×C6, C2×A4 [×3], C22×A4, C24⋊A4

Character table of C24⋊A4

 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 ζ32 ζ3 ζ32 ζ3 ζ3 linear of order 3 ρ7 1 1 -1 1 -1 1 ζ32 ζ3 -1 -1 1 ζ65 ζ3 ζ6 ζ65 ζ6 ζ32 linear of order 6 ρ8 1 1 1 -1 -1 1 ζ3 ζ32 -1 1 -1 ζ6 ζ6 ζ3 ζ32 ζ65 ζ65 linear of order 6 ρ9 1 1 -1 1 -1 1 ζ3 ζ32 -1 -1 1 ζ6 ζ32 ζ65 ζ6 ζ65 ζ3 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 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 linear of order 3 ρ12 1 1 -1 -1 1 1 ζ3 ζ32 1 -1 -1 ζ32 ζ6 ζ65 ζ6 ζ3 ζ65 linear of order 6 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⋊A4
On 16 points - transitive group 16T417
Generators in S16
(1 4)(2 3)(5 15)(6 16)(7 14)(8 11)(9 12)(10 13)
(1 2)(3 4)(5 11)(6 12)(7 13)(8 15)(9 16)(10 14)
(1 5)(2 11)(3 8)(4 15)(6 7)(9 10)(12 13)(14 16)
(1 6)(2 12)(3 9)(4 16)(5 7)(8 10)(11 13)(14 15)
(1 5)(3 10)(4 16)(6 7)(8 9)(14 15)
(1 6)(3 8)(4 14)(5 7)(9 10)(15 16)
(5 6 7)(8 9 10)(11 12 13)(14 15 16)

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

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

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

G:=TransitiveGroup(16,417);

On 16 points - transitive group 16T419
Generators in S16
(5 15)(6 16)(7 14)(8 12)(9 13)(10 11)
(5 13)(6 11)(7 12)(8 14)(9 15)(10 16)
(1 4)(2 3)(5 13)(6 16)(7 8)(9 15)(10 11)(12 14)
(1 2)(3 4)(5 9)(6 11)(7 14)(8 12)(10 16)(13 15)
(1 5)(2 9)(3 15)(4 13)(6 7)(8 16)(10 12)(11 14)
(1 6)(2 11)(3 10)(4 16)(5 7)(8 13)(9 14)(12 15)
(2 3 4)(5 6 7)(8 9 10)(11 12 13)(14 15 16)

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

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

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

G:=TransitiveGroup(16,419);

On 24 points - transitive group 24T369
Generators in S24
(4 8)(5 9)(6 7)(16 19)(17 20)(18 21)
(10 15)(11 13)(12 14)(16 19)(17 20)(18 21)
(2 24)(3 22)(5 9)(6 7)(10 15)(11 13)(17 20)(18 21)
(1 23)(3 22)(4 8)(6 7)(11 13)(12 14)(16 19)(18 21)
(1 8)(2 20)(3 11)(4 23)(5 15)(6 18)(7 21)(9 10)(12 19)(13 22)(14 16)(17 24)
(1 12)(2 9)(3 21)(4 16)(5 24)(6 13)(7 11)(8 19)(10 20)(14 23)(15 17)(18 22)
(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)| (4,8)(5,9)(6,7)(16,19)(17,20)(18,21), (10,15)(11,13)(12,14)(16,19)(17,20)(18,21), (2,24)(3,22)(5,9)(6,7)(10,15)(11,13)(17,20)(18,21), (1,23)(3,22)(4,8)(6,7)(11,13)(12,14)(16,19)(18,21), (1,8)(2,20)(3,11)(4,23)(5,15)(6,18)(7,21)(9,10)(12,19)(13,22)(14,16)(17,24), (1,12)(2,9)(3,21)(4,16)(5,24)(6,13)(7,11)(8,19)(10,20)(14,23)(15,17)(18,22), (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( (4,8)(5,9)(6,7)(16,19)(17,20)(18,21), (10,15)(11,13)(12,14)(16,19)(17,20)(18,21), (2,24)(3,22)(5,9)(6,7)(10,15)(11,13)(17,20)(18,21), (1,23)(3,22)(4,8)(6,7)(11,13)(12,14)(16,19)(18,21), (1,8)(2,20)(3,11)(4,23)(5,15)(6,18)(7,21)(9,10)(12,19)(13,22)(14,16)(17,24), (1,12)(2,9)(3,21)(4,16)(5,24)(6,13)(7,11)(8,19)(10,20)(14,23)(15,17)(18,22), (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([(4,8),(5,9),(6,7),(16,19),(17,20),(18,21)], [(10,15),(11,13),(12,14),(16,19),(17,20),(18,21)], [(2,24),(3,22),(5,9),(6,7),(10,15),(11,13),(17,20),(18,21)], [(1,23),(3,22),(4,8),(6,7),(11,13),(12,14),(16,19),(18,21)], [(1,8),(2,20),(3,11),(4,23),(5,15),(6,18),(7,21),(9,10),(12,19),(13,22),(14,16),(17,24)], [(1,12),(2,9),(3,21),(4,16),(5,24),(6,13),(7,11),(8,19),(10,20),(14,23),(15,17),(18,22)], [(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(24,369);

