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## G = C42⋊2C12order 192 = 26·3

### 2nd semidirect product of C42 and C12 acting via C12/C2=C6

Aliases: C422C12, C429C4⋊C3, C42⋊C34C4, C22.4(C4×A4), (C2×C42).2C6, (C22×C4).2A4, C23.13(C2×A4), C2.1(C23.A4), (C2×C42⋊C3).2C2, SmallGroup(192,193)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C42⋊2C12
 Chief series C1 — C22 — C42 — C2×C42 — C2×C42⋊C3 — C42⋊2C12
 Lower central C42 — C42⋊2C12
 Upper central C1 — C2

Generators and relations for C422C12
G = < a,b,c | a4=b4=c12=1, ab=ba, cac-1=a-1b, cbc-1=ab2 >

Character table of C422C12

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 12A 12B 12C 12D size 1 1 3 3 16 16 4 4 6 6 6 6 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 1 1 1 trivial ρ2 1 1 1 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 ζ3 ζ32 -1 -1 1 1 1 1 -1 -1 ζ32 ζ3 ζ65 ζ6 ζ65 ζ6 linear of order 6 ρ4 1 1 1 1 ζ3 ζ32 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ5 1 1 1 1 ζ32 ζ3 -1 -1 1 1 1 1 -1 -1 ζ3 ζ32 ζ6 ζ65 ζ6 ζ65 linear of order 6 ρ6 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ7 1 -1 1 -1 1 1 -i i -1 1 -1 1 i -i -1 -1 -i -i i i linear of order 4 ρ8 1 -1 1 -1 1 1 i -i -1 1 -1 1 -i i -1 -1 i i -i -i linear of order 4 ρ9 1 -1 1 -1 ζ32 ζ3 -i i -1 1 -1 1 i -i ζ65 ζ6 ζ43ζ32 ζ43ζ3 ζ4ζ32 ζ4ζ3 linear of order 12 ρ10 1 -1 1 -1 ζ3 ζ32 -i i -1 1 -1 1 i -i ζ6 ζ65 ζ43ζ3 ζ43ζ32 ζ4ζ3 ζ4ζ32 linear of order 12 ρ11 1 -1 1 -1 ζ32 ζ3 i -i -1 1 -1 1 -i i ζ65 ζ6 ζ4ζ32 ζ4ζ3 ζ43ζ32 ζ43ζ3 linear of order 12 ρ12 1 -1 1 -1 ζ3 ζ32 i -i -1 1 -1 1 -i i ζ6 ζ65 ζ4ζ3 ζ4ζ32 ζ43ζ3 ζ43ζ32 linear of order 12 ρ13 3 3 3 3 0 0 3 3 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ14 3 3 3 3 0 0 -3 -3 -1 -1 -1 -1 1 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ15 3 -3 3 -3 0 0 3i -3i 1 -1 1 -1 i -i 0 0 0 0 0 0 complex lifted from C4×A4 ρ16 3 -3 3 -3 0 0 -3i 3i 1 -1 1 -1 -i i 0 0 0 0 0 0 complex lifted from C4×A4 ρ17 6 6 -2 -2 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from C23.A4 ρ18 6 6 -2 -2 0 0 0 0 2 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from C23.A4 ρ19 6 -6 -2 2 0 0 0 0 2 2 -2 -2 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ20 6 -6 -2 2 0 0 0 0 -2 -2 2 2 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Permutation representations of C422C12
On 24 points - transitive group 24T311
Generators in S24
```(2 16 8 22)(3 17 9 23)(5 13 11 19)(6 14 12 20)
(1 15 7 21)(2 8)(3 17 9 23)(4 24 10 18)(5 11)(6 14 12 20)(13 19)(16 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)| (2,16,8,22)(3,17,9,23)(5,13,11,19)(6,14,12,20), (1,15,7,21)(2,8)(3,17,9,23)(4,24,10,18)(5,11)(6,14,12,20)(13,19)(16,22), (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( (2,16,8,22)(3,17,9,23)(5,13,11,19)(6,14,12,20), (1,15,7,21)(2,8)(3,17,9,23)(4,24,10,18)(5,11)(6,14,12,20)(13,19)(16,22), (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([[(2,16,8,22),(3,17,9,23),(5,13,11,19),(6,14,12,20)], [(1,15,7,21),(2,8),(3,17,9,23),(4,24,10,18),(5,11),(6,14,12,20),(13,19),(16,22)], [(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(24,311);`

Matrix representation of C422C12 in GL6(𝔽3)

 0 2 1 0 1 0 1 0 1 0 2 0 2 1 2 0 2 2 2 1 2 0 0 0 1 1 1 1 2 0 1 1 0 0 2 0
,
 0 0 2 0 2 2 2 0 2 2 0 1 2 0 2 2 0 2 2 0 0 1 0 1 2 0 0 2 0 0 0 2 0 1 0 2
,
 1 1 1 2 0 2 1 2 1 1 2 2 2 2 1 1 1 2 2 0 0 2 0 2 1 2 1 2 0 1 0 2 2 2 2 0

`G:=sub<GL(6,GF(3))| [0,1,2,2,1,1,2,0,1,1,1,1,1,1,2,2,1,0,0,0,0,0,1,0,1,2,2,0,2,2,0,0,2,0,0,0],[0,2,2,2,2,0,0,0,0,0,0,2,2,2,2,0,0,0,0,2,2,1,2,1,2,0,0,0,0,0,2,1,2,1,0,2],[1,1,2,2,1,0,1,2,2,0,2,2,1,1,1,0,1,2,2,1,1,2,2,2,0,2,1,0,0,2,2,2,2,2,1,0] >;`

C422C12 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_2C_{12}`
`% in TeX`

`G:=Group("C4^2:2C12");`
`// GroupNames label`

`G:=SmallGroup(192,193);`
`// by ID`

`G=gap.SmallGroup(192,193);`
`# by ID`

`G:=PCGroup([7,-2,-3,-2,-2,2,-2,2,42,4371,346,360,2524,2321,102,2028,3541]);`
`// Polycyclic`

`G:=Group<a,b,c|a^4=b^4=c^12=1,a*b=b*a,c*a*c^-1=a^-1*b,c*b*c^-1=a*b^2>;`
`// generators/relations`

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