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## G = C8.D4order 64 = 26

### 1st non-split extension by C8 of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C8.D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×M4(2) — C8.D4
 Lower central C1 — C2 — C2×C4 — C8.D4
 Upper central C1 — C22 — C22×C4 — C8.D4
 Jennings C1 — C2 — C2 — C2×C4 — C8.D4

Generators and relations for C8.D4
G = < a,b,c | a8=b4=1, c2=a4, bab-1=a3, cac-1=a-1, cbc-1=a4b-1 >

Character table of C8.D4

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D size 1 1 1 1 4 2 2 4 8 8 8 8 4 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 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 linear of order 2 ρ7 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 linear of order 2 ρ9 2 -2 2 -2 0 2 -2 0 0 0 0 0 -2 0 2 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 -2 2 -2 0 2 -2 0 0 0 0 0 2 0 -2 0 orthogonal lifted from D4 ρ13 2 -2 2 -2 0 -2 2 0 0 0 0 0 0 2i 0 -2i complex lifted from C4○D4 ρ14 2 -2 2 -2 0 -2 2 0 0 0 0 0 0 -2i 0 2i complex lifted from C4○D4 ρ15 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ16 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C8.D4
On 32 points
Generators in S32
```(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 14 19 32)(2 9 20 27)(3 12 21 30)(4 15 22 25)(5 10 23 28)(6 13 24 31)(7 16 17 26)(8 11 18 29)
(1 28 5 32)(2 27 6 31)(3 26 7 30)(4 25 8 29)(9 24 13 20)(10 23 14 19)(11 22 15 18)(12 21 16 17)```

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

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

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

C8.D4 is a maximal subgroup of
C23.2SD16  C23.10SD16  C24.110D4  M4(2)⋊15D4  C8.D4⋊C2  (C2×C8)⋊13D4  M4(2)⋊17D4  C42.260D4  C42.262D4  C24.128D4  C24.129D4  C24.130D4  C4.162+ 1+4  C4.172+ 1+4  C4.192+ 1+4  C42.300D4  C42.303D4  C42.304D4
C23.D4p: C23.2D8  D8.D4  C24.4D4  C40.4D4  C56.4D4 ...
C4⋊C4.D2p: C42.385C23  C42.389C23  C42.390C23  C42.25C23  C42.28C23  C42.30C23  D810D4  Q169D4 ...
(C2p×Q16)⋊C2: C42.256D4  C24.36D4  C40.36D4  C56.36D4 ...
C8.D4 is a maximal quotient of
C8.D4p: C8.D8  C8.2D12  C8.2D20  C8.2D28 ...
C23.D4p: C23.12D8  C24.4D4  C40.4D4  C56.4D4 ...
C4⋊C4.D2p: C8.SD16  C8.8SD16  C8.3Q16  C42.254C23  C42.255C23  C232Q16  C24.85D4  (C2×Q8).8Q8 ...
(C2×C8).D2p: C24.67D4  C24.75D4  C2.(C8⋊D4)  C8⋊(C4⋊C4)  (C2×Q16)⋊10C4  (C2×C4)⋊3Q16  (C2×C4).26D8  C24.36D4 ...

Matrix representation of C8.D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 7 11 15 0 0 0 11 7 0 15 0 0 0 1 10 6 0 0 0 0 6 10
,
 16 15 0 0 0 0 1 1 0 0 0 0 0 0 14 3 7 10 0 0 12 12 10 10 0 0 7 6 3 5 0 0 6 5 14 5
,
 16 15 0 0 0 0 0 1 0 0 0 0 0 0 14 3 7 10 0 0 12 12 10 10 0 0 2 11 3 5 0 0 11 10 14 5

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,7,11,0,0,0,0,11,7,1,0,0,0,15,0,10,6,0,0,0,15,6,10],[16,1,0,0,0,0,15,1,0,0,0,0,0,0,14,12,7,6,0,0,3,12,6,5,0,0,7,10,3,14,0,0,10,10,5,5],[16,0,0,0,0,0,15,1,0,0,0,0,0,0,14,12,2,11,0,0,3,12,11,10,0,0,7,10,3,14,0,0,10,10,5,5] >;`

C8.D4 in GAP, Magma, Sage, TeX

`C_8.D_4`
`% in TeX`

`G:=Group("C8.D4");`
`// GroupNames label`

`G:=SmallGroup(64,151);`
`// by ID`

`G=gap.SmallGroup(64,151);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,247,362,332,963,117]);`
`// Polycyclic`

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

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