On 24 points - transitive group 24T371
Generators in S24
(1 16)(2 17)(3 18)(4 13)(5 14)(6 15)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
(1 10)(2 11)(3 12)(4 19)(5 20)(6 21)(7 13)(8 14)(9 15)(16 22)(17 23)(18 24)
(2 8)(3 9)(5 23)(6 24)(11 14)(12 15)(17 20)(18 21)
(1 7)(3 9)(4 22)(6 24)(10 13)(12 15)(16 19)(18 21)
(4 22)(6 24)(11 14)(12 15)(16 19)(17 20)
(4 22)(5 23)(10 13)(12 15)(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)| (1,16)(2,17)(3,18)(4,13)(5,14)(6,15)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (1,10)(2,11)(3,12)(4,19)(5,20)(6,21)(7,13)(8,14)(9,15)(16,22)(17,23)(18,24), (2,8)(3,9)(5,23)(6,24)(11,14)(12,15)(17,20)(18,21), (1,7)(3,9)(4,22)(6,24)(10,13)(12,15)(16,19)(18,21), (4,22)(6,24)(11,14)(12,15)(16,19)(17,20), (4,22)(5,23)(10,13)(12,15)(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( (1,16)(2,17)(3,18)(4,13)(5,14)(6,15)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (1,10)(2,11)(3,12)(4,19)(5,20)(6,21)(7,13)(8,14)(9,15)(16,22)(17,23)(18,24), (2,8)(3,9)(5,23)(6,24)(11,14)(12,15)(17,20)(18,21), (1,7)(3,9)(4,22)(6,24)(10,13)(12,15)(16,19)(18,21), (4,22)(6,24)(11,14)(12,15)(16,19)(17,20), (4,22)(5,23)(10,13)(12,15)(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([(1,16),(2,17),(3,18),(4,13),(5,14),(6,15),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)], [(1,10),(2,11),(3,12),(4,19),(5,20),(6,21),(7,13),(8,14),(9,15),(16,22),(17,23),(18,24)], [(2,8),(3,9),(5,23),(6,24),(11,14),(12,15),(17,20),(18,21)], [(1,7),(3,9),(4,22),(6,24),(10,13),(12,15),(16,19),(18,21)], [(4,22),(6,24),(11,14),(12,15),(16,19),(17,20)], [(4,22),(5,23),(10,13),(12,15),(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(24,371);

On 24 points - transitive group 24T376
Generators in S24
(7 19)(8 20)(9 21)(13 16)(14 17)(15 18)
(1 4)(2 5)(3 6)(7 17)(8 18)(9 16)(10 22)(11 23)(12 24)(13 21)(14 19)(15 20)
(1 10)(3 12)(4 22)(6 24)(7 19)(8 20)(14 17)(15 18)
(1 10)(2 11)(4 22)(5 23)(8 20)(9 21)(13 16)(15 18)
(2 13)(3 14)(4 22)(5 21)(6 7)(8 20)(9 23)(11 16)(12 17)(19 24)
(1 15)(3 14)(4 8)(5 23)(6 19)(7 24)(9 21)(10 18)(12 17)(20 22)
(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)| (7,19)(8,20)(9,21)(13,16)(14,17)(15,18), (1,4)(2,5)(3,6)(7,17)(8,18)(9,16)(10,22)(11,23)(12,24)(13,21)(14,19)(15,20), (1,10)(3,12)(4,22)(6,24)(7,19)(8,20)(14,17)(15,18), (1,10)(2,11)(4,22)(5,23)(8,20)(9,21)(13,16)(15,18), (2,13)(3,14)(4,22)(5,21)(6,7)(8,20)(9,23)(11,16)(12,17)(19,24), (1,15)(3,14)(4,8)(5,23)(6,19)(7,24)(9,21)(10,18)(12,17)(20,22), (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( (7,19)(8,20)(9,21)(13,16)(14,17)(15,18), (1,4)(2,5)(3,6)(7,17)(8,18)(9,16)(10,22)(11,23)(12,24)(13,21)(14,19)(15,20), (1,10)(3,12)(4,22)(6,24)(7,19)(8,20)(14,17)(15,18), (1,10)(2,11)(4,22)(5,23)(8,20)(9,21)(13,16)(15,18), (2,13)(3,14)(4,22)(5,21)(6,7)(8,20)(9,23)(11,16)(12,17)(19,24), (1,15)(3,14)(4,8)(5,23)(6,19)(7,24)(9,21)(10,18)(12,17)(20,22), (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([(7,19),(8,20),(9,21),(13,16),(14,17),(15,18)], [(1,4),(2,5),(3,6),(7,17),(8,18),(9,16),(10,22),(11,23),(12,24),(13,21),(14,19),(15,20)], [(1,10),(3,12),(4,22),(6,24),(7,19),(8,20),(14,17),(15,18)], [(1,10),(2,11),(4,22),(5,23),(8,20),(9,21),(13,16),(15,18)], [(2,13),(3,14),(4,22),(5,21),(6,7),(8,20),(9,23),(11,16),(12,17),(19,24)], [(1,15),(3,14),(4,8),(5,23),(6,19),(7,24),(9,21),(10,18),(12,17),(20,22)], [(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(24,376);

