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## G = C8.4Q8order 64 = 26

### 3rd non-split extension by C8 of Q8 acting via Q8/C4=C2

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

Aliases: C8.4Q8, C16.1C4, C4.19D8, C22.1Q16, C4.9(C4⋊C4), (C2×C16).5C2, C8.16(C2×C4), (C2×C4).65D4, C2.5(C2.D8), C8.C4.3C2, (C2×C8).87C22, SmallGroup(64,49)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C8.4Q8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×C16 — C8.4Q8
 Lower central C1 — C2 — C4 — C8 — C8.4Q8
 Upper central C1 — C4 — C2×C4 — C2×C8 — C8.4Q8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C8.4Q8

Generators and relations for C8.4Q8
G = < a,b,c | a8=1, b4=a2, c2=ab2, ab=ba, cac-1=a3, cbc-1=a6b3 >

Character table of C8.4Q8

 class 1 2A 2B 4A 4B 4C 8A 8B 8C 8D 8E 8F 8G 8H 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 2 1 1 2 2 2 2 2 8 8 8 8 2 2 2 2 2 2 2 2 ρ1 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 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 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 linear of order 2 ρ5 1 1 -1 -1 -1 1 -1 -1 1 1 -i -i i i -1 1 1 -1 -1 -1 1 1 linear of order 4 ρ6 1 1 -1 -1 -1 1 -1 -1 1 1 i i -i -i -1 1 1 -1 -1 -1 1 1 linear of order 4 ρ7 1 1 -1 -1 -1 1 -1 -1 1 1 -i i -i i 1 -1 -1 1 1 1 -1 -1 linear of order 4 ρ8 1 1 -1 -1 -1 1 -1 -1 1 1 i -i i -i 1 -1 -1 1 1 1 -1 -1 linear of order 4 ρ9 2 2 2 2 2 2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 2 2 -2 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ11 2 2 -2 2 2 -2 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ12 2 2 -2 -2 -2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ13 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 -√2 -√2 √2 √2 -√2 √2 √2 -√2 symplectic lifted from Q16, Schur index 2 ρ14 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 √2 √2 -√2 -√2 √2 -√2 -√2 √2 symplectic lifted from Q16, Schur index 2 ρ15 2 -2 0 -2i 2i 0 √-2 -√-2 -√2 √2 0 0 0 0 ζ1615+ζ169 ζ167-ζ16 ζ165-ζ163 ζ1613+ζ1611 ζ167+ζ16 ζ165+ζ163 -ζ165+ζ163 -ζ167+ζ16 complex faithful ρ16 2 -2 0 2i -2i 0 -√-2 √-2 -√2 √2 0 0 0 0 ζ167+ζ16 ζ167-ζ16 ζ165-ζ163 ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 -ζ165+ζ163 -ζ167+ζ16 complex faithful ρ17 2 -2 0 -2i 2i 0 -√-2 √-2 √2 -√2 0 0 0 0 ζ165+ζ163 ζ165-ζ163 -ζ167+ζ16 ζ1615+ζ169 ζ1613+ζ1611 ζ167+ζ16 ζ167-ζ16 -ζ165+ζ163 complex faithful ρ18 2 -2 0 2i -2i 0 -√-2 √-2 -√2 √2 0 0 0 0 ζ1615+ζ169 -ζ167+ζ16 -ζ165+ζ163 ζ1613+ζ1611 ζ167+ζ16 ζ165+ζ163 ζ165-ζ163 ζ167-ζ16 complex faithful ρ19 2 -2 0 -2i 2i 0 √-2 -√-2 -√2 √2 0 0 0 0 ζ167+ζ16 -ζ167+ζ16 -ζ165+ζ163 ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 ζ165-ζ163 ζ167-ζ16 complex faithful ρ20 2 -2 0 2i -2i 0 √-2 -√-2 √2 -√2 0 0 0 0 ζ1613+ζ1611 ζ165-ζ163 -ζ167+ζ16 ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 ζ167-ζ16 -ζ165+ζ163 complex faithful ρ21 2 -2 0 2i -2i 0 √-2 -√-2 √2 -√2 0 0 0 0 ζ165+ζ163 -ζ165+ζ163 ζ167-ζ16 ζ1615+ζ169 ζ1613+ζ1611 ζ167+ζ16 -ζ167+ζ16 ζ165-ζ163 complex faithful ρ22 2 -2 0 -2i 2i 0 -√-2 √-2 √2 -√2 0 0 0 0 ζ1613+ζ1611 -ζ165+ζ163 ζ167-ζ16 ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 -ζ167+ζ16 ζ165-ζ163 complex faithful

Smallest permutation representation of C8.4Q8
On 32 points
Generators in S32
```(1 11 5 15 9 3 13 7)(2 12 6 16 10 4 14 8)(17 19 21 23 25 27 29 31)(18 20 22 24 26 28 30 32)
(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)
(1 27 13 31 9 19 5 23)(2 26 14 30 10 18 6 22)(3 25 15 29 11 17 7 21)(4 24 16 28 12 32 8 20)```

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

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

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

C8.4Q8 is a maximal subgroup of
D16.C4  M6(2)⋊C2  C16.18D4  M5(2).1C4  C8○D16  D165C4  D4.3D8  D4.4D8  D4.5D8
C16p.C4: C32.C4  C48.C4  C80.6C4  C16.F5  C80.2C4  C112.C4 ...
C8p.Q8: C8.Q16  C24.7Q8  C40.7Q8  C8.7Dic14 ...
C8.4Q8 is a maximal quotient of
C164C8
C4.D8p: C163C8  C48.C4  C80.6C4  C112.C4 ...
C4p.(C4⋊C4): C8.8C42  C24.7Q8  C40.7Q8  C16.F5  C80.2C4  C8.7Dic14 ...

Matrix representation of C8.4Q8 in GL2(𝔽17) generated by

 9 0 0 15
,
 5 0 0 7
,
 0 1 4 0
`G:=sub<GL(2,GF(17))| [9,0,0,15],[5,0,0,7],[0,4,1,0] >;`

C8.4Q8 in GAP, Magma, Sage, TeX

`C_8._4Q_8`
`% in TeX`

`G:=Group("C8.4Q8");`
`// GroupNames label`

`G:=SmallGroup(64,49);`
`// by ID`

`G=gap.SmallGroup(64,49);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,127,362,230,117,1444,88]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=1,b^4=a^2,c^2=a*b^2,a*b=b*a,c*a*c^-1=a^3,c*b*c^-1=a^6*b^3>;`
`// generators/relations`

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