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## G = C23.Q8order 64 = 26

### 3rd non-split extension by C23 of Q8 acting via Q8/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C23.Q8
 Chief series C1 — C2 — C22 — C23 — C24 — C2×C22⋊C4 — C23.Q8
 Lower central C1 — C23 — C23.Q8
 Upper central C1 — C23 — C23.Q8
 Jennings C1 — C23 — C23.Q8

Generators and relations for C23.Q8
G = < a,b,c,d,e | a2=b2=c2=d4=1, e2=bd2, eae-1=ab=ba, dad-1=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 165 in 93 conjugacy classes, 39 normal (7 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C22⋊C4, C4⋊C4, C22×C4, C24, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C23.Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C4⋊D4, C22⋊Q8, C422C2, C23.Q8

Character table of C23.Q8

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L size 1 1 1 1 1 1 1 1 4 4 4 4 4 4 4 4 4 4 4 4 4 4 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 1 -1 -1 -1 -1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ6 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 ρ7 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 ρ8 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 ρ9 2 2 -2 -2 2 2 -2 -2 0 0 0 0 0 0 0 -2 2 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 -2 2 -2 2 2 -2 0 0 0 0 0 0 2 0 0 0 0 0 0 -2 orthogonal lifted from D4 ρ11 2 2 -2 -2 2 2 -2 -2 0 0 0 0 0 0 0 2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 -2 -2 2 -2 2 2 -2 0 0 0 0 0 0 -2 0 0 0 0 0 0 2 orthogonal lifted from D4 ρ13 2 2 2 -2 -2 -2 2 -2 0 0 0 -2 0 0 0 0 0 0 2 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 -2 -2 -2 2 -2 0 0 0 2 0 0 0 0 0 0 -2 0 0 0 orthogonal lifted from D4 ρ15 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 ρ16 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 ρ17 2 -2 -2 -2 2 -2 2 2 0 0 2i 0 0 0 0 0 0 -2i 0 0 0 0 complex lifted from C4○D4 ρ18 2 -2 2 -2 -2 2 -2 2 0 0 0 0 -2i 0 0 0 0 0 0 2i 0 0 complex lifted from C4○D4 ρ19 2 2 -2 2 -2 -2 -2 2 0 0 0 0 0 2i 0 0 0 0 0 0 -2i 0 complex lifted from C4○D4 ρ20 2 2 -2 2 -2 -2 -2 2 0 0 0 0 0 -2i 0 0 0 0 0 0 2i 0 complex lifted from C4○D4 ρ21 2 -2 2 -2 -2 2 -2 2 0 0 0 0 2i 0 0 0 0 0 0 -2i 0 0 complex lifted from C4○D4 ρ22 2 -2 -2 -2 2 -2 2 2 0 0 -2i 0 0 0 0 0 0 2i 0 0 0 0 complex lifted from C4○D4

Smallest permutation representation of C23.Q8
On 32 points
Generators in S32
```(2 26)(4 28)(5 32)(6 16)(7 30)(8 14)(9 15)(10 29)(11 13)(12 31)(18 24)(20 22)
(1 21)(2 22)(3 23)(4 24)(5 15)(6 16)(7 13)(8 14)(9 32)(10 29)(11 30)(12 31)(17 27)(18 28)(19 25)(20 26)
(1 25)(2 26)(3 27)(4 28)(5 9)(6 10)(7 11)(8 12)(13 30)(14 31)(15 32)(16 29)(17 23)(18 24)(19 21)(20 22)
(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 29 23 12)(2 32 24 11)(3 31 21 10)(4 30 22 9)(5 28 13 20)(6 27 14 19)(7 26 15 18)(8 25 16 17)```

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

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

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

Matrix representation of C23.Q8 in GL6(𝔽5)

 1 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 0 1 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 3 0 0 0 0 0 0 2 0 0 0 0 0 0 0 4 0 0 0 0 4 0 0 0 0 0 0 0 0 2 0 0 0 0 2 0

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

C23.Q8 in GAP, Magma, Sage, TeX

`C_2^3.Q_8`
`% in TeX`

`G:=Group("C2^3.Q8");`
`// GroupNames label`

`G:=SmallGroup(64,77);`
`// by ID`

`G=gap.SmallGroup(64,77);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,2,144,121,55,362,332]);`
`// Polycyclic`

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

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