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## G = C3⋊C8.22D6order 288 = 25·32

### 11st non-split extension by C3⋊C8 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3⋊C8.22D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C3⋊C8 — C12.29D6 — C3⋊C8.22D6
 Lower central C32 — C3×C6 — C3⋊C8.22D6
 Upper central C1 — C4 — C2×C4

Generators and relations for C3⋊C8.22D6
G = < a,b,c,d | a3=b8=1, c6=b2, d2=b4, bab-1=cac-1=dad-1=a-1, cbc-1=dbd-1=b5, dcd-1=b4c5 >

Subgroups: 506 in 144 conjugacy classes, 52 normal (36 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×6], C6 [×2], C6 [×4], C8 [×4], C2×C4, C2×C4 [×2], D4 [×3], Q8, C32, Dic3 [×6], C12 [×4], C12 [×2], D6 [×6], C2×C6 [×2], C2×C6, C2×C8 [×3], M4(2) [×3], C4○D4, C3⋊S3 [×2], C3×C6, C3×C6, C3⋊C8 [×2], C3⋊C8 [×2], C24 [×4], Dic6 [×3], C4×S3 [×6], D12 [×3], C3⋊D4 [×6], C2×C12 [×2], C2×C12, C8○D4, C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×2], C62, S3×C8 [×4], C8⋊S3 [×4], C2×C3⋊C8, C4.Dic3, C2×C24, C3×M4(2), C4○D12 [×3], C3×C3⋊C8 [×2], C3×C3⋊C8 [×2], C324Q8, C4×C3⋊S3 [×2], C12⋊S3, C327D4 [×2], C6×C12, C8○D12, D12.C4, C12.29D6 [×2], C12.31D6 [×2], C6×C3⋊C8, C3×C4.Dic3, C12.59D6, C3⋊C8.22D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], C23, D6 [×6], C22×C4, C4×S3 [×4], C22×S3 [×2], C8○D4, S32, S3×C2×C4 [×2], C6.D6 [×2], C2×S32, C8○D12, D12.C4, C2×C6.D6, C3⋊C8.22D6

Smallest permutation representation of C3⋊C8.22D6
On 48 points
Generators in S48
```(1 9 17)(2 18 10)(3 11 19)(4 20 12)(5 13 21)(6 22 14)(7 15 23)(8 24 16)(25 33 41)(26 42 34)(27 35 43)(28 44 36)(29 37 45)(30 46 38)(31 39 47)(32 48 40)
(1 36 7 42 13 48 19 30)(2 25 8 31 14 37 20 43)(3 38 9 44 15 26 21 32)(4 27 10 33 16 39 22 45)(5 40 11 46 17 28 23 34)(6 29 12 35 18 41 24 47)
(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 15 13 3)(2 8 14 20)(4 18 16 6)(5 11 17 23)(7 21 19 9)(10 24 22 12)(25 43 37 31)(26 36 38 48)(27 29 39 41)(28 46 40 34)(30 32 42 44)(33 35 45 47)```

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 6G 6H 8A 8B 8C 8D 8E ··· 8J 12A ··· 12F 12G ··· 12K 24A ··· 24H 24I 24J 24K 24L 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 8 ··· 8 12 ··· 12 12 ··· 12 24 ··· 24 24 24 24 24 size 1 1 2 18 18 2 2 4 1 1 2 18 18 2 2 2 2 4 4 4 4 3 3 3 3 6 ··· 6 2 ··· 2 4 ··· 4 6 ··· 6 12 12 12 12

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 S3 S3 D6 D6 C4×S3 C4×S3 C8○D4 C8○D12 S32 C6.D6 C2×S32 C6.D6 D12.C4 C3⋊C8.22D6 kernel C3⋊C8.22D6 C12.29D6 C12.31D6 C6×C3⋊C8 C3×C4.Dic3 C12.59D6 C32⋊4Q8 C12⋊S3 C32⋊7D4 C2×C3⋊C8 C4.Dic3 C3⋊C8 C2×C12 C12 C2×C6 C32 C3 C2×C4 C4 C4 C22 C3 C1 # reps 1 2 2 1 1 1 2 2 4 1 1 4 2 4 4 4 8 1 1 1 1 2 4

Matrix representation of C3⋊C8.22D6 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 46 0 0 0 0 0 0 46 0 0 0 0 0 0 0 51 0 0 0 0 22 0 0 0 0 0 0 0 46 27 0 0 0 0 0 27
,
 0 46 0 0 0 0 27 27 0 0 0 0 0 0 22 0 0 0 0 0 0 51 0 0 0 0 0 0 27 46 0 0 0 0 0 46
,
 0 72 0 0 0 0 72 0 0 0 0 0 0 0 27 0 0 0 0 0 0 46 0 0 0 0 0 0 72 1 0 0 0 0 0 1

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,22,0,0,0,0,51,0,0,0,0,0,0,0,46,0,0,0,0,0,27,27],[0,27,0,0,0,0,46,27,0,0,0,0,0,0,22,0,0,0,0,0,0,51,0,0,0,0,0,0,27,0,0,0,0,0,46,46],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,27,0,0,0,0,0,0,46,0,0,0,0,0,0,72,0,0,0,0,0,1,1] >;`

C3⋊C8.22D6 in GAP, Magma, Sage, TeX

`C_3\rtimes C_8._{22}D_6`
`% in TeX`

`G:=Group("C3:C8.22D6");`
`// GroupNames label`

`G:=SmallGroup(288,465);`
`// by ID`

`G=gap.SmallGroup(288,465);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,64,219,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^3=b^8=1,c^6=b^2,d^2=b^4,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^5,d*c*d^-1=b^4*c^5>;`
`// generators/relations`

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