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## G = D12.32D6order 288 = 25·32

### 7th non-split extension by D12 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12.32D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — Dic6⋊S3 — D12.32D6
 Lower central C32 — C3×C6 — C3×C12 — D12.32D6
 Upper central C1 — C2 — C2×C4

Generators and relations for D12.32D6
G = < a,b,c,d | a12=b2=1, c6=d2=a6, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a3b, dcd-1=a6c5 >

Subgroups: 402 in 130 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×3], C22, C22, S3, C6 [×2], C6 [×5], C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×4], C32, Dic3 [×3], C12 [×4], C12 [×5], D6, C2×C6 [×2], C2×C6 [×2], M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8 [×6], Dic6, Dic6 [×2], Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12 [×2], C2×C12 [×3], C3×D4 [×2], C3×Q8 [×4], C8.C22, C3×Dic3 [×3], C3×C12 [×2], S3×C6, C62, C4.Dic3 [×3], D4.S3 [×2], Q82S3 [×2], C3⋊Q16 [×4], C2×Dic6, C4○D12, C6×Q8, C3×C4○D4, C324C8 [×2], C3×Dic6, C3×Dic6 [×2], C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C6×C12, Q8.11D6, Q8.14D6, Dic6⋊S3 [×2], C322Q16 [×2], C12.58D6, C6×Dic6, C3×C4○D12, D12.32D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C3⋊D4 [×4], C22×S3 [×2], C8.C22, S32, C2×C3⋊D4 [×2], D6⋊S3 [×2], C2×S32, Q8.11D6, Q8.14D6, C2×D6⋊S3, D12.32D6

Smallest permutation representation of D12.32D6
On 48 points
Generators in S48
```(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 28)(2 27)(3 26)(4 25)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 40)(20 39)(21 38)(22 37)(23 48)(24 47)
(1 12 11 10 9 8 7 6 5 4 3 2)(13 20 15 22 17 24 19 14 21 16 23 18)(25 26 27 28 29 30 31 32 33 34 35 36)(37 42 47 40 45 38 43 48 41 46 39 44)
(1 18 7 24)(2 13 8 19)(3 20 9 14)(4 15 10 21)(5 22 11 16)(6 17 12 23)(25 41 31 47)(26 48 32 42)(27 43 33 37)(28 38 34 44)(29 45 35 39)(30 40 36 46)```

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + - + - + - - image C1 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 C3⋊D4 C3⋊D4 C8.C22 S32 D6⋊S3 C2×S32 D6⋊S3 Q8.11D6 Q8.14D6 D12.32D6 kernel D12.32D6 Dic6⋊S3 C32⋊2Q16 C12.58D6 C6×Dic6 C3×C4○D12 C2×Dic6 C4○D12 C3×C12 C62 Dic6 D12 C2×C12 C12 C2×C6 C32 C2×C4 C4 C4 C22 C3 C3 C1 # reps 1 2 2 1 1 1 1 1 1 1 3 1 2 4 4 1 1 1 1 1 2 2 4

Matrix representation of D12.32D6 in GL8(𝔽73)

 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0
,
 51 32 0 0 0 0 0 0 10 22 0 0 0 0 0 0 0 0 30 60 0 0 0 0 0 0 13 43 0 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0
,
 30 13 0 0 0 0 0 0 60 43 0 0 0 0 0 0 0 0 70 42 0 0 0 0 0 0 45 3 0 0 0 0 0 0 0 0 11 30 0 0 0 0 0 0 30 62 0 0 0 0 0 0 0 0 30 62 0 0 0 0 0 0 62 43

`G:=sub<GL(8,GF(73))| [1,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0],[51,10,0,0,0,0,0,0,32,22,0,0,0,0,0,0,0,0,30,13,0,0,0,0,0,0,60,43,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0],[30,60,0,0,0,0,0,0,13,43,0,0,0,0,0,0,0,0,70,45,0,0,0,0,0,0,42,3,0,0,0,0,0,0,0,0,11,30,0,0,0,0,0,0,30,62,0,0,0,0,0,0,0,0,30,62,0,0,0,0,0,0,62,43] >;`

D12.32D6 in GAP, Magma, Sage, TeX

`D_{12}._{32}D_6`
`% in TeX`

`G:=Group("D12.32D6");`
`// GroupNames label`

`G:=SmallGroup(288,475);`
`// by ID`

`G=gap.SmallGroup(288,475);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,120,219,100,675,346,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=1,c^6=d^2=a^6,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=a^6*c^5>;`
`// generators/relations`

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