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## G = C12.71D12order 288 = 25·32

### 2nd non-split extension by C12 of D12 acting via D12/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C12.71D12
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C3×C4.Dic3 — C12.71D12
 Lower central C32 — C3×C6 — C62 — C12.71D12
 Upper central C1 — C2 — C2×C4

Generators and relations for C12.71D12
G = < a,b,c | a12=1, b12=c2=a6, bab-1=cac-1=a-1, cbc-1=a9b11 >

Subgroups: 354 in 95 conjugacy classes, 32 normal (10 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, C4.10D4, C3⋊Dic3, C3×C12, C62, C4.Dic3, C3×M4(2), C2×Dic6, C3×C3⋊C8, C324Q8, C2×C3⋊Dic3, C6×C12, C12.47D4, C3×C4.Dic3, C2×C324Q8, C12.71D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C4×S3, D12, C3⋊D4, C4.10D4, S32, D6⋊C4, C6.D6, C3⋊D12, C12.47D4, C6.D12, C12.71D12

Smallest permutation representation of C12.71D12
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 35 45 31 41 27 37 47 33 43 29 39)(26 40 30 44 34 48 38 28 42 32 46 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 37 38 39 40 41 42 43 44 45 46 47 48)
(1 27 13 39)(2 32 14 44)(3 25 15 37)(4 30 16 42)(5 47 17 35)(6 28 18 40)(7 45 19 33)(8 26 20 38)(9 43 21 31)(10 48 22 36)(11 41 23 29)(12 46 24 34)```

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + - + + + - - image C1 C2 C2 C4 S3 D4 D6 D12 C3⋊D4 C4×S3 C4.10D4 S32 C3⋊D12 C6.D6 C12.47D4 C12.71D12 kernel C12.71D12 C3×C4.Dic3 C2×C32⋊4Q8 C2×C3⋊Dic3 C4.Dic3 C3×C12 C2×C12 C12 C12 C2×C6 C32 C2×C4 C4 C22 C3 C1 # reps 1 2 1 4 2 2 2 4 4 4 1 1 2 1 4 4

Matrix representation of C12.71D12 in GL4(𝔽73) generated by

 59 66 0 0 7 66 0 0 0 0 59 66 0 0 7 66
,
 0 0 22 10 0 0 61 51 68 14 0 0 19 5 0 0
,
 68 14 0 0 19 5 0 0 0 0 14 19 0 0 5 59
`G:=sub<GL(4,GF(73))| [59,7,0,0,66,66,0,0,0,0,59,7,0,0,66,66],[0,0,68,19,0,0,14,5,22,61,0,0,10,51,0,0],[68,19,0,0,14,5,0,0,0,0,14,5,0,0,19,59] >;`

C12.71D12 in GAP, Magma, Sage, TeX

`C_{12}._{71}D_{12}`
`% in TeX`

`G:=Group("C12.71D12");`
`// GroupNames label`

`G:=SmallGroup(288,209);`
`// by ID`

`G=gap.SmallGroup(288,209);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,204,219,100,675,346,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=1,b^12=c^2=a^6,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^9*b^11>;`
`// generators/relations`

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