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## G = C4.D8order 64 = 26

### 1st non-split extension by C4 of D8 acting via D8/D4=C2

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

Aliases: C4.9D8, C4.11SD16, C42.3C22, C4⋊C82C2, (C2×D4).2C4, C41D4.1C2, (C2×C4).108D4, C2.4(D4⋊C4), C2.4(C4.D4), C22.39(C22⋊C4), (C2×C4).12(C2×C4), SmallGroup(64,12)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C4.D8
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4⋊1D4 — C4.D8
 Lower central C1 — C22 — C2×C4 — C4.D8
 Upper central C1 — C22 — C42 — C4.D8
 Jennings C1 — C22 — C22 — C42 — C4.D8

Generators and relations for C4.D8
G = < a,b,c | a4=b8=1, c2=a, bab-1=a-1, ac=ca, cbc-1=ab-1 >

Character table of C4.D8

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 8 8 2 2 2 2 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 trivial ρ2 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 linear of order 2 ρ4 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 -i -i i i -i i -i i linear of order 4 ρ6 1 1 1 1 -1 1 -1 -1 -1 -1 1 i i -i -i i -i i -i linear of order 4 ρ7 1 1 1 1 1 -1 -1 -1 -1 -1 1 i -i i -i i i -i -i linear of order 4 ρ8 1 1 1 1 1 -1 -1 -1 -1 -1 1 -i i -i i -i -i i i linear of order 4 ρ9 2 2 2 2 0 0 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 -2 2 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 -2 2 0 0 -2 0 0 2 0 0 √2 √2 0 0 -√2 -√2 0 orthogonal lifted from D8 ρ12 2 2 -2 -2 0 0 0 2 -2 0 0 √2 0 0 √2 -√2 0 0 -√2 orthogonal lifted from D8 ρ13 2 -2 -2 2 0 0 -2 0 0 2 0 0 -√2 -√2 0 0 √2 √2 0 orthogonal lifted from D8 ρ14 2 2 -2 -2 0 0 0 2 -2 0 0 -√2 0 0 -√2 √2 0 0 √2 orthogonal lifted from D8 ρ15 2 2 -2 -2 0 0 0 -2 2 0 0 -√-2 0 0 √-2 √-2 0 0 -√-2 complex lifted from SD16 ρ16 2 -2 -2 2 0 0 2 0 0 -2 0 0 √-2 -√-2 0 0 √-2 -√-2 0 complex lifted from SD16 ρ17 2 -2 -2 2 0 0 2 0 0 -2 0 0 -√-2 √-2 0 0 -√-2 √-2 0 complex lifted from SD16 ρ18 2 2 -2 -2 0 0 0 -2 2 0 0 √-2 0 0 -√-2 -√-2 0 0 √-2 complex lifted from SD16 ρ19 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4.D4

Smallest permutation representation of C4.D8
On 32 points
Generators in S32
```(1 24 29 9)(2 10 30 17)(3 18 31 11)(4 12 32 19)(5 20 25 13)(6 14 26 21)(7 22 27 15)(8 16 28 23)
(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 16 24 28 29 23 9 8)(2 7 10 22 30 27 17 15)(3 14 18 26 31 21 11 6)(4 5 12 20 32 25 19 13)```

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

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

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

Matrix representation of C4.D8 in GL4(𝔽17) generated by

 16 15 0 0 1 1 0 0 0 0 1 0 0 0 0 1
,
 7 7 0 0 5 10 0 0 0 0 6 6 0 0 14 0
,
 7 7 0 0 5 0 0 0 0 0 6 6 0 0 14 11
`G:=sub<GL(4,GF(17))| [16,1,0,0,15,1,0,0,0,0,1,0,0,0,0,1],[7,5,0,0,7,10,0,0,0,0,6,14,0,0,6,0],[7,5,0,0,7,0,0,0,0,0,6,14,0,0,6,11] >;`

C4.D8 in GAP, Magma, Sage, TeX

`C_4.D_8`
`% in TeX`

`G:=Group("C4.D8");`
`// GroupNames label`

`G:=SmallGroup(64,12);`
`// by ID`

`G=gap.SmallGroup(64,12);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,362,332,158,681,165]);`
`// Polycyclic`

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

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