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## G = C23order 8 = 23

### Elementary abelian group of type [2,2,2]

Aliases: C23, SmallGroup(8,5)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23
 Chief series C1 — C2 — C22 — C23
 Lower central C1 — C23
 Upper central C1 — C23
 Jennings C1 — C23

Generators and relations for C23
G = < a,b,c | a2=b2=c2=1, ab=ba, ac=ca, bc=cb >

Character table of C23

 class 1 2A 2B 2C 2D 2E 2F 2G size 1 1 1 1 1 1 1 1 ρ1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 -1 -1 -1 1 linear of order 2 ρ3 1 -1 -1 -1 1 1 -1 1 linear of order 2 ρ4 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ5 1 -1 1 -1 1 -1 1 -1 linear of order 2 ρ6 1 1 1 -1 -1 1 -1 -1 linear of order 2 ρ7 1 1 -1 1 1 -1 -1 -1 linear of order 2 ρ8 1 -1 -1 1 -1 1 1 -1 linear of order 2

Permutation representations of C23
Regular action on 8 points - transitive group 8T3
Generators in S8
```(1 2)(3 4)(5 6)(7 8)
(1 3)(2 4)(5 7)(6 8)
(1 7)(2 8)(3 5)(4 6)```

`G:=sub<Sym(8)| (1,2)(3,4)(5,6)(7,8), (1,3)(2,4)(5,7)(6,8), (1,7)(2,8)(3,5)(4,6)>;`

`G:=Group( (1,2)(3,4)(5,6)(7,8), (1,3)(2,4)(5,7)(6,8), (1,7)(2,8)(3,5)(4,6) );`

`G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8)], [(1,3),(2,4),(5,7),(6,8)], [(1,7),(2,8),(3,5),(4,6)]])`

`G:=TransitiveGroup(8,3);`

C23 is a maximal subgroup of   C22⋊C4  F8
C23 is a maximal quotient of   C4○D4

Polynomial with Galois group C23 over ℚ
actionf(x)Disc(f)
8T3x8-x4+1216·34

Matrix representation of C23 in GL3(ℤ) generated by

 1 0 0 0 1 0 0 0 -1
,
 1 0 0 0 -1 0 0 0 -1
,
 -1 0 0 0 -1 0 0 0 1
`G:=sub<GL(3,Integers())| [1,0,0,0,1,0,0,0,-1],[1,0,0,0,-1,0,0,0,-1],[-1,0,0,0,-1,0,0,0,1] >;`

C23 in GAP, Magma, Sage, TeX

`C_2^3`
`% in TeX`

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

`G:=SmallGroup(8,5);`
`// by ID`

`G=gap.SmallGroup(8,5);`
`# by ID`

`G:=PCGroup([3,-2,2,2]);`
`// Polycyclic`

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

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