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## G = C62⋊8D4order 288 = 25·32

### 5th semidirect product of C62 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62⋊8D4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×C3⋊D12 — C62⋊8D4
 Lower central C32 — C62 — C62⋊8D4
 Upper central C1 — C22 — C23

Generators and relations for C628D4
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a-1b3, dad=a-1, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 1634 in 327 conjugacy classes, 56 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×7], C3 [×2], C3, C4 [×3], C22, C22 [×2], C22 [×21], S3 [×17], C6 [×2], C6 [×4], C6 [×12], C2×C4 [×3], D4 [×6], C23, C23 [×9], C32, Dic3 [×3], C12 [×3], D6 [×63], C2×C6 [×2], C2×C6 [×4], C2×C6 [×14], C22⋊C4 [×3], C2×D4 [×3], C24, C3×S3, C3⋊S3 [×4], C3×C6, C3×C6 [×2], C3×C6 [×2], D12 [×4], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×2], C22×S3, C22×S3 [×30], C22×C6 [×2], C22×C6 [×2], C22≀C2, C3×Dic3 [×3], S3×C6 [×3], C2×C3⋊S3 [×4], C2×C3⋊S3 [×12], C62, C62 [×2], C62 [×2], D6⋊C4 [×4], C6.D4, C3×C22⋊C4, C2×D12 [×2], C2×C3⋊D4, C2×C3⋊D4 [×2], C6×D4, S3×C23 [×3], C3⋊D12 [×4], C6×Dic3, C6×Dic3 [×2], C3×C3⋊D4 [×2], S3×C2×C6, C22×C3⋊S3 [×2], C22×C3⋊S3 [×6], C2×C62, D6⋊D4, C232D6, C6.D12 [×2], C3×C6.D4, C2×C3⋊D12 [×2], C6×C3⋊D4, C23×C3⋊S3, C628D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×6], C23, D6 [×6], C2×D4 [×3], D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C22≀C2, S32, C2×D12, S3×D4 [×4], C2×C3⋊D4, C3⋊D12 [×2], C2×S32, D6⋊D4, C232D6, C2×C3⋊D12, Dic3⋊D6 [×2], C628D4

Permutation representations of C628D4
On 24 points - transitive group 24T672
Generators in S24
```(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 4 3 6 2 5)(7 11 9 10 8 12)(13 18 17 16 15 14)(19 24 23 22 21 20)
(1 15 12 22)(2 17 10 24)(3 13 11 20)(4 14 7 21)(5 16 8 23)(6 18 9 19)
(1 2)(4 6)(7 9)(10 12)(13 20)(14 19)(15 24)(16 23)(17 22)(18 21)```

`G:=sub<Sym(24)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,4,3,6,2,5)(7,11,9,10,8,12)(13,18,17,16,15,14)(19,24,23,22,21,20), (1,15,12,22)(2,17,10,24)(3,13,11,20)(4,14,7,21)(5,16,8,23)(6,18,9,19), (1,2)(4,6)(7,9)(10,12)(13,20)(14,19)(15,24)(16,23)(17,22)(18,21)>;`

`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), (1,4,3,6,2,5)(7,11,9,10,8,12)(13,18,17,16,15,14)(19,24,23,22,21,20), (1,15,12,22)(2,17,10,24)(3,13,11,20)(4,14,7,21)(5,16,8,23)(6,18,9,19), (1,2)(4,6)(7,9)(10,12)(13,20)(14,19)(15,24)(16,23)(17,22)(18,21) );`

`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)], [(1,4,3,6,2,5),(7,11,9,10,8,12),(13,18,17,16,15,14),(19,24,23,22,21,20)], [(1,15,12,22),(2,17,10,24),(3,13,11,20),(4,14,7,21),(5,16,8,23),(6,18,9,19)], [(1,2),(4,6),(7,9),(10,12),(13,20),(14,19),(15,24),(16,23),(17,22),(18,21)])`

`G:=TransitiveGroup(24,672);`

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 3A 3B 3C 4A 4B 4C 6A ··· 6F 6G ··· 6Q 6R 6S 12A ··· 12F order 1 2 2 2 2 2 2 2 2 2 2 3 3 3 4 4 4 6 ··· 6 6 ··· 6 6 6 12 ··· 12 size 1 1 1 1 2 2 12 18 18 18 18 2 2 4 12 12 12 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 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 D12 C3⋊D4 S32 S3×D4 C3⋊D12 C2×S32 Dic3⋊D6 kernel C62⋊8D4 C6.D12 C3×C6.D4 C2×C3⋊D12 C6×C3⋊D4 C23×C3⋊S3 C6.D4 C2×C3⋊D4 C2×C3⋊S3 C62 C2×Dic3 C22×S3 C22×C6 C2×C6 C2×C6 C23 C6 C22 C22 C2 # reps 1 2 1 2 1 1 1 1 4 2 3 1 2 4 4 1 4 2 1 4

Matrix representation of C628D4 in GL6(𝔽13)

 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 10 6 0 0 0 0 7 3 0 0 0 0 0 0 9 2 0 0 0 0 11 4
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 12 0

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,10,7,0,0,0,0,6,3,0,0,0,0,0,0,9,11,0,0,0,0,2,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;`

C628D4 in GAP, Magma, Sage, TeX

`C_6^2\rtimes_8D_4`
`% in TeX`

`G:=Group("C6^2:8D4");`
`// GroupNames label`

`G:=SmallGroup(288,629);`
`// by ID`

`G=gap.SmallGroup(288,629);`
`# by ID`

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

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

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