On 24 points - transitive group 24T381
Generators in S24
(1 12)(2 10)(3 11)(4 19)(5 20)(6 21)(7 18)(8 16)(9 17)(13 22)(14 23)(15 24)
(1 8)(2 9)(3 7)(4 23)(5 24)(6 22)(10 17)(11 18)(12 16)(13 21)(14 19)(15 20)
(2 24)(3 22)(5 9)(6 7)(10 15)(11 13)(17 20)(18 21)
(1 23)(3 22)(4 8)(6 7)(11 13)(12 14)(16 19)(18 21)
(1 8)(2 20)(3 11)(4 23)(5 15)(6 18)(7 21)(9 10)(12 19)(13 22)(14 16)(17 24)
(1 12)(2 9)(3 21)(4 16)(5 24)(6 13)(7 11)(8 19)(10 20)(14 23)(15 17)(18 22)
(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)| (1,12)(2,10)(3,11)(4,19)(5,20)(6,21)(7,18)(8,16)(9,17)(13,22)(14,23)(15,24), (1,8)(2,9)(3,7)(4,23)(5,24)(6,22)(10,17)(11,18)(12,16)(13,21)(14,19)(15,20), (2,24)(3,22)(5,9)(6,7)(10,15)(11,13)(17,20)(18,21), (1,23)(3,22)(4,8)(6,7)(11,13)(12,14)(16,19)(18,21), (1,8)(2,20)(3,11)(4,23)(5,15)(6,18)(7,21)(9,10)(12,19)(13,22)(14,16)(17,24), (1,12)(2,9)(3,21)(4,16)(5,24)(6,13)(7,11)(8,19)(10,20)(14,23)(15,17)(18,22), (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( (1,12)(2,10)(3,11)(4,19)(5,20)(6,21)(7,18)(8,16)(9,17)(13,22)(14,23)(15,24), (1,8)(2,9)(3,7)(4,23)(5,24)(6,22)(10,17)(11,18)(12,16)(13,21)(14,19)(15,20), (2,24)(3,22)(5,9)(6,7)(10,15)(11,13)(17,20)(18,21), (1,23)(3,22)(4,8)(6,7)(11,13)(12,14)(16,19)(18,21), (1,8)(2,20)(3,11)(4,23)(5,15)(6,18)(7,21)(9,10)(12,19)(13,22)(14,16)(17,24), (1,12)(2,9)(3,21)(4,16)(5,24)(6,13)(7,11)(8,19)(10,20)(14,23)(15,17)(18,22), (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([(1,12),(2,10),(3,11),(4,19),(5,20),(6,21),(7,18),(8,16),(9,17),(13,22),(14,23),(15,24)], [(1,8),(2,9),(3,7),(4,23),(5,24),(6,22),(10,17),(11,18),(12,16),(13,21),(14,19),(15,20)], [(2,24),(3,22),(5,9),(6,7),(10,15),(11,13),(17,20),(18,21)], [(1,23),(3,22),(4,8),(6,7),(11,13),(12,14),(16,19),(18,21)], [(1,8),(2,20),(3,11),(4,23),(5,15),(6,18),(7,21),(9,10),(12,19),(13,22),(14,16),(17,24)], [(1,12),(2,9),(3,21),(4,16),(5,24),(6,13),(7,11),(8,19),(10,20),(14,23),(15,17),(18,22)], [(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(24,381);

Matrix representation of C24⋊A4 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 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 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 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 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 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 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 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 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 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 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 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

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],[0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0],[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,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-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] >;

C24⋊A4 in GAP, Magma, Sage, TeX

C_2^4\rtimes A_4
% in TeX

G:=Group("C2^4:A4");
// GroupNames label

G:=SmallGroup(192,1009);
// by ID

G=gap.SmallGroup(192,1009);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-2,2,4371,850,185,2524,2111,333,1027,1784]);
// Polycyclic

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

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