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## G = C62⋊23D6order 432 = 24·33

### 4th semidirect product of C62 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C62⋊23D6
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C2×S3×C3⋊S3 — C62⋊23D6
 Lower central C33 — C32×C6 — C62⋊23D6
 Upper central C1 — C2 — C22

Generators and relations for C6223D6
G = < a,b,c,d | a6=b6=c6=d2=1, ab=ba, cac-1=a-1b3, dad=a-1, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 3296 in 452 conjugacy classes, 70 normal (32 characteristic)
C1, C2, C2 [×6], C3, C3 [×4], C3 [×4], C4 [×2], C22, C22 [×8], S3 [×36], C6, C6 [×4], C6 [×22], C2×C4, D4 [×4], C23 [×2], C32, C32 [×4], C32 [×4], Dic3, Dic3 [×4], C12 [×5], D6, D6 [×58], C2×C6, C2×C6 [×4], C2×C6 [×9], C2×D4, C3×S3 [×8], C3⋊S3 [×32], C3×C6, C3×C6 [×4], C3×C6 [×18], C4×S3 [×5], D12 [×5], C3⋊D4, C3⋊D4 [×9], C3×D4 [×5], C22×S3 [×14], C33, C3×Dic3 [×4], C3×Dic3 [×4], C3⋊Dic3, C3×C12, S32 [×8], S3×C6 [×4], S3×C6 [×4], C2×C3⋊S3, C2×C3⋊S3 [×46], C62, C62 [×4], C62 [×5], S3×D4 [×5], S3×C32, C3×C3⋊S3, C33⋊C2 [×2], C33⋊C2, C32×C6, C32×C6, C6.D6 [×4], C3⋊D12 [×8], C3×C3⋊D4 [×4], C3×C3⋊D4 [×4], C4×C3⋊S3, C12⋊S3, C327D4, C327D4, D4×C32, C2×S32 [×4], C22×C3⋊S3 [×10], C32×Dic3, C3×C3⋊Dic3, S3×C3⋊S3 [×2], S3×C3×C6, C6×C3⋊S3, C2×C33⋊C2 [×2], C2×C33⋊C2 [×2], C3×C62, Dic3⋊D6 [×4], D4×C3⋊S3, C338(C2×C4), C337D4, C338D4, C32×C3⋊D4, C3×C327D4, C2×S3×C3⋊S3, C22×C33⋊C2, C6223D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×5], D4 [×2], C23, D6 [×15], C2×D4, C3⋊S3, C22×S3 [×5], S32 [×4], C2×C3⋊S3 [×3], S3×D4 [×5], C2×S32 [×4], C22×C3⋊S3, S3×C3⋊S3, Dic3⋊D6 [×4], D4×C3⋊S3, C2×S3×C3⋊S3, C6223D6

Smallest permutation representation of C6223D6
On 36 points
Generators in S36
```(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)
(1 13 9 10 4 16)(2 14 7 11 5 17)(3 15 8 12 6 18)(19 34 26 22 31 29)(20 35 27 23 32 30)(21 36 28 24 33 25)
(1 30 2 26 3 28)(4 35 5 31 6 33)(7 19 8 21 9 23)(10 27 11 29 12 25)(13 32 14 34 15 36)(16 20 17 22 18 24)
(1 2)(4 7)(5 9)(6 8)(10 11)(13 17)(14 16)(15 18)(19 33)(20 32)(21 31)(22 36)(23 35)(24 34)(25 29)(26 28)```

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

`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), (1,13,9,10,4,16)(2,14,7,11,5,17)(3,15,8,12,6,18)(19,34,26,22,31,29)(20,35,27,23,32,30)(21,36,28,24,33,25), (1,30,2,26,3,28)(4,35,5,31,6,33)(7,19,8,21,9,23)(10,27,11,29,12,25)(13,32,14,34,15,36)(16,20,17,22,18,24), (1,2)(4,7)(5,9)(6,8)(10,11)(13,17)(14,16)(15,18)(19,33)(20,32)(21,31)(22,36)(23,35)(24,34)(25,29)(26,28) );`

`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)], [(1,13,9,10,4,16),(2,14,7,11,5,17),(3,15,8,12,6,18),(19,34,26,22,31,29),(20,35,27,23,32,30),(21,36,28,24,33,25)], [(1,30,2,26,3,28),(4,35,5,31,6,33),(7,19,8,21,9,23),(10,27,11,29,12,25),(13,32,14,34,15,36),(16,20,17,22,18,24)], [(1,2),(4,7),(5,9),(6,8),(10,11),(13,17),(14,16),(15,18),(19,33),(20,32),(21,31),(22,36),(23,35),(24,34),(25,29),(26,28)])`

51 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3E 3F 3G 3H 3I 4A 4B 6A ··· 6E 6F ··· 6V 6W 6X 6Y 6Z 6AA 12A 12B 12C 12D 12E order 1 2 2 2 2 2 2 2 3 ··· 3 3 3 3 3 4 4 6 ··· 6 6 ··· 6 6 6 6 6 6 12 12 12 12 12 size 1 1 2 6 18 27 27 54 2 ··· 2 4 4 4 4 6 18 2 ··· 2 4 ··· 4 12 12 12 12 36 12 12 12 12 36

51 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 D6 D6 S32 S3×D4 C2×S32 Dic3⋊D6 kernel C62⋊23D6 C33⋊8(C2×C4) C33⋊7D4 C33⋊8D4 C32×C3⋊D4 C3×C32⋊7D4 C2×S3×C3⋊S3 C22×C33⋊C2 C3×C3⋊D4 C32⋊7D4 C33⋊C2 C3×Dic3 C3⋊Dic3 S3×C6 C2×C3⋊S3 C62 C2×C6 C32 C6 C3 # reps 1 1 1 1 1 1 1 1 4 1 2 4 1 4 1 5 4 5 4 8

Matrix representation of C6223D6 in GL8(𝔽13)

 1 0 0 0 0 0 0 0 3 12 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 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 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 0 1 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 8 0 0 0 0 0 0 0 12 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 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0
,
 1 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 12 12 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 12 0 0 0 0 0 0 0 12 1

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

C6223D6 in GAP, Magma, Sage, TeX

`C_6^2\rtimes_{23}D_6`
`% in TeX`

`G:=Group("C6^2:23D6");`
`// GroupNames label`

`G:=SmallGroup(432,686);`
`// by ID`

`G=gap.SmallGroup(432,686);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,135,571,2028,14118]);`
`// Polycyclic`

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

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