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## G = C23.4Q8order 64 = 26

### 4th 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 — C23.4Q8
 Chief series C1 — C2 — C22 — C23 — C24 — C2×C22⋊C4 — C23.4Q8
 Lower central C1 — C23 — C23.4Q8
 Upper central C1 — C23 — C23.4Q8
 Jennings C1 — C23 — C23.4Q8

Generators and relations for C23.4Q8
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=bd-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, C23.4Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C22⋊Q8, C22.D4, C41D4, C23.4Q8

Character table of C23.4Q8

 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 -2 0 0 0 0 0 0 2 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 -2 0 0 0 0 0 0 2 orthogonal lifted from D4 ρ12 2 -2 -2 -2 2 -2 2 2 0 0 2 0 0 0 0 0 0 -2 0 0 0 0 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 -2 0 0 0 0 0 0 2 0 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 0 0 0 0 0 2i -2i 0 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 0 0 -2i 2i 0 0 0 0 0 complex lifted from C4○D4 ρ21 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 ρ22 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

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

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

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

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

Matrix representation of C23.4Q8 in GL6(𝔽5)

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

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

C23.4Q8 in GAP, Magma, Sage, TeX

`C_2^3._4Q_8`
`% in TeX`

`G:=Group("C2^3.4Q8");`
`// GroupNames label`

`G:=SmallGroup(64,80);`
`// by ID`

`G=gap.SmallGroup(64,80);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,2,48,121,151,362,332]);`
`// 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=b*d^-1>;`
`// generators/relations`

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