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## G = C12.14D12order 288 = 25·32

### 14th non-split extension by C12 of D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C12.14D12
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C6×Dic6 — C12.14D12
 Lower central C32 — C3×C6 — C62 — C12.14D12
 Upper central C1 — C2 — C2×C4

Generators and relations for C12.14D12
G = < a,b,c | a12=1, b12=a6, c2=a3, bab-1=a-1, cac-1=a5, cbc-1=a3b11 >

Subgroups: 242 in 86 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C32, Dic3, C12, C12, C2×C6, C2×C6, M4(2), C2×Q8, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C4.10D4, C3×Dic3, C3×C12, C62, C4.Dic3, C4.Dic3, C3×M4(2), C2×Dic6, C6×Q8, C3×C3⋊C8, C324C8, C3×Dic6, C6×Dic3, C6×C12, C12.47D4, C12.10D4, C3×C4.Dic3, C12.58D6, C6×Dic6, C12.14D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C4×S3, D12, C2×Dic3, C3⋊D4, C4.10D4, S32, D6⋊C4, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, C12.47D4, C12.10D4, D6⋊Dic3, C12.14D12

Smallest permutation representation of C12.14D12
On 48 points
Generators in S48
```(1 11 21 7 17 3 13 23 9 19 5 15)(2 16 6 20 10 24 14 4 18 8 22 12)(25 47 45 43 41 39 37 35 33 31 29 27)(26 28 30 32 34 36 38 40 42 44 46 48)
(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)
(1 45 7 39 13 33 19 27)(2 38 20 44 14 26 8 32)(3 43 9 37 15 31 21 25)(4 36 22 42 16 48 10 30)(5 41 11 35 17 29 23 47)(6 34 24 40 18 46 12 28)```

`G:=sub<Sym(48)| (1,11,21,7,17,3,13,23,9,19,5,15)(2,16,6,20,10,24,14,4,18,8,22,12)(25,47,45,43,41,39,37,35,33,31,29,27)(26,28,30,32,34,36,38,40,42,44,46,48), (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), (1,45,7,39,13,33,19,27)(2,38,20,44,14,26,8,32)(3,43,9,37,15,31,21,25)(4,36,22,42,16,48,10,30)(5,41,11,35,17,29,23,47)(6,34,24,40,18,46,12,28)>;`

`G:=Group( (1,11,21,7,17,3,13,23,9,19,5,15)(2,16,6,20,10,24,14,4,18,8,22,12)(25,47,45,43,41,39,37,35,33,31,29,27)(26,28,30,32,34,36,38,40,42,44,46,48), (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), (1,45,7,39,13,33,19,27)(2,38,20,44,14,26,8,32)(3,43,9,37,15,31,21,25)(4,36,22,42,16,48,10,30)(5,41,11,35,17,29,23,47)(6,34,24,40,18,46,12,28) );`

`G=PermutationGroup([[(1,11,21,7,17,3,13,23,9,19,5,15),(2,16,6,20,10,24,14,4,18,8,22,12),(25,47,45,43,41,39,37,35,33,31,29,27),(26,28,30,32,34,36,38,40,42,44,46,48)], [(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)], [(1,45,7,39,13,33,19,27),(2,38,20,44,14,26,8,32),(3,43,9,37,15,31,21,25),(4,36,22,42,16,48,10,30),(5,41,11,35,17,29,23,47),(6,34,24,40,18,46,12,28)]])`

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + - + + - + - + - - image C1 C2 C2 C2 C4 S3 S3 D4 Dic3 D6 D12 C3⋊D4 C4×S3 C4.10D4 S32 D6⋊S3 C3⋊D12 S3×Dic3 C12.47D4 C12.10D4 C12.14D12 kernel C12.14D12 C3×C4.Dic3 C12.58D6 C6×Dic6 C6×Dic3 C4.Dic3 C2×Dic6 C3×C12 C2×Dic3 C2×C12 C12 C12 C2×C6 C32 C2×C4 C4 C4 C22 C3 C3 C1 # reps 1 1 1 1 4 1 1 2 2 2 2 6 2 1 1 1 1 1 2 2 4

Matrix representation of C12.14D12 in GL4(𝔽73) generated by

 3 21 20 8 0 49 0 0 0 0 70 0 0 0 0 24
,
 8 12 21 20 21 65 70 65 0 0 0 64 0 0 49 0
,
 65 61 65 40 0 0 49 0 0 65 0 0 52 8 3 8
`G:=sub<GL(4,GF(73))| [3,0,0,0,21,49,0,0,20,0,70,0,8,0,0,24],[8,21,0,0,12,65,0,0,21,70,0,49,20,65,64,0],[65,0,0,52,61,0,65,8,65,49,0,3,40,0,0,8] >;`

C12.14D12 in GAP, Magma, Sage, TeX

`C_{12}._{14}D_{12}`
`% in TeX`

`G:=Group("C12.14D12");`
`// GroupNames label`

`G:=SmallGroup(288,208);`
`// by ID`

`G=gap.SmallGroup(288,208);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,36,422,100,346,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^3,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^3*b^11>;`
`// generators/relations`

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