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## G = C4.Q8order 32 = 25

### 1st non-split extension by C4 of Q8 acting via Q8/C4=C2

p-group, metacyclic, nilpotent (class 3), monomial

Aliases: C82C4, C4.1Q8, C2.3SD16, C22.10D4, C4⋊C4.2C2, (C2×C8).6C2, C4.6(C2×C4), C2.3(C4⋊C4), (C2×C4).17C22, SmallGroup(32,13)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C4.Q8
 Chief series C1 — C2 — C22 — C2×C4 — C2×C8 — C4.Q8
 Lower central C1 — C2 — C4 — C4.Q8
 Upper central C1 — C22 — C2×C4 — C4.Q8
 Jennings C1 — C2 — C2 — C2×C4 — C4.Q8

Generators and relations for C4.Q8
G = < a,b,c | a4=1, b4=a2, c2=a-1b2, ab=ba, cac-1=a-1, cbc-1=b3 >

Character table of C4.Q8

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D size 1 1 1 1 2 2 4 4 4 4 2 2 2 2 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 -1 1 -1 1 -1 i -i i -i -1 1 -1 1 linear of order 4 ρ6 1 -1 1 -1 1 -1 -i -i i i 1 -1 1 -1 linear of order 4 ρ7 1 -1 1 -1 1 -1 i i -i -i 1 -1 1 -1 linear of order 4 ρ8 1 -1 1 -1 1 -1 -i i -i i -1 1 -1 1 linear of order 4 ρ9 2 2 2 2 -2 -2 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 symplectic lifted from Q8, Schur index 2 ρ11 2 2 -2 -2 0 0 0 0 0 0 -√-2 √-2 √-2 -√-2 complex lifted from SD16 ρ12 2 -2 -2 2 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 complex lifted from SD16 ρ13 2 2 -2 -2 0 0 0 0 0 0 √-2 -√-2 -√-2 √-2 complex lifted from SD16 ρ14 2 -2 -2 2 0 0 0 0 0 0 √-2 √-2 -√-2 -√-2 complex lifted from SD16

Smallest permutation representation of C4.Q8
Regular action on 32 points
Generators in S32
```(1 21 5 17)(2 22 6 18)(3 23 7 19)(4 24 8 20)(9 28 13 32)(10 29 14 25)(11 30 15 26)(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)
(1 11 19 28)(2 14 20 31)(3 9 21 26)(4 12 22 29)(5 15 23 32)(6 10 24 27)(7 13 17 30)(8 16 18 25)```

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

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

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

Matrix representation of C4.Q8 in GL3(𝔽17) generated by

 16 0 0 0 0 16 0 1 0
,
 1 0 0 0 5 12 0 5 5
,
 4 0 0 0 12 5 0 5 5
`G:=sub<GL(3,GF(17))| [16,0,0,0,0,1,0,16,0],[1,0,0,0,5,5,0,12,5],[4,0,0,0,12,5,0,5,5] >;`

C4.Q8 in GAP, Magma, Sage, TeX

`C_4.Q_8`
`% in TeX`

`G:=Group("C4.Q8");`
`// GroupNames label`

`G:=SmallGroup(32,13);`
`// by ID`

`G=gap.SmallGroup(32,13);`
`# by ID`

`G:=PCGroup([5,-2,2,-2,2,-2,40,61,26,302,72]);`
`// Polycyclic`

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

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