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## G = C42.C8order 128 = 27

### 1st non-split extension by C42 of C8 acting via C8/C2=C4

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.C8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×M4(2) — C4⋊M4(2) — C42.C8
 Lower central C1 — C2 — C22 — C2×C4 — C42.C8
 Upper central C1 — C4 — C2×C4 — C2×M4(2) — C42.C8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C2×C4 — C2×M4(2) — C42.C8

Generators and relations for C42.C8
G = < a,b,c | a4=b4=1, c8=b2, ab=ba, cac-1=a-1b-1, cbc-1=a2b >

Character table of C42.C8

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E 8F 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 2 4 1 1 2 4 4 4 4 4 4 4 4 4 8 8 8 8 8 8 8 8 8 8 ρ1 1 1 1 1 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 -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 -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 1 -1 -1 -1 linear of order 2 ρ5 1 1 1 1 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 -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 -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 -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 -1 -1 1 1 -i -i i i -i i ζ87 ζ83 ζ85 ζ8 ζ8 ζ85 ζ87 ζ83 linear of order 8 ρ10 1 1 1 -1 -1 -1 -1 -1 1 1 -1 1 i i -i -i -i i ζ85 ζ8 ζ87 ζ87 ζ83 ζ83 ζ8 ζ85 linear of order 8 ρ11 1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 -i -i i i -i i ζ83 ζ87 ζ8 ζ85 ζ85 ζ8 ζ83 ζ87 linear of order 8 ρ12 1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 i i -i -i i -i ζ8 ζ85 ζ83 ζ87 ζ87 ζ83 ζ8 ζ85 linear of order 8 ρ13 1 1 1 -1 -1 -1 -1 -1 1 1 -1 1 i i -i -i -i i ζ8 ζ85 ζ83 ζ83 ζ87 ζ87 ζ85 ζ8 linear of order 8 ρ14 1 1 1 -1 -1 -1 -1 -1 1 1 -1 1 -i -i i i i -i ζ83 ζ87 ζ8 ζ8 ζ85 ζ85 ζ87 ζ83 linear of order 8 ρ15 1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 i i -i -i i -i ζ85 ζ8 ζ87 ζ83 ζ83 ζ87 ζ85 ζ8 linear of order 8 ρ16 1 1 1 -1 -1 -1 -1 -1 1 1 -1 1 -i -i i i i -i ζ87 ζ83 ζ85 ζ85 ζ8 ζ8 ζ83 ζ87 linear of order 8 ρ17 2 2 2 -2 2 2 2 0 0 0 0 -2 -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 2 0 0 0 0 -2 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 -2 0 0 0 0 -2 -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 -2 0 0 0 0 -2 2i -2i -2i 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ21 4 4 -4 0 4 4 -4 0 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 0 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C4.10D4, Schur index 2 ρ23 4 -4 0 0 4i -4i 0 -2 2i -2i 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ24 4 -4 0 0 -4i 4i 0 -2 -2i 2i 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ25 4 -4 0 0 -4i 4i 0 2 2i -2i -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ26 4 -4 0 0 4i -4i 0 2 -2i 2i -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C42.C8
On 16 points - transitive group 16T242
Generators in S16
```(1 13 9 5)(2 10)(3 7 11 15)(4 12)(6 14)(8 16)
(1 13 9 5)(2 6 10 14)(3 7 11 15)(4 16 12 8)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)```

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

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

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

`G:=TransitiveGroup(16,242);`

Matrix representation of C42.C8 in GL4(𝔽5) generated by

 4 0 0 0 0 4 0 2 0 0 4 0 0 4 0 1
,
 4 0 2 0 0 4 0 2 4 0 1 0 0 4 0 1
,
 0 0 0 2 1 0 0 0 0 1 0 0 0 0 1 0
`G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,4,0,0,4,0,0,2,0,1],[4,0,4,0,0,4,0,4,2,0,1,0,0,2,0,1],[0,1,0,0,0,0,1,0,0,0,0,1,2,0,0,0] >;`

C42.C8 in GAP, Magma, Sage, TeX

`C_4^2.C_8`
`% in TeX`

`G:=Group("C4^2.C8");`
`// GroupNames label`

`G:=SmallGroup(128,59);`
`// by ID`

`G=gap.SmallGroup(128,59);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,120,422,352,1242,521,136,2804,124]);`
`// Polycyclic`

`G:=Group<a,b,c|a^4=b^4=1,c^8=b^2,a*b=b*a,c*a*c^-1=a^-1*b^-1,c*b*c^-1=a^2*b>;`
`// generators/relations`

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