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## G = D12.30D6order 288 = 25·32

### 5th non-split extension by D12 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12.30D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — C32⋊2D8 — D12.30D6
 Lower central C32 — C3×C6 — C3×C12 — D12.30D6
 Upper central C1 — C4 — C2×C4

Generators and relations for D12.30D6
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=c5 >

Subgroups: 434 in 135 conjugacy classes, 44 normal (18 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×2], C6 [×2], C6 [×7], C8 [×2], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×2], C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×2], C2×C6 [×2], C2×C6 [×3], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3×S3 [×2], C3×C6, C3×C6, C3⋊C8 [×6], Dic6 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×3], C3×D4 [×4], C3×Q8 [×2], C4○D8, C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×2], C62, C2×C3⋊C8 [×3], D4⋊S3 [×2], D4.S3 [×2], Q82S3 [×2], C3⋊Q16 [×2], C4○D12 [×2], C3×C4○D4 [×2], C324C8 [×2], C3×Dic6 [×2], S3×C12 [×2], C3×D12 [×2], C3×C3⋊D4 [×2], C6×C12, Q8.13D6 [×2], C322D8, Dic6⋊S3 [×2], C322Q16, C2×C324C8, C3×C4○D12 [×2], D12.30D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C3⋊D4 [×4], C22×S3 [×2], C4○D8, S32, C2×C3⋊D4 [×2], D6⋊S3 [×2], C2×S32, Q8.13D6 [×2], C2×D6⋊S3, D12.30D6

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 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 C3⋊D4 C3⋊D4 C4○D8 S32 D6⋊S3 C2×S32 D6⋊S3 Q8.13D6 D12.30D6 kernel D12.30D6 C32⋊2D8 Dic6⋊S3 C32⋊2Q16 C2×C32⋊4C8 C3×C4○D12 C4○D12 C3×C12 C62 Dic6 D12 C2×C12 C12 C2×C6 C32 C2×C4 C4 C4 C22 C3 C1 # reps 1 1 2 1 1 2 2 1 1 2 2 2 4 4 4 1 1 1 1 4 4

Matrix representation of D12.30D6 in GL4(𝔽5) generated by

 2 3 3 2 3 1 2 0 3 1 3 0 3 0 2 4
,
 4 2 0 4 4 4 4 2 3 0 2 4 3 1 3 0
,
 4 0 2 0 3 0 1 3 4 0 3 0 0 2 1 2
,
 3 4 2 2 2 0 4 2 1 4 4 0 0 0 0 3
`G:=sub<GL(4,GF(5))| [2,3,3,3,3,1,1,0,3,2,3,2,2,0,0,4],[4,4,3,3,2,4,0,1,0,4,2,3,4,2,4,0],[4,3,4,0,0,0,0,2,2,1,3,1,0,3,0,2],[3,2,1,0,4,0,4,0,2,4,4,0,2,2,0,3] >;`

D12.30D6 in GAP, Magma, Sage, TeX

`D_{12}._{30}D_6`
`% in TeX`

`G:=Group("D12.30D6");`
`// GroupNames label`

`G:=SmallGroup(288,470);`
`// by ID`

`G=gap.SmallGroup(288,470);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,100,675,346,80,1356,9414]);`
`// 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=c^5>;`
`// generators/relations`

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