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## G = D12.34D6order 288 = 25·32

### The non-split extension by D12 of D6 acting through Inn(D12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — D12.34D6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — S3×Dic3 — S3×Dic6 — D12.34D6
 Lower central C32 — C3×C6 — D12.34D6
 Upper central C1 — C2 — C2×C4

Generators and relations for D12.34D6
G = < a,b,c,d | a12=b2=1, c6=d2=a6, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a6c5 >

Subgroups: 994 in 311 conjugacy classes, 108 normal (10 characteristic)
C1, C2, C2 [×5], C3 [×2], C3, C4 [×2], C4 [×8], C22, C22 [×4], S3 [×4], C6 [×2], C6 [×9], C2×C4, C2×C4 [×14], D4 [×10], Q8 [×10], C32, Dic3 [×4], Dic3 [×12], C12 [×4], C12 [×6], D6 [×4], C2×C6 [×2], C2×C6 [×5], C2×Q8 [×5], C4○D4 [×10], C3×S3 [×4], C3×C6, C3×C6, Dic6 [×2], Dic6 [×20], C4×S3 [×4], C4×S3 [×8], D12 [×2], C2×Dic3 [×14], C3⋊D4 [×4], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×5], C3×D4 [×6], C3×Q8 [×2], 2- 1+4, C3×Dic3 [×4], C3⋊Dic3 [×4], C3×C12 [×2], S3×C6 [×4], C62, C2×Dic6 [×7], C4○D12 [×2], C4○D12 [×4], D42S3 [×12], S3×Q8 [×4], C3×C4○D4 [×2], S3×Dic3 [×8], D6⋊S3 [×4], C322Q8 [×4], C3×Dic6 [×2], S3×C12 [×4], C3×D12 [×2], C3×C3⋊D4 [×4], C324Q8 [×4], C2×C3⋊Dic3 [×2], C6×C12, Q8○D12 [×2], S3×Dic6 [×4], D125S3 [×4], D6.4D6 [×4], C3×C4○D12 [×2], C2×C324Q8, D12.34D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C24, C22×S3 [×14], 2- 1+4, S32, S3×C23 [×2], C2×S32 [×3], Q8○D12 [×2], C22×S32, D12.34D6

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

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

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

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

45 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 ··· 6G 6H 6I 6J 6K 12A 12B 12C 12D 12E ··· 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 2 6 6 6 6 2 2 4 2 2 6 6 6 6 18 18 18 18 2 2 4 ··· 4 12 12 12 12 2 2 2 2 4 ··· 4 12 12 12 12

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + - + + + - - image C1 C2 C2 C2 C2 C2 S3 D6 D6 D6 D6 D6 2- 1+4 S32 C2×S32 C2×S32 Q8○D12 D12.34D6 kernel D12.34D6 S3×Dic6 D12⋊5S3 D6.4D6 C3×C4○D12 C2×C32⋊4Q8 C4○D12 Dic6 C4×S3 D12 C3⋊D4 C2×C12 C32 C2×C4 C4 C22 C3 C1 # reps 1 4 4 4 2 1 2 2 4 2 4 2 1 1 2 1 4 4

Matrix representation of D12.34D6 in GL4(𝔽13) generated by

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

D12.34D6 in GAP, Magma, Sage, TeX

`D_{12}._{34}D_6`
`% in TeX`

`G:=Group("D12.34D6");`
`// GroupNames label`

`G:=SmallGroup(288,946);`
`// by ID`

`G=gap.SmallGroup(288,946);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,100,675,80,1356,9414]);`
`// 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,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^6*c^5>;`
`// generators/relations`

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