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## G = C42⋊A4order 192 = 26·3

### The semidirect product of C42 and A4 acting faithfully

Aliases: C42⋊A4, C244A4, C24⋊C222C3, C22.2(C22⋊A4), SmallGroup(192,1023)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C24⋊C22 — C42⋊A4
 Chief series C1 — C22 — C24 — C24⋊C22 — C42⋊A4
 Lower central C24⋊C22 — C42⋊A4
 Upper central C1

Generators and relations for C42⋊A4
G = < a,b,c,d,e | a4=b4=c2=d2=e3=1, ab=ba, cac=ab2, dad=a-1, eae-1=a-1b-1, cbc=a2b, dbd=a2b-1, ebe-1=a, ece-1=cd=dc, ede-1=c >

Subgroups: 482 in 70 conjugacy classes, 9 normal (4 characteristic)
C1, C2 [×3], C3, C4 [×3], C22, C22 [×14], C2×C4 [×3], D4 [×3], Q8, C23 [×4], A4 [×9], C42 [×3], C22⋊C4 [×6], C2×D4 [×3], C2×Q8, C24 [×2], C22≀C2 [×2], C4.4D4 [×3], C42⋊C3 [×3], C22⋊A4 [×2], C24⋊C22, C42⋊A4
Quotients: C1, C3, A4 [×5], C22⋊A4, C42⋊A4

Character table of C42⋊A4

 class 1 2A 2B 2C 3A 3B 4A 4B 4C size 1 3 12 12 64 64 12 12 12 ρ1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 ζ32 ζ3 1 1 1 linear of order 3 ρ3 1 1 1 1 ζ3 ζ32 1 1 1 linear of order 3 ρ4 3 3 3 -1 0 0 -1 -1 -1 orthogonal lifted from A4 ρ5 3 3 -1 -1 0 0 -1 -1 3 orthogonal lifted from A4 ρ6 3 3 -1 3 0 0 -1 -1 -1 orthogonal lifted from A4 ρ7 3 3 -1 -1 0 0 3 -1 -1 orthogonal lifted from A4 ρ8 3 3 -1 -1 0 0 -1 3 -1 orthogonal lifted from A4 ρ9 12 -4 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C42⋊A4
On 16 points - transitive group 16T440
Generators in S16
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 9 15 5)(2 10 16 6)(3 11 13 7)(4 12 14 8)
(1 3)(2 14)(4 16)(6 10)(8 12)(13 15)
(1 15)(2 14)(3 13)(4 16)(5 7)(9 11)
(2 9 8)(3 15 13)(4 5 10)(6 16 7)(11 12 14)```

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

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

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

`G:=TransitiveGroup(16,440);`

On 24 points - transitive group 24T372
Generators in S24
```(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 6 3)(2 7 5 4)(9 14)(10 15)(11 16)(12 13)(17 21 19 23)(18 22 20 24)
(2 5)(4 7)(9 11)(10 12)(18 20)(21 23)
(1 6)(2 5)(9 11)(13 15)(17 19)(21 23)
(1 10 21)(2 15 20)(3 16 24)(4 11 19)(5 13 18)(6 12 23)(7 9 17)(8 14 22)```

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

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

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

`G:=TransitiveGroup(24,372);`

On 24 points - transitive group 24T391
Generators in S24
```(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 6 3 8)(2 5 4 7)(9 13)(10 14)(11 15)(12 16)(17 22 19 24)(18 23 20 21)
(1 3)(6 8)(9 16)(10 13)(11 14)(12 15)(17 18)(19 20)(21 22)(23 24)
(1 8)(2 7)(3 6)(4 5)(9 11)(14 16)(17 18)(19 20)(21 24)(22 23)
(1 9 24)(2 13 18)(3 11 22)(4 15 20)(5 12 19)(6 16 21)(7 10 17)(8 14 23)```

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

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

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

`G:=TransitiveGroup(24,391);`

On 24 points - transitive group 24T392
Generators in S24
```(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 3 2 4)(5 7 6 8)(9 12 11 10)(13 14 15 16)(21 23)(22 24)
(1 5)(2 6)(3 7)(4 8)(9 10)(11 12)(13 14)(15 16)(17 24)(18 21)(19 22)(20 23)
(1 3)(2 4)(5 7)(6 8)(9 14)(10 13)(11 16)(12 15)(17 23)(18 22)(19 21)(20 24)
(1 19 9)(2 17 11)(3 18 10)(4 20 12)(5 21 13)(6 23 15)(7 22 14)(8 24 16)```

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

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

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

`G:=TransitiveGroup(24,392);`

Matrix representation of C42⋊A4 in GL12(ℤ)

 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 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 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 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 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 -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 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 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 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 -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 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 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
,
 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())| [0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-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,-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,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-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,-1,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0],[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,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,0,0,0,-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,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-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],[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] >;`

C42⋊A4 in GAP, Magma, Sage, TeX

`C_4^2\rtimes A_4`
`% in TeX`

`G:=Group("C4^2:A4");`
`// GroupNames label`

`G:=SmallGroup(192,1023);`
`// by ID`

`G=gap.SmallGroup(192,1023);`
`# by ID`

`G:=PCGroup([7,-3,-2,2,-2,2,-2,2,85,191,675,570,745,1264,1971,718,4037,7062]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^2=e^3=1,a*b=b*a,c*a*c=a*b^2,d*a*d=a^-1,e*a*e^-1=a^-1*b^-1,c*b*c=a^2*b,d*b*d=a^2*b^-1,e*b*e^-1=a,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;`
`// generators/relations`

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