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## G = C8.Q8order 64 = 26

### The non-split extension by C8 of Q8 acting via Q8/C2=C22

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

Aliases: C8.Q8, C161C4, C4.9SD16, M5(2).1C2, C22.5SD16, C4.6(C4⋊C4), C8.19(C2×C4), (C2×C4).13D4, C4.Q8.1C2, C2.3(C4.Q8), C8.C4.2C2, (C2×C8).12C22, 2-Sylow(AGammaL(1,81)), SmallGroup(64,46)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C8.Q8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — M5(2) — C8.Q8
 Lower central C1 — C2 — C4 — C8 — C8.Q8
 Upper central C1 — C2 — C2×C4 — C2×C8 — C8.Q8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C8.Q8

Generators and relations for C8.Q8
G = < a,b,c | a8=1, b4=a2, c2=a-1b2, bab-1=a5, cac-1=a3, cbc-1=a4b3 >

Character table of C8.Q8

 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 i -i 1 1 -1 -i i 1 -1 -1 1 linear of order 4 ρ6 1 1 -1 1 -1 -i i 1 1 -1 -i i -1 1 1 -1 linear of order 4 ρ7 1 1 -1 1 -1 i -i 1 1 -1 i -i -1 1 1 -1 linear of order 4 ρ8 1 1 -1 1 -1 -i i 1 1 -1 i -i 1 -1 -1 1 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 symplectic lifted from Q8, Schur index 2 ρ11 2 2 -2 -2 2 0 0 0 0 0 0 0 √-2 -√-2 √-2 -√-2 complex lifted from SD16 ρ12 2 2 2 -2 -2 0 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 complex lifted from SD16 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 complex faithful ρ16 4 -4 0 0 0 0 0 -2√-2 2√-2 0 0 0 0 0 0 0 complex faithful

Permutation representations of C8.Q8
On 16 points - transitive group 16T136
Generators in S16
```(1 3 5 7 9 11 13 15)(2 12 6 16 10 4 14 8)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)
(2 4 10 12)(3 7)(5 13)(6 16 14 8)(11 15)```

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

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

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

`G:=TransitiveGroup(16,136);`

C8.Q8 is a maximal subgroup of
M5(2)⋊3C4  Q32⋊C4  D16⋊C4  M5(2).C22  C23.10SD16  C804C4  C805C4
C4p.SD16: D83Q8  D8.2Q8  C8.Dic6  C24.6Q8  C24.Q8  C8.Dic10  C40.6Q8  C40.Q8 ...
C8.Q8 is a maximal quotient of
C161C8
C4p.(C4⋊C4): C8.11C42  C8.Dic6  C24.6Q8  C24.Q8  C8.Dic10  C40.6Q8  C40.Q8  C804C4 ...

Matrix representation of C8.Q8 in GL4(𝔽3) generated by

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

C8.Q8 in GAP, Magma, Sage, TeX

`C_8.Q_8`
`% in TeX`

`G:=Group("C8.Q8");`
`// GroupNames label`

`G:=SmallGroup(64,46);`
`// by ID`

`G=gap.SmallGroup(64,46);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,31,362,86,489,1444,88]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=1,b^4=a^2,c^2=a^-1*b^2,b*a*b^-1=a^5,c*a*c^-1=a^3,c*b*c^-1=a^4*b^3>;`
`// generators/relations`

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