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## G = C23.C8order 64 = 26

### The non-split extension by C23 of C8 acting via C8/C2=C4

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

Aliases: C23.C8, C8.24D4, M5(2)⋊3C2, C4.7M4(2), (C2×C4).C8, (C2×C8).3C4, (C22×C4).4C4, C22.4(C2×C8), C2.7(C22⋊C8), (C2×C8).42C22, C4.29(C22⋊C4), (C2×M4(2)).10C2, (C2×C4).67(C2×C4), SmallGroup(64,30)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C23.C8
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×M4(2) — C23.C8
 Lower central C1 — C2 — C22 — C23.C8
 Upper central C1 — C4 — C2×C8 — C23.C8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C23.C8

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

Character table of C23.C8

 class 1 2A 2B 2C 4A 4B 4C 4D 8A 8B 8C 8D 8E 8F 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 2 4 1 1 2 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 ζ8 ζ83 ζ8 ζ85 ζ83 ζ87 ζ87 ζ85 linear of order 8 ρ10 1 1 1 1 -1 -1 -1 -1 i -i -i i i -i ζ85 ζ83 ζ8 ζ85 ζ87 ζ87 ζ83 ζ8 linear of order 8 ρ11 1 1 1 -1 -1 -1 -1 1 -i i i -i i -i ζ83 ζ8 ζ83 ζ87 ζ8 ζ85 ζ85 ζ87 linear of order 8 ρ12 1 1 1 -1 -1 -1 -1 1 -i i i -i i -i ζ87 ζ85 ζ87 ζ83 ζ85 ζ8 ζ8 ζ83 linear of order 8 ρ13 1 1 1 1 -1 -1 -1 -1 -i i i -i -i i ζ83 ζ85 ζ87 ζ83 ζ8 ζ8 ζ85 ζ87 linear of order 8 ρ14 1 1 1 1 -1 -1 -1 -1 i -i -i i i -i ζ8 ζ87 ζ85 ζ8 ζ83 ζ83 ζ87 ζ85 linear of order 8 ρ15 1 1 1 -1 -1 -1 -1 1 i -i -i i -i i ζ85 ζ87 ζ85 ζ8 ζ87 ζ83 ζ83 ζ8 linear of order 8 ρ16 1 1 1 1 -1 -1 -1 -1 -i i i -i -i i ζ87 ζ8 ζ83 ζ87 ζ85 ζ85 ζ8 ζ83 linear of order 8 ρ17 2 2 -2 0 2 2 -2 0 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 0 2 2 -2 0 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 0 -2 -2 2 0 2i 2i -2i -2i 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ20 2 2 -2 0 -2 -2 2 0 -2i -2i 2i 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ21 4 -4 0 0 -4i 4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ22 4 -4 0 0 4i -4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C23.C8
On 16 points - transitive group 16T104
Generators in S16
```(2 10)(3 11)(6 14)(7 15)
(2 10)(4 12)(6 14)(8 16)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)```

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

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

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

`G:=TransitiveGroup(16,104);`

C23.C8 is a maximal subgroup of
C24.C8  C23.1M4(2)  C42.C8  C22⋊C4.C8  C23.D8  C23.2D8  C23.SD16  C23.2SD16  M5(2).19C22  D8⋊D4  D8.D4  M5(2).C22  C23.10SD16
C4p.M4(2): C8.5M4(2)  C8.19M4(2)  C8.25D12  C24.D4  C8.25D20  C40.D4  C20.10M4(2)  C20.29M4(2) ...
C23.C8 is a maximal quotient of
C23⋊C16  C22.M5(2)  C8.17Q16  C20.10M4(2)
C8.D4p: C8.31D8  C8.25D12  C8.25D20  M5(2)⋊D7 ...
(C2×C4p).C8: M5(2)⋊C4  C24.D4  C40.D4  C20.29M4(2)  C56.D4 ...

Matrix representation of C23.C8 in GL4(𝔽5) generated by

 4 0 0 0 0 4 0 0 0 0 1 0 0 0 0 1
,
 4 0 0 0 0 1 0 0 0 0 4 0 0 0 0 1
,
 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 0 0 0 3 1 0 0 0 0 1 0 0 0 0 1 0
`G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,1,0,0,0,0,4,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,1,0,0,0,0,1,0,0,0,0,1,3,0,0,0] >;`

C23.C8 in GAP, Magma, Sage, TeX

`C_2^3.C_8`
`% in TeX`

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

`G:=SmallGroup(64,30);`
`// by ID`

`G=gap.SmallGroup(64,30);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,650,489,69,88]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^2=b^2=c^2=1,d^8=c,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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