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## G = C8⋊C4⋊C4order 128 = 27

### 1st semidirect product of C8⋊C4 and C4 acting faithfully

p-group, metabelian, nilpotent (class 5), monomial

Aliases: C8⋊C41C4, C41D41C4, (C2×D4).3D4, (C2×Q8).3D4, C83D4.1C2, C42.3(C2×C4), C423C42C2, C42.C45C2, C2.6(C42⋊C4), C4.4D4.3C22, C22.16(C23⋊C4), (C2×C4).32(C22⋊C4), SmallGroup(128,138)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42 — C8⋊C4⋊C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×D4 — C4.4D4 — C8⋊3D4 — C8⋊C4⋊C4
 Lower central C1 — C2 — C22 — C2×C4 — C42 — C8⋊C4⋊C4
 Upper central C1 — C2 — C22 — C2×C4 — C4.4D4 — C8⋊C4⋊C4
 Jennings C1 — C2 — C2 — C22 — C2×C4 — C4.4D4 — C8⋊C4⋊C4

Generators and relations for C8⋊C4⋊C4
G = < a,b,c | a8=b4=c4=1, bab-1=a5, cac-1=a5b, cbc-1=a6b-1 >

Character table of C8⋊C4⋊C4

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 8A 8B 8C 8D size 1 1 2 8 16 4 8 8 16 16 8 8 16 16 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 1 1 -1 1 1 -1 1 i -i -1 -1 -i i linear of order 4 ρ6 1 1 1 -1 -1 1 -1 1 -i i 1 1 -i i linear of order 4 ρ7 1 1 1 -1 1 1 -1 1 -i i -1 -1 i -i linear of order 4 ρ8 1 1 1 -1 -1 1 -1 1 i -i 1 1 i -i linear of order 4 ρ9 2 2 2 2 0 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 -2 0 2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 4 0 0 -4 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ12 4 4 -4 0 0 0 0 0 0 0 2 -2 0 0 orthogonal lifted from C42⋊C4 ρ13 4 4 -4 0 0 0 0 0 0 0 -2 2 0 0 orthogonal lifted from C42⋊C4 ρ14 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C8⋊C4⋊C4
On 16 points - transitive group 16T362
Generators in S16
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 6)(4 8)(9 15 13 11)(10 12 14 16)
(1 14 6 11)(2 15 5 10)(3 12 4 13)(7 16 8 9)```

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

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

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

`G:=TransitiveGroup(16,362);`

Matrix representation of C8⋊C4⋊C4 in GL8(ℤ)

 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0
,
 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0
,
 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

`G:=sub<GL(8,Integers())| [0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0],[0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;`

C8⋊C4⋊C4 in GAP, Magma, Sage, TeX

`C_8\rtimes C_4\rtimes C_4`
`% in TeX`

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

`G:=SmallGroup(128,138);`
`// by ID`

`G=gap.SmallGroup(128,138);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,422,387,184,1690,745,1684,1411,375,172,4037]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=b^4=c^4=1,b*a*b^-1=a^5,c*a*c^-1=a^5*b,c*b*c^-1=a^6*b^-1>;`
`// generators/relations`

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