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## G = C42⋊2A4order 192 = 26·3

### The semidirect product of C42 and A4 acting via A4/C22=C3

Aliases: C422A4, C24.10A4, C22⋊(C42⋊C3), (C22×C42)⋊3C3, C22.1(C22⋊A4), SmallGroup(192,1020)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C42 — C42⋊2A4
 Chief series C1 — C22 — C24 — C22×C42 — C42⋊2A4
 Lower central C22×C42 — C42⋊2A4
 Upper central C1

Generators and relations for C422A4
G = < a,b,c,d,e | a4=b4=c2=d2=e3=1, ab=ba, ac=ca, ad=da, eae-1=ab-1, bc=cb, bd=db, ebe-1=a-1b2, ece-1=cd=dc, ede-1=c >

Subgroups: 414 in 103 conjugacy classes, 13 normal (5 characteristic)
C1, C2, C3, C4, C22, C22, C22, C2×C4, C23, A4, C42, C42, C22×C4, C24, C2×C42, C23×C4, C42⋊C3, C22⋊A4, C22×C42, C422A4
Quotients: C1, C3, A4, C42⋊C3, C22⋊A4, C422A4

Character table of C422A4

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P size 1 3 3 3 3 3 64 64 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 ρ1 1 1 1 1 1 1 1 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 ζ32 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ3 1 1 1 1 1 1 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ4 3 -1 -1 -1 -1 3 0 0 -1 -1 3 3 3 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from A4 ρ5 3 -1 -1 -1 -1 3 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 3 3 3 -1 -1 orthogonal lifted from A4 ρ6 3 -1 -1 -1 -1 3 0 0 -1 -1 -1 -1 -1 -1 3 3 3 3 -1 -1 -1 -1 -1 -1 orthogonal lifted from A4 ρ7 3 3 3 3 3 3 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from A4 ρ8 3 -1 -1 -1 -1 3 0 0 3 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 3 3 orthogonal lifted from A4 ρ9 3 3 -1 -1 -1 -1 0 0 -1+2i 1 1 -1+2i -1-2i 1 -1-2i 1 1 -1+2i -1+2i 1 1 -1-2i 1 -1-2i complex lifted from C42⋊C3 ρ10 3 -1 3 -1 -1 -1 0 0 1 -1-2i -1+2i 1 1 -1-2i -1+2i 1 1 -1-2i -1+2i 1 1 -1-2i -1+2i 1 complex lifted from C42⋊C3 ρ11 3 -1 -1 -1 3 -1 0 0 -1-2i 1 -1-2i 1 1 -1+2i 1 -1-2i -1+2i 1 -1+2i 1 1 -1-2i 1 -1+2i complex lifted from C42⋊C3 ρ12 3 3 -1 -1 -1 -1 0 0 1 -1-2i -1-2i 1 1 -1+2i 1 -1+2i -1-2i 1 1 -1-2i -1+2i 1 -1+2i 1 complex lifted from C42⋊C3 ρ13 3 -1 3 -1 -1 -1 0 0 -1+2i 1 1 -1-2i -1+2i 1 1 -1-2i -1+2i 1 1 -1-2i -1+2i 1 1 -1-2i complex lifted from C42⋊C3 ρ14 3 -1 -1 3 -1 -1 0 0 1 -1-2i 1 -1+2i -1-2i 1 1 -1-2i -1+2i 1 -1-2i 1 1 -1+2i -1+2i 1 complex lifted from C42⋊C3 ρ15 3 3 -1 -1 -1 -1 0 0 1 -1+2i -1+2i 1 1 -1-2i 1 -1-2i -1+2i 1 1 -1+2i -1-2i 1 -1-2i 1 complex lifted from C42⋊C3 ρ16 3 -1 -1 -1 3 -1 0 0 -1+2i 1 -1+2i 1 1 -1-2i 1 -1+2i -1-2i 1 -1-2i 1 1 -1+2i 1 -1-2i complex lifted from C42⋊C3 ρ17 3 3 -1 -1 -1 -1 0 0 -1-2i 1 1 -1-2i -1+2i 1 -1+2i 1 1 -1-2i -1-2i 1 1 -1+2i 1 -1+2i complex lifted from C42⋊C3 ρ18 3 -1 -1 3 -1 -1 0 0 -1-2i 1 -1+2i 1 1 -1-2i -1-2i 1 1 -1+2i 1 -1-2i -1+2i 1 1 -1+2i complex lifted from C42⋊C3 ρ19 3 -1 -1 3 -1 -1 0 0 -1+2i 1 -1-2i 1 1 -1+2i -1+2i 1 1 -1-2i 1 -1+2i -1-2i 1 1 -1-2i complex lifted from C42⋊C3 ρ20 3 -1 -1 -1 3 -1 0 0 1 -1+2i 1 -1+2i -1-2i 1 -1+2i 1 1 -1-2i 1 -1-2i -1+2i 1 -1-2i 1 complex lifted from C42⋊C3 ρ21 3 -1 3 -1 -1 -1 0 0 -1-2i 1 1 -1+2i -1-2i 1 1 -1+2i -1-2i 1 1 -1+2i -1-2i 1 1 -1+2i complex lifted from C42⋊C3 ρ22 3 -1 -1 3 -1 -1 0 0 1 -1+2i 1 -1-2i -1+2i 1 1 -1+2i -1-2i 1 -1+2i 1 1 -1-2i -1-2i 1 complex lifted from C42⋊C3 ρ23 3 -1 3 -1 -1 -1 0 0 1 -1+2i -1-2i 1 1 -1+2i -1-2i 1 1 -1+2i -1-2i 1 1 -1+2i -1-2i 1 complex lifted from C42⋊C3 ρ24 3 -1 -1 -1 3 -1 0 0 1 -1-2i 1 -1-2i -1+2i 1 -1-2i 1 1 -1+2i 1 -1+2i -1-2i 1 -1+2i 1 complex lifted from C42⋊C3

Permutation representations of C422A4
On 24 points - transitive group 24T388
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 5 2 6)(3 7 4 8)(9 12 11 10)(17 20 19 18)
(1 4)(2 3)(5 8)(6 7)(9 18)(10 19)(11 20)(12 17)(13 15)(14 16)(21 23)(22 24)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 23)(14 24)(15 21)(16 22)(17 19)(18 20)
(1 24 20)(2 22 18)(3 16 11)(4 14 9)(5 21 17)(6 23 19)(7 13 12)(8 15 10)```

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

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

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

`G:=TransitiveGroup(24,388);`

Matrix representation of C422A4 in GL6(𝔽13)

 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 5 0 0 0 0 0 0 8
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0

`G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0] >;`

C422A4 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_2A_4`
`% in TeX`

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

`G:=SmallGroup(192,1020);`
`// by ID`

`G=gap.SmallGroup(192,1020);`
`# by ID`

`G:=PCGroup([7,-3,-2,2,-2,2,-2,2,85,680,2207,184,675,1264,4037,7062]);`
`// Polycyclic`

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

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