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## G = C8.17D4order 64 = 26

### 4th non-split extension by C8 of D4 acting via D4/C22=C2

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

Aliases: C8.17D4, C4.12D8, Q16.2C4, M5(2).3C2, C22.4SD16, C8.3(C2×C4), (C2×C4).12D4, (C2×Q16).5C2, C8.C4.1C2, C4.6(C22⋊C4), (C2×C8).11C22, C2.11(D4⋊C4), SmallGroup(64,43)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C8.17D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×Q16 — C8.17D4
 Lower central C1 — C2 — C4 — C8 — C8.17D4
 Upper central C1 — C2 — C2×C4 — C2×C8 — C8.17D4
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C8.17D4

Generators and relations for C8.17D4
G = < a,b,c | a8=1, b4=a4, c2=bab-1=a-1, ac=ca, cbc-1=a-1b3 >

Character table of C8.17D4

 class 1 2A 2B 4A 4B 4C 4D 8A 8B 8C 8D 8E 16A 16B 16C 16D size 1 1 2 2 2 8 8 2 2 4 8 8 4 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 1 -1 1 -1 1 -1 -1 -1 1 i -i i -i i -i linear of order 4 ρ6 1 1 -1 1 -1 -1 1 -1 -1 1 i -i -i i -i i linear of order 4 ρ7 1 1 -1 1 -1 1 -1 -1 -1 1 -i i -i i -i i linear of order 4 ρ8 1 1 -1 1 -1 -1 1 -1 -1 1 -i i i -i i -i linear of order 4 ρ9 2 2 -2 2 -2 0 0 2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 0 0 -2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 -2 2 0 0 0 0 0 0 0 -√2 √2 √2 -√2 orthogonal lifted from D8 ρ12 2 2 -2 -2 2 0 0 0 0 0 0 0 √2 -√2 -√2 √2 orthogonal lifted from D8 ρ13 2 2 2 -2 -2 0 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 complex lifted from SD16 ρ14 2 2 2 -2 -2 0 0 0 0 0 0 0 √-2 √-2 -√-2 -√-2 complex lifted from SD16 ρ15 4 -4 0 0 0 0 0 -2√2 2√2 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ16 4 -4 0 0 0 0 0 2√2 -2√2 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C8.17D4
On 32 points
Generators in S32
```(1 15 13 11 9 7 5 3)(2 16 14 12 10 8 6 4)(17 31 29 27 25 23 21 19)(18 32 30 28 26 24 22 20)
(1 22 13 26 9 30 5 18)(2 25 6 21 10 17 14 29)(3 20 15 24 11 28 7 32)(4 23 8 19 12 31 16 27)
(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)| (1,15,13,11,9,7,5,3)(2,16,14,12,10,8,6,4)(17,31,29,27,25,23,21,19)(18,32,30,28,26,24,22,20), (1,22,13,26,9,30,5,18)(2,25,6,21,10,17,14,29)(3,20,15,24,11,28,7,32)(4,23,8,19,12,31,16,27), (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( (1,15,13,11,9,7,5,3)(2,16,14,12,10,8,6,4)(17,31,29,27,25,23,21,19)(18,32,30,28,26,24,22,20), (1,22,13,26,9,30,5,18)(2,25,6,21,10,17,14,29)(3,20,15,24,11,28,7,32)(4,23,8,19,12,31,16,27), (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([(1,15,13,11,9,7,5,3),(2,16,14,12,10,8,6,4),(17,31,29,27,25,23,21,19),(18,32,30,28,26,24,22,20)], [(1,22,13,26,9,30,5,18),(2,25,6,21,10,17,14,29),(3,20,15,24,11,28,7,32),(4,23,8,19,12,31,16,27)], [(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)])`

C8.17D4 is a maximal subgroup of
C23.21SD16  Q32⋊C4  Q16.D4  D8.12D4  D4.4D8  D4.5D8  C23.10SD16  Dic20.C4
C4p.D8: C8.3D8  C8.5D8  C24.8D4  C12.4D8  Q16.Dic3  C40.8D4  C20.4D8  Q16.Dic5 ...
C8.17D4 is a maximal quotient of
Q16⋊C8  C23.13SD16  C4.10D16  C8.2C42  Dic20.C4
C4p.D8: C8.27D8  C24.8D4  C12.4D8  Q16.Dic3  C40.8D4  C20.4D8  Q16.Dic5  C56.8D4 ...

Matrix representation of C8.17D4 in GL4(𝔽7) generated by

 2 0 5 1 1 2 2 1 1 6 6 5 5 5 1 5
,
 4 0 5 5 6 2 2 5 1 1 0 5 0 3 4 1
,
 0 2 5 5 2 2 0 1 6 1 4 4 2 1 2 1
`G:=sub<GL(4,GF(7))| [2,1,1,5,0,2,6,5,5,2,6,1,1,1,5,5],[4,6,1,0,0,2,1,3,5,2,0,4,5,5,5,1],[0,2,6,2,2,2,1,1,5,0,4,2,5,1,4,1] >;`

C8.17D4 in GAP, Magma, Sage, TeX

`C_8._{17}D_4`
`% in TeX`

`G:=Group("C8.17D4");`
`// GroupNames label`

`G:=SmallGroup(64,43);`
`// by ID`

`G=gap.SmallGroup(64,43);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,199,362,476,86,489,117,1444,730,88]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=1,b^4=a^4,c^2=b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b^3>;`
`// generators/relations`

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