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## G = C23⋊C8order 64 = 26

### The semidirect product of C23 and C8 acting via C8/C2=C4

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C23⋊C8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C22⋊C4 — C23⋊C8
 Lower central C1 — C2 — C22 — C23⋊C8
 Upper central C1 — C22 — C22×C4 — C23⋊C8
 Jennings C1 — C2 — C22 — C22×C4 — C23⋊C8

Generators and relations for C23⋊C8
G = < a,b,c,d | a2=b2=c2=d8=1, ab=ba, ac=ca, dad-1=abc, dbd-1=bc=cb, cd=dc >

Character table of C23⋊C8

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 2 2 4 4 2 2 2 2 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 1 linear of order 2 ρ4 1 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 linear of order 2 ρ5 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 i i -i i -i -i i -i linear of order 4 ρ6 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 -i -i i -i i i -i i linear of order 4 ρ7 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 i i -i -i i i -i -i linear of order 4 ρ8 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -i -i i i -i -i i i linear of order 4 ρ9 1 1 -1 -1 -1 1 1 -1 i -i i -i -i i ζ85 ζ8 ζ87 ζ87 ζ85 ζ8 ζ83 ζ83 linear of order 8 ρ10 1 1 -1 -1 -1 1 1 -1 -i i -i i i -i ζ83 ζ87 ζ8 ζ8 ζ83 ζ87 ζ85 ζ85 linear of order 8 ρ11 1 1 -1 -1 -1 1 1 -1 i -i i -i -i i ζ8 ζ85 ζ83 ζ83 ζ8 ζ85 ζ87 ζ87 linear of order 8 ρ12 1 1 -1 -1 -1 1 1 -1 -i i -i i i -i ζ87 ζ83 ζ85 ζ85 ζ87 ζ83 ζ8 ζ8 linear of order 8 ρ13 1 1 -1 -1 -1 1 -1 1 i -i i -i i -i ζ85 ζ8 ζ87 ζ83 ζ8 ζ85 ζ87 ζ83 linear of order 8 ρ14 1 1 -1 -1 -1 1 -1 1 -i i -i i -i i ζ87 ζ83 ζ85 ζ8 ζ83 ζ87 ζ85 ζ8 linear of order 8 ρ15 1 1 -1 -1 -1 1 -1 1 i -i i -i i -i ζ8 ζ85 ζ83 ζ87 ζ85 ζ8 ζ83 ζ87 linear of order 8 ρ16 1 1 -1 -1 -1 1 -1 1 -i i -i i -i i ζ83 ζ87 ζ8 ζ85 ζ87 ζ83 ζ8 ζ85 linear of order 8 ρ17 2 2 2 2 -2 -2 0 0 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 -2 -2 0 0 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 -2 2 -2 0 0 -2i 2i 2i -2i 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ20 2 2 -2 -2 2 -2 0 0 2i -2i -2i 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ21 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ22 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4.D4

Permutation representations of C23⋊C8
On 16 points - transitive group 16T84
Generators in S16
```(2 10)(3 15)(4 8)(6 14)(7 11)(12 16)
(1 5)(2 10)(3 7)(4 12)(6 14)(8 16)(9 13)(11 15)
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)```

`G:=sub<Sym(16)| (2,10)(3,15)(4,8)(6,14)(7,11)(12,16), (1,5)(2,10)(3,7)(4,12)(6,14)(8,16)(9,13)(11,15), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)>;`

`G:=Group( (2,10)(3,15)(4,8)(6,14)(7,11)(12,16), (1,5)(2,10)(3,7)(4,12)(6,14)(8,16)(9,13)(11,15), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16) );`

`G=PermutationGroup([[(2,10),(3,15),(4,8),(6,14),(7,11),(12,16)], [(1,5),(2,10),(3,7),(4,12),(6,14),(8,16),(9,13),(11,15)], [(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)]])`

`G:=TransitiveGroup(16,84);`

On 16 points - transitive group 16T85
Generators in S16
```(1 5)(2 10)(3 11)(4 8)(6 14)(7 15)(9 13)(12 16)
(2 14)(4 16)(6 10)(8 12)
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)```

`G:=sub<Sym(16)| (1,5)(2,10)(3,11)(4,8)(6,14)(7,15)(9,13)(12,16), (2,14)(4,16)(6,10)(8,12), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)>;`

`G:=Group( (1,5)(2,10)(3,11)(4,8)(6,14)(7,15)(9,13)(12,16), (2,14)(4,16)(6,10)(8,12), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16) );`

`G=PermutationGroup([[(1,5),(2,10),(3,11),(4,8),(6,14),(7,15),(9,13),(12,16)], [(2,14),(4,16),(6,10),(8,12)], [(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)]])`

`G:=TransitiveGroup(16,85);`

Matrix representation of C23⋊C8 in GL6(𝔽17)

 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 9 1 0 0 0 0 0 0 1 0 0 0 0 0 8 16
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 0 1 0 0 0 0 13 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 13 1 0 0 0 0 0 4 0 0

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,9,0,0,0,0,0,1,0,0,0,0,0,0,1,8,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,1,4,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C23⋊C8 in GAP, Magma, Sage, TeX

`C_2^3\rtimes C_8`
`% in TeX`

`G:=Group("C2^3:C8");`
`// GroupNames label`

`G:=SmallGroup(64,4);`
`// by ID`

`G=gap.SmallGroup(64,4);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,362,297,117]);`
`// Polycyclic`

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

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