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## G = C22⋊C4.C8order 128 = 27

### The non-split extension by C22⋊C4 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 — C23 — C22⋊C4.C8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×M4(2) — C42.6C22 — C22⋊C4.C8
 Lower central C1 — C2 — C22 — C23 — C22⋊C4.C8
 Upper central C1 — C4 — C2×C4 — C2×M4(2) — C22⋊C4.C8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C2×C4 — C2×M4(2) — C22⋊C4.C8

Generators and relations for C22⋊C4.C8
G = < a,b,c,d | a2=b2=c4=1, d8=b, cac-1=ab=ba, dad-1=abc2, bc=cb, bd=db, dcd-1=abc >

Character table of C22⋊C4.C8

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 8E 8F 8G 8H 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 2 4 1 1 2 4 8 8 4 4 4 4 4 4 4 4 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 i i i -i -i -i i -i ζ8 ζ8 ζ87 ζ83 ζ83 ζ85 ζ87 ζ85 linear of order 8 ρ10 1 1 1 1 -1 -1 -1 -1 -1 1 -i i i -i i -i -i i ζ8 ζ85 ζ83 ζ83 ζ87 ζ8 ζ87 ζ85 linear of order 8 ρ11 1 1 1 1 -1 -1 -1 -1 1 -1 i i i -i -i -i i -i ζ85 ζ85 ζ83 ζ87 ζ87 ζ8 ζ83 ζ8 linear of order 8 ρ12 1 1 1 1 -1 -1 -1 -1 1 -1 -i -i -i i i i -i i ζ83 ζ83 ζ85 ζ8 ζ8 ζ87 ζ85 ζ87 linear of order 8 ρ13 1 1 1 1 -1 -1 -1 -1 -1 1 -i i i -i i -i -i i ζ85 ζ8 ζ87 ζ87 ζ83 ζ85 ζ83 ζ8 linear of order 8 ρ14 1 1 1 1 -1 -1 -1 -1 -1 1 i -i -i i -i i i -i ζ83 ζ87 ζ8 ζ8 ζ85 ζ83 ζ85 ζ87 linear of order 8 ρ15 1 1 1 1 -1 -1 -1 -1 -1 1 i -i -i i -i i i -i ζ87 ζ83 ζ85 ζ85 ζ8 ζ87 ζ8 ζ83 linear of order 8 ρ16 1 1 1 1 -1 -1 -1 -1 1 -1 -i -i -i i i i -i i ζ87 ζ87 ζ8 ζ85 ζ85 ζ83 ζ8 ζ83 linear of order 8 ρ17 2 2 2 -2 2 2 2 -2 0 0 0 2 -2 2 0 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 -2 2 2 2 -2 0 0 0 -2 2 -2 0 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 -2 -2 -2 -2 2 0 0 0 2i -2i -2i 0 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ20 2 2 2 -2 -2 -2 -2 2 0 0 0 -2i 2i 2i 0 -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 0 0 0 2ζ87 0 0 0 2ζ8 0 2ζ83 2ζ85 0 0 0 0 0 0 0 0 complex faithful ρ24 4 -4 0 0 4i -4i 0 0 0 0 2ζ85 0 0 0 2ζ83 0 2ζ8 2ζ87 0 0 0 0 0 0 0 0 complex faithful ρ25 4 -4 0 0 -4i 4i 0 0 0 0 2ζ83 0 0 0 2ζ85 0 2ζ87 2ζ8 0 0 0 0 0 0 0 0 complex faithful ρ26 4 -4 0 0 4i -4i 0 0 0 0 2ζ8 0 0 0 2ζ87 0 2ζ85 2ζ83 0 0 0 0 0 0 0 0 complex faithful

Smallest permutation representation of C22⋊C4.C8
On 32 points
Generators in S32
```(1 17)(2 18)(3 27)(4 28)(5 21)(6 22)(7 31)(8 32)(9 25)(10 26)(11 19)(12 20)(13 29)(14 30)(15 23)(16 24)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(1 5 9 13)(2 22)(3 7 11 15)(4 32)(6 26)(8 20)(10 30)(12 24)(14 18)(16 28)(17 29 25 21)(19 31 27 23)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32)```

`G:=sub<Sym(32)| (1,17)(2,18)(3,27)(4,28)(5,21)(6,22)(7,31)(8,32)(9,25)(10,26)(11,19)(12,20)(13,29)(14,30)(15,23)(16,24), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,5,9,13)(2,22)(3,7,11,15)(4,32)(6,26)(8,20)(10,30)(12,24)(14,18)(16,28)(17,29,25,21)(19,31,27,23), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)>;`

`G:=Group( (1,17)(2,18)(3,27)(4,28)(5,21)(6,22)(7,31)(8,32)(9,25)(10,26)(11,19)(12,20)(13,29)(14,30)(15,23)(16,24), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,5,9,13)(2,22)(3,7,11,15)(4,32)(6,26)(8,20)(10,30)(12,24)(14,18)(16,28)(17,29,25,21)(19,31,27,23), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32) );`

`G=PermutationGroup([(1,17),(2,18),(3,27),(4,28),(5,21),(6,22),(7,31),(8,32),(9,25),(10,26),(11,19),(12,20),(13,29),(14,30),(15,23),(16,24)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32)], [(1,5,9,13),(2,22),(3,7,11,15),(4,32),(6,26),(8,20),(10,30),(12,24),(14,18),(16,28),(17,29,25,21),(19,31,27,23)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)])`

Matrix representation of C22⋊C4.C8 in GL4(𝔽17) generated by

 1 2 0 0 0 16 0 0 0 0 16 15 0 0 0 1
,
 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 13 0 0 0 4 4 0 0 0 0 13 9 0 0 4 4
,
 0 0 1 0 0 0 0 1 9 0 0 0 8 8 0 0
`G:=sub<GL(4,GF(17))| [1,0,0,0,2,16,0,0,0,0,16,0,0,0,15,1],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[13,4,0,0,0,4,0,0,0,0,13,4,0,0,9,4],[0,0,9,8,0,0,0,8,1,0,0,0,0,1,0,0] >;`

C22⋊C4.C8 in GAP, Magma, Sage, TeX

`C_2^2\rtimes C_4.C_8`
`% in TeX`

`G:=Group("C2^2:C4.C8");`
`// GroupNames label`

`G:=SmallGroup(128,60);`
`// by ID`

`G=gap.SmallGroup(128,60);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,120,422,723,346,521,136,2804,124]);`
`// Polycyclic`

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

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