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G = C17⋊C8order 136 = 23·17

The semidirect product of C17 and C8 acting faithfully

Aliases: C17⋊C8, D17.C4, C17⋊C4.C2, SmallGroup(136,12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C17 — C17⋊C8
 Chief series C1 — C17 — D17 — C17⋊C4 — C17⋊C8
 Lower central C17 — C17⋊C8
 Upper central C1

Generators and relations for C17⋊C8
G = < a,b | a17=b8=1, bab-1=a2 >

Character table of C17⋊C8

 class 1 2 4A 4B 8A 8B 8C 8D 17A 17B size 1 17 17 17 17 17 17 17 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ3 1 1 -1 -1 i -i -i i 1 1 linear of order 4 ρ4 1 1 -1 -1 -i i i -i 1 1 linear of order 4 ρ5 1 -1 -i i ζ87 ζ8 ζ85 ζ83 1 1 linear of order 8 ρ6 1 -1 -i i ζ83 ζ85 ζ8 ζ87 1 1 linear of order 8 ρ7 1 -1 i -i ζ8 ζ87 ζ83 ζ85 1 1 linear of order 8 ρ8 1 -1 i -i ζ85 ζ83 ζ87 ζ8 1 1 linear of order 8 ρ9 8 0 0 0 0 0 0 0 -1+√17/2 -1-√17/2 orthogonal faithful ρ10 8 0 0 0 0 0 0 0 -1-√17/2 -1+√17/2 orthogonal faithful

Permutation representations of C17⋊C8
On 17 points: primitive - transitive group 17T4
Generators in S17
```(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17)
(2 10 14 16 17 9 5 3)(4 11 6 12 15 8 13 7)```

`G:=sub<Sym(17)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17), (2,10,14,16,17,9,5,3)(4,11,6,12,15,8,13,7)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17), (2,10,14,16,17,9,5,3)(4,11,6,12,15,8,13,7) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)], [(2,10,14,16,17,9,5,3),(4,11,6,12,15,8,13,7)]])`

`G:=TransitiveGroup(17,4);`

C17⋊C8 is a maximal subgroup of   F17  C51⋊C8
C17⋊C8 is a maximal quotient of   C34.C8  C51⋊C8

Matrix representation of C17⋊C8 in GL8(𝔽2)

 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0
,
 1 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 1 1

`G:=sub<GL(8,GF(2))| [0,0,0,1,0,0,0,0,0,1,1,0,0,1,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,1,1,1,0,1,0,1,1,1,0,0,1,1,1,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,1,0,0,1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,1,1,1,1,0,1,0,0,0,1,1] >;`

C17⋊C8 in GAP, Magma, Sage, TeX

`C_{17}\rtimes C_8`
`% in TeX`

`G:=Group("C17:C8");`
`// GroupNames label`

`G:=SmallGroup(136,12);`
`// by ID`

`G=gap.SmallGroup(136,12);`
`# by ID`

`G:=PCGroup([4,-2,-2,-2,-17,8,21,1155,839,523]);`
`// Polycyclic`

`G:=Group<a,b|a^17=b^8=1,b*a*b^-1=a^2>;`
`// generators/relations`

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