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## G = C12.30D12order 288 = 25·32

### 30th non-split extension by C12 of D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C12.30D12
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — Dic3⋊Dic3 — C12.30D12
 Lower central C32 — C62 — C12.30D12
 Upper central C1 — C22 — C2×C4

Generators and relations for C12.30D12
G = < a,b,c | a12=b12=1, c2=a6, bab-1=a-1, cac-1=a5, cbc-1=b-1 >

Subgroups: 690 in 175 conjugacy classes, 54 normal (32 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], S3 [×8], C6 [×6], C6 [×3], C2×C4, C2×C4 [×7], Q8 [×2], C23, C32, Dic3 [×8], C12 [×4], C12 [×6], D6 [×14], C2×C6 [×2], C2×C6, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, C3⋊S3 [×2], C3×C6 [×3], Dic6 [×2], C4×S3 [×8], C2×Dic3 [×4], C2×Dic3 [×3], C2×C12 [×2], C2×C12 [×5], C3×Q8 [×2], C22×S3 [×3], C22⋊Q8, C3×Dic3 [×4], C3⋊Dic3, C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, Dic3⋊C4 [×2], C4⋊Dic3, C4⋊Dic3 [×2], D6⋊C4 [×4], C3×C4⋊C4, C2×Dic6, S3×C2×C4 [×3], C6×Q8, C3×Dic6 [×2], C6×Dic3 [×4], C4×C3⋊S3 [×2], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C4.D12, D63Q8, C6.D12 [×2], Dic3⋊Dic3 [×2], C3×C4⋊Dic3, C6×Dic6, C2×C4×C3⋊S3, C12.30D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], Q8 [×2], C23, D6 [×6], C2×D4, C2×Q8, C4○D4, D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C22⋊Q8, S32, C2×D12, D42S3, S3×Q8 [×2], Q83S3, C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C4.D12, D63Q8, D12⋊S3, Dic3.D6, C2×C3⋊D12, C12.30D12

Smallest permutation representation of C12.30D12
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 20 34 44 9 24 30 48 5 16 26 40)(2 19 35 43 10 23 31 47 6 15 27 39)(3 18 36 42 11 22 32 46 7 14 28 38)(4 17 25 41 12 21 33 45 8 13 29 37)
(1 25 7 31)(2 30 8 36)(3 35 9 29)(4 28 10 34)(5 33 11 27)(6 26 12 32)(13 18 19 24)(14 23 20 17)(15 16 21 22)(37 38 43 44)(39 48 45 42)(40 41 46 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), (1,20,34,44,9,24,30,48,5,16,26,40)(2,19,35,43,10,23,31,47,6,15,27,39)(3,18,36,42,11,22,32,46,7,14,28,38)(4,17,25,41,12,21,33,45,8,13,29,37), (1,25,7,31)(2,30,8,36)(3,35,9,29)(4,28,10,34)(5,33,11,27)(6,26,12,32)(13,18,19,24)(14,23,20,17)(15,16,21,22)(37,38,43,44)(39,48,45,42)(40,41,46,47)>;`

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G 6H 6I 12A ··· 12H 12I ··· 12P order 1 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 12 ··· 12 12 ··· 12 size 1 1 1 1 18 18 2 2 4 2 2 12 12 12 12 18 18 2 ··· 2 4 4 4 4 ··· 4 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 4 4 type + + + + + + + + + - + + + + - - + + + image C1 C2 C2 C2 C2 C2 S3 S3 D4 Q8 D6 D6 C4○D4 D12 C3⋊D4 S32 D4⋊2S3 S3×Q8 Q8⋊3S3 C3⋊D12 C2×S32 D12⋊S3 Dic3.D6 kernel C12.30D12 C6.D12 Dic3⋊Dic3 C3×C4⋊Dic3 C6×Dic6 C2×C4×C3⋊S3 C4⋊Dic3 C2×Dic6 C3×C12 C2×C3⋊S3 C2×Dic3 C2×C12 C3×C6 C12 C12 C2×C4 C6 C6 C6 C4 C22 C2 C2 # reps 1 2 2 1 1 1 1 1 2 2 4 2 2 4 4 1 1 2 1 2 1 2 2

Matrix representation of C12.30D12 in GL8(𝔽13)

 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 9 3 0 0 0 0 0 0 3 4 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0
,
 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1

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

C12.30D12 in GAP, Magma, Sage, TeX

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

`G:=Group("C12.30D12");`
`// GroupNames label`

`G:=SmallGroup(288,519);`
`// by ID`

`G=gap.SmallGroup(288,519);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,64,422,219,100,1356,9414]);`
`// Polycyclic`

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

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