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## G = C36.C12order 432 = 24·33

### 1st non-split extension by C36 of C12 acting via C12/C2=C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C36.C12
 Chief series C1 — C3 — C9 — C18 — C36 — C4×3- 1+2 — C9⋊C24 — C36.C12
 Lower central C9 — C18 — C36.C12
 Upper central C1 — C4 — C2×C4

Generators and relations for C36.C12
G = < a,b | a36=1, b12=a18, bab-1=a23 >

Subgroups: 158 in 68 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C2×C4, C9, C9, C32, C12, C12, C2×C6, C2×C6, M4(2), C18, C18, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, 3- 1+2, C36, C36, C2×C18, C2×C18, C3×C12, C62, C4.Dic3, C3×M4(2), C2×3- 1+2, C2×3- 1+2, C9⋊C8, C2×C36, C2×C36, C3×C3⋊C8, C6×C12, C4×3- 1+2, C22×3- 1+2, C4.Dic9, C3×C4.Dic3, C9⋊C24, C2×C4×3- 1+2, C36.C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Dic3, C12, D6, C2×C6, M4(2), C3×S3, C2×Dic3, C2×C12, C3×Dic3, S3×C6, C4.Dic3, C3×M4(2), C9⋊C6, C6×Dic3, C9⋊C12, C2×C9⋊C6, C3×C4.Dic3, C2×C9⋊C12, C36.C12

Smallest permutation representation of C36.C12
On 72 points
Generators in S72
```(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)(37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)
(1 47 10 38 19 65 28 56)(2 58 23 37 8 52 29 67 14 46 35 61 20 40 5 55 26 70 11 49 32 64 17 43)(3 69 36 72 33 39 30 42 27 45 24 48 21 51 18 54 15 57 12 60 9 63 6 66)(4 44 13 71 22 62 31 53)(7 41 16 68 25 59 34 50)```

`G:=sub<Sym(72)| (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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,47,10,38,19,65,28,56)(2,58,23,37,8,52,29,67,14,46,35,61,20,40,5,55,26,70,11,49,32,64,17,43)(3,69,36,72,33,39,30,42,27,45,24,48,21,51,18,54,15,57,12,60,9,63,6,66)(4,44,13,71,22,62,31,53)(7,41,16,68,25,59,34,50)>;`

`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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,47,10,38,19,65,28,56)(2,58,23,37,8,52,29,67,14,46,35,61,20,40,5,55,26,70,11,49,32,64,17,43)(3,69,36,72,33,39,30,42,27,45,24,48,21,51,18,54,15,57,12,60,9,63,6,66)(4,44,13,71,22,62,31,53)(7,41,16,68,25,59,34,50) );`

`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),(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)], [(1,47,10,38,19,65,28,56),(2,58,23,37,8,52,29,67,14,46,35,61,20,40,5,55,26,70,11,49,32,64,17,43),(3,69,36,72,33,39,30,42,27,45,24,48,21,51,18,54,15,57,12,60,9,63,6,66),(4,44,13,71,22,62,31,53),(7,41,16,68,25,59,34,50)]])`

62 conjugacy classes

 class 1 2A 2B 3A 3B 3C 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 9A 9B 9C 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 18A ··· 18I 24A ··· 24H 36A ··· 36L order 1 2 2 3 3 3 4 4 4 6 6 6 6 6 6 6 8 8 8 8 9 9 9 12 12 12 12 12 12 12 12 12 12 18 ··· 18 24 ··· 24 36 ··· 36 size 1 1 2 2 3 3 1 1 2 2 2 2 3 3 6 6 18 18 18 18 6 6 6 2 2 2 2 3 3 3 3 6 6 6 ··· 6 18 ··· 18 6 ··· 6

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 6 6 6 6 6 type + + + + - + - + - + - image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 S3 Dic3 D6 Dic3 M4(2) C3×S3 C3×Dic3 S3×C6 C3×Dic3 C3×M4(2) C4.Dic3 C3×C4.Dic3 C9⋊C6 C9⋊C12 C2×C9⋊C6 C9⋊C12 C36.C12 kernel C36.C12 C9⋊C24 C2×C4×3- 1+2 C4.Dic9 C4×3- 1+2 C22×3- 1+2 C9⋊C8 C2×C36 C36 C2×C18 C6×C12 C3×C12 C3×C12 C62 3- 1+2 C2×C12 C12 C12 C2×C6 C9 C32 C3 C2×C4 C4 C4 C22 C1 # reps 1 2 1 2 2 2 4 2 4 4 1 1 1 1 2 2 2 2 2 4 4 8 1 1 1 1 4

Matrix representation of C36.C12 in GL6(𝔽73)

 70 30 0 0 0 0 0 3 70 0 0 0 24 3 0 0 0 0 21 9 0 0 0 3 22 0 0 24 0 0 21 0 9 0 24 0
,
 20 0 0 71 0 0 0 0 20 29 60 0 59 14 0 29 0 42 4 0 0 53 0 0 19 29 0 0 0 59 37 0 68 0 53 0

`G:=sub<GL(6,GF(73))| [70,0,24,21,22,21,30,3,3,9,0,0,0,70,0,0,0,9,0,0,0,0,24,0,0,0,0,0,0,24,0,0,0,3,0,0],[20,0,59,4,19,37,0,0,14,0,29,0,0,20,0,0,0,68,71,29,29,53,0,0,0,60,0,0,0,53,0,0,42,0,59,0] >;`

C36.C12 in GAP, Magma, Sage, TeX

`C_{36}.C_{12}`
`% in TeX`

`G:=Group("C36.C12");`
`// GroupNames label`

`G:=SmallGroup(432,143);`
`// by ID`

`G=gap.SmallGroup(432,143);`
`# by ID`

`G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,80,10085,2035,292,14118]);`
`// Polycyclic`

`G:=Group<a,b|a^36=1,b^12=a^18,b*a*b^-1=a^23>;`
`// generators/relations`

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