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## G = D12.2D6order 288 = 25·32

### 2nd non-split extension by D12 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12.2D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D12⋊S3 — D12.2D6
 Lower central C32 — C3×C6 — C3×C12 — D12.2D6
 Upper central C1 — C2 — C4 — C8

Generators and relations for D12.2D6
G = < a,b,c,d | a12=b2=1, c6=d2=a6, bab=cac-1=a-1, dad-1=a5, cbc-1=ab, dbd-1=a4b, dcd-1=c5 >

Subgroups: 562 in 135 conjugacy classes, 38 normal (16 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], S3 [×6], C6 [×2], C6 [×3], C8, C8, C2×C4 [×3], D4 [×4], Q8 [×2], C32, Dic3 [×5], C12 [×2], C12 [×3], D6 [×5], C2×C6 [×2], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3×S3 [×2], C3⋊S3, C3×C6, C3⋊C8 [×3], C24 [×2], C24, Dic6 [×2], C4×S3 [×5], D12 [×2], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×2], C3×D4 [×2], C3×Q8 [×2], C4○D8, C3×Dic3 [×2], C3⋊Dic3, C3×C12, S3×C6 [×2], C2×C3⋊S3, S3×C8 [×3], C24⋊C2 [×2], D4⋊S3 [×2], C3⋊Q16 [×2], C3×SD16 [×2], D42S3 [×2], Q83S3 [×2], C324C8, C3×C24, S3×Dic3 [×2], C3⋊D12 [×2], C3×Dic6 [×2], C3×D12 [×2], C4×C3⋊S3, Q8.7D6 [×2], C322D8, C322Q16, C3×C24⋊C2 [×2], C8×C3⋊S3, D12⋊S3 [×2], D12.2D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C22×S3 [×2], C4○D8, S32, S3×D4 [×2], C2×S32, Q8.7D6 [×2], D6⋊D6, D12.2D6

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 C4○D8 S32 S3×D4 C2×S32 Q8.7D6 D6⋊D6 D12.2D6 kernel D12.2D6 C32⋊2D8 C32⋊2Q16 C3×C24⋊C2 C8×C3⋊S3 D12⋊S3 C24⋊C2 C3⋊Dic3 C2×C3⋊S3 C24 Dic6 D12 C32 C8 C6 C4 C3 C2 C1 # reps 1 1 1 2 1 2 2 1 1 2 2 2 4 1 2 1 4 2 4

Matrix representation of D12.2D6 in GL6(𝔽73)

 46 0 0 0 0 0 0 27 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 72
,
 0 10 0 0 0 0 22 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 1 0 0 0 0 72 0 0 0 0 0 0 0 1 72 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 72 72
,
 27 0 0 0 0 0 0 27 0 0 0 0 0 0 72 0 0 0 0 0 72 1 0 0 0 0 0 0 1 0 0 0 0 0 72 72

`G:=sub<GL(6,GF(73))| [46,0,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[0,22,0,0,0,0,10,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;`

D12.2D6 in GAP, Magma, Sage, TeX

`D_{12}._2D_6`
`% in TeX`

`G:=Group("D12.2D6");`
`// GroupNames label`

`G:=SmallGroup(288,457);`
`// by ID`

`G=gap.SmallGroup(288,457);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,135,58,675,346,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=1,c^6=d^2=a^6,b*a*b=c*a*c^-1=a^-1,d*a*d^-1=a^5,c*b*c^-1=a*b,d*b*d^-1=a^4*b,d*c*d^-1=c^5>;`
`// generators/relations`

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