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## G = D12.25D6order 288 = 25·32

### 10th non-split extension by D12 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — D12.25D6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — S3×Dic3 — S3×Dic6 — D12.25D6
 Lower central C32 — C3×C6 — D12.25D6
 Upper central C1 — C2 — Q8

Generators and relations for D12.25D6
G = < a,b,c,d | a12=b2=1, c6=d2=a6, bab=a-1, cac-1=a7, ad=da, cbc-1=a6b, bd=db, dcd-1=a6c5 >

Subgroups: 1026 in 311 conjugacy classes, 108 normal (24 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, Q8, C32, Dic3, Dic3, Dic3, C12, C12, D6, D6, D6, C2×C6, C2×Q8, C4○D4, C3×S3, C3⋊S3, C3×C6, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C3×Q8, 2- 1+4, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C2×Dic6, C4○D12, D42S3, S3×Q8, S3×Q8, Q83S3, Q83S3, C6×Q8, C3×C4○D4, S3×Dic3, D6⋊S3, C3⋊D12, C3⋊D12, C322Q8, C3×Dic6, S3×C12, C3×D12, C324Q8, C4×C3⋊S3, Q8×C32, Q8.15D6, Q8○D12, S3×Dic6, D125S3, D12⋊S3, D6.D6, C3×S3×Q8, C3×Q83S3, Q8×C3⋊S3, D12.25D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2- 1+4, S32, S3×C23, C2×S32, Q8.15D6, Q8○D12, C22×S32, D12.25D6

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

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 6D 6E 6F 6G 6H 12A ··· 12F 12G 12H 12I 12J 12K 12L 12M 12N order 1 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 6 12 ··· 12 12 12 12 12 12 12 12 12 size 1 1 6 6 6 6 18 2 2 4 2 2 2 6 6 6 6 18 18 18 2 2 4 6 6 12 12 12 4 ··· 4 6 6 8 8 8 12 12 12

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 8 type + + + + + + + + + + + + + + - + + - - image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 D6 2- 1+4 S32 C2×S32 Q8.15D6 Q8○D12 D12.25D6 kernel D12.25D6 S3×Dic6 D12⋊5S3 D12⋊S3 D6.D6 C3×S3×Q8 C3×Q8⋊3S3 Q8×C3⋊S3 S3×Q8 Q8⋊3S3 Dic6 C4×S3 D12 C3×Q8 C32 Q8 C4 C3 C3 C1 # reps 1 3 3 3 3 1 1 1 1 1 3 6 3 2 1 1 3 2 2 1

Matrix representation of D12.25D6 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 5 8 0 0 0 0 5 0 0 0 5 8 0 0 0 0 5 0 0 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 9 2 0 0 0 0 11 4 0 0 4 11 0 0 0 0 2 9 0 0
,
 0 12 0 0 0 0 1 1 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 3 7 0 0 0 0 6 10 0 0 3 7 0 0 0 0 6 10 0 0

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,5,5,0,0,0,0,8,0,0,0,5,5,0,0,0,0,8,0,0,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,4,2,0,0,0,0,11,9,0,0,9,11,0,0,0,0,2,4,0,0],[0,1,0,0,0,0,12,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,6,0,0,0,0,7,10,0,0,3,6,0,0,0,0,7,10,0,0] >;`

D12.25D6 in GAP, Magma, Sage, TeX

`D_{12}._{25}D_6`
`% in TeX`

`G:=Group("D12.25D6");`
`// GroupNames label`

`G:=SmallGroup(288,963);`
`// by ID`

`G=gap.SmallGroup(288,963);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,100,675,185,80,1356,9414]);`
`// Polycyclic`

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

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