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## G = C42⋊C12order 192 = 26·3

### 1st semidirect product of C42 and C12 acting via C12/C2=C6

Aliases: C421C12, C425C4⋊C3, C42⋊C32C4, C22.3(C4×A4), (C2×C42).1C6, (C22×C4).1A4, C23.12(C2×A4), C2.1(C42⋊C6), (C2×C42⋊C3).1C2, SmallGroup(192,192)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C42⋊C12
G = < a,b,c | a4=b4=c12=1, ab=ba, cac-1=a-1b-1, cbc-1=a-1 >

Character table of C42⋊C12

 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 2i -2i -2i 2i 0 0 0 0 0 0 0 0 complex lifted from C42⋊C6 ρ18 6 -6 -2 2 0 0 0 0 2i 2i -2i -2i 0 0 0 0 0 0 0 0 complex faithful ρ19 6 -6 -2 2 0 0 0 0 -2i -2i 2i 2i 0 0 0 0 0 0 0 0 complex faithful ρ20 6 6 -2 -2 0 0 0 0 -2i 2i 2i -2i 0 0 0 0 0 0 0 0 complex lifted from C42⋊C6

Permutation representations of C42⋊C12
On 24 points - transitive group 24T299
Generators in S24
```(1 4 7 10)(2 22 8 16)(3 14)(5 19 11 13)(6 23)(9 20)(12 17)(15 24 21 18)
(1 18)(2 11 8 5)(3 17 9 23)(4 15)(6 14 12 20)(7 24)(10 21)(13 16 19 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)| (1,4,7,10)(2,22,8,16)(3,14)(5,19,11,13)(6,23)(9,20)(12,17)(15,24,21,18), (1,18)(2,11,8,5)(3,17,9,23)(4,15)(6,14,12,20)(7,24)(10,21)(13,16,19,22), (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( (1,4,7,10)(2,22,8,16)(3,14)(5,19,11,13)(6,23)(9,20)(12,17)(15,24,21,18), (1,18)(2,11,8,5)(3,17,9,23)(4,15)(6,14,12,20)(7,24)(10,21)(13,16,19,22), (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([[(1,4,7,10),(2,22,8,16),(3,14),(5,19,11,13),(6,23),(9,20),(12,17),(15,24,21,18)], [(1,18),(2,11,8,5),(3,17,9,23),(4,15),(6,14,12,20),(7,24),(10,21),(13,16,19,22)], [(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(24,299);`

On 24 points - transitive group 24T305
Generators in S24
```(1 10)(2 22 8 16)(3 14 9 20)(4 7)(5 13 11 19)(6 23 12 17)(15 24)(18 21)
(1 24 7 18)(2 11)(3 17 9 23)(4 21 10 15)(5 8)(6 20 12 14)(13 16)(19 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)| (1,10)(2,22,8,16)(3,14,9,20)(4,7)(5,13,11,19)(6,23,12,17)(15,24)(18,21), (1,24,7,18)(2,11)(3,17,9,23)(4,21,10,15)(5,8)(6,20,12,14)(13,16)(19,22), (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( (1,10)(2,22,8,16)(3,14,9,20)(4,7)(5,13,11,19)(6,23,12,17)(15,24)(18,21), (1,24,7,18)(2,11)(3,17,9,23)(4,21,10,15)(5,8)(6,20,12,14)(13,16)(19,22), (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([[(1,10),(2,22,8,16),(3,14,9,20),(4,7),(5,13,11,19),(6,23,12,17),(15,24),(18,21)], [(1,24,7,18),(2,11),(3,17,9,23),(4,21,10,15),(5,8),(6,20,12,14),(13,16),(19,22)], [(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(24,305);`

On 24 points - transitive group 24T308
Generators in S24
```(2 13 8 19)(3 20 9 14)(4 10)(5 16 11 22)(6 17 12 23)(15 21)
(1 18 7 24)(3 20 9 14)(4 15 10 21)(5 11)(6 23 12 17)(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)| (2,13,8,19)(3,20,9,14)(4,10)(5,16,11,22)(6,17,12,23)(15,21), (1,18,7,24)(3,20,9,14)(4,15,10,21)(5,11)(6,23,12,17)(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( (2,13,8,19)(3,20,9,14)(4,10)(5,16,11,22)(6,17,12,23)(15,21), (1,18,7,24)(3,20,9,14)(4,15,10,21)(5,11)(6,23,12,17)(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([[(2,13,8,19),(3,20,9,14),(4,10),(5,16,11,22),(6,17,12,23),(15,21)], [(1,18,7,24),(3,20,9,14),(4,15,10,21),(5,11),(6,23,12,17),(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(24,308);`

Matrix representation of C42⋊C12 in GL6(𝔽13)

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

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

C42⋊C12 in GAP, Magma, Sage, TeX

`C_4^2\rtimes C_{12}`
`% in TeX`

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

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

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

`G:=PCGroup([7,-2,-3,-2,-2,2,-2,2,42,1683,346,360,4204,2321,102,2028,3541]);`
`// Polycyclic`

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

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