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## G = C122.C3order 432 = 24·33

### 2nd non-split extension by C122 of C3 acting faithfully

Aliases: C62.2A4, C122.2C3, C4223- 1+2, C42⋊C92C3, C32.(C42⋊C3), (C4×C12).4C32, C22.(C32.A4), (C2×C6).9(C3×A4), C3.4(C3×C42⋊C3), SmallGroup(432,102)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4×C12 — C122.C3
 Chief series C1 — C22 — C42 — C4×C12 — C42⋊C9 — C122.C3
 Lower central C42 — C4×C12 — C122.C3
 Upper central C1 — C3 — C32

Generators and relations for C122.C3
G = < a,b,c | a12=b12=1, c3=b4, ab=ba, cac-1=ab-1, cbc-1=a3b10 >

Smallest permutation representation of C122.C3
On 36 points
Generators in S36
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 24 18 23 17 22 16 21 15 20 14 19)(25 36 35 34 33 32 31 30 29 28 27 26)
(1 26 14 9 34 18 5 30 16)(2 35 21 10 31 19 6 27 23)(3 32 17 11 28 15 7 36 13)(4 29 24 12 25 22 8 33 20)```

`G:=sub<Sym(36)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,24,18,23,17,22,16,21,15,20,14,19)(25,36,35,34,33,32,31,30,29,28,27,26), (1,26,14,9,34,18,5,30,16)(2,35,21,10,31,19,6,27,23)(3,32,17,11,28,15,7,36,13)(4,29,24,12,25,22,8,33,20)>;`

`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)(25,26,27,28,29,30,31,32,33,34,35,36), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,24,18,23,17,22,16,21,15,20,14,19)(25,36,35,34,33,32,31,30,29,28,27,26), (1,26,14,9,34,18,5,30,16)(2,35,21,10,31,19,6,27,23)(3,32,17,11,28,15,7,36,13)(4,29,24,12,25,22,8,33,20) );`

`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),(25,26,27,28,29,30,31,32,33,34,35,36)], [(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,24,18,23,17,22,16,21,15,20,14,19),(25,36,35,34,33,32,31,30,29,28,27,26)], [(1,26,14,9,34,18,5,30,16),(2,35,21,10,31,19,6,27,23),(3,32,17,11,28,15,7,36,13),(4,29,24,12,25,22,8,33,20)]])`

56 conjugacy classes

 class 1 2 3A 3B 3C 3D 4A 4B 4C 4D 6A ··· 6H 9A ··· 9F 12A ··· 12AF order 1 2 3 3 3 3 4 4 4 4 6 ··· 6 9 ··· 9 12 ··· 12 size 1 3 1 1 3 3 3 3 3 3 3 ··· 3 48 ··· 48 3 ··· 3

56 irreducible representations

 dim 1 1 1 3 3 3 3 3 3 3 type + + image C1 C3 C3 A4 3- 1+2 C3×A4 C42⋊C3 C32.A4 C3×C42⋊C3 C122.C3 kernel C122.C3 C42⋊C9 C122 C62 C42 C2×C6 C32 C22 C3 C1 # reps 1 6 2 1 2 2 4 6 8 24

Matrix representation of C122.C3 in GL3(𝔽13) generated by

 7 0 0 0 2 0 0 0 1
,
 10 0 0 0 2 0 0 0 2
,
 0 0 9 9 0 0 0 1 0
`G:=sub<GL(3,GF(13))| [7,0,0,0,2,0,0,0,1],[10,0,0,0,2,0,0,0,2],[0,9,0,0,0,1,9,0,0] >;`

C122.C3 in GAP, Magma, Sage, TeX

`C_{12}^2.C_3`
`% in TeX`

`G:=Group("C12^2.C3");`
`// GroupNames label`

`G:=SmallGroup(432,102);`
`// by ID`

`G=gap.SmallGroup(432,102);`
`# by ID`

`G:=PCGroup([7,-3,-3,-3,-2,2,-2,2,63,169,1515,360,10399,102,9077,15882]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^12=1,c^3=b^4,a*b=b*a,c*a*c^-1=a*b^-1,c*b*c^-1=a^3*b^10>;`
`// generators/relations`

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