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## G = C8⋊3D4order 64 = 26

### 3rd semidirect product of C8 and D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C8⋊3D4
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C8⋊C4 — C8⋊3D4
 Lower central C1 — C2 — C2×C4 — C8⋊3D4
 Upper central C1 — C22 — C42 — C8⋊3D4
 Jennings C1 — C2 — C2 — C2×C4 — C8⋊3D4

Generators and relations for C83D4
G = < a,b,c | a8=b4=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >

Subgroups: 161 in 72 conjugacy classes, 29 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C22⋊C4, C2×C8, D8, SD16, C2×D4, C2×D4, C2×D4, C2×Q8, C8⋊C4, C4.4D4, C41D4, C2×D8, C2×SD16, C83D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C41D4, C8⋊C22, C83D4

Character table of C83D4

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 8A 8B 8C 8D size 1 1 1 1 8 8 8 2 2 4 4 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 0 0 2 -2 0 0 0 -2 2 0 0 orthogonal lifted from D4 ρ10 2 -2 2 -2 0 0 0 2 -2 0 0 0 2 -2 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 0 0 0 -2 2 0 0 0 0 0 -2 2 orthogonal lifted from D4 ρ12 2 2 2 2 0 0 0 -2 -2 2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 0 0 0 -2 -2 -2 2 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 -2 2 -2 0 0 0 -2 2 0 0 0 0 0 2 -2 orthogonal lifted from D4 ρ15 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ16 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22

Smallest permutation representation of C83D4
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 21 31 16)(2 18 32 13)(3 23 25 10)(4 20 26 15)(5 17 27 12)(6 22 28 9)(7 19 29 14)(8 24 30 11)
(1 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 32)(9 15)(10 14)(11 13)(18 24)(19 23)(20 22)```

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

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

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

C83D4 is a maximal subgroup of
C42.531C23
C42.D2p: C42.247D4  M4(2)⋊7D4  M4(2)⋊9D4  C42.255D4  C42.257D4  C42.271D4  C42.272D4  C42.275D4 ...
(C2p×D8)⋊C2: C42.391C23  C42.72C23  C42.533C23  C2411D4  C4011D4  C5611D4 ...
C8pD4⋊C2: C42.C23  C42.2C23  M4(2)⋊10D4  M4(2)⋊11D4  C42.406C23  C42.407C23  C42.410C23  D89D4 ...
(C2×D4).D2p: C8⋊C4⋊C4  C42.3C23  C249D4  C409D4  C569D4 ...
C83D4 is a maximal quotient of
(C2×C4)⋊2D8  (C2×C8)⋊20D4
C8⋊D4p: C83D8  C8⋊D12  C8⋊D20  C8⋊D28 ...
C42.D2p: C42.110D4  C42.112D4  C42.64D6  C42.74D6  C42.64D10  C42.74D10  C42.64D14  C42.74D14 ...
(C2×C8).D2p: C85SD16  C86SD16  C42.664C23  C42.666C23  C42.667C23  C83Q16  C42.26Q8  (C2×D8)⋊10C4 ...

Matrix representation of C83D4 in GL6(𝔽17)

 0 16 0 0 0 0 1 0 0 0 0 0 0 0 5 5 4 4 0 0 12 5 13 4 0 0 13 13 12 12 0 0 4 13 5 12
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16

`G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,5,12,13,4,0,0,5,5,13,13,0,0,4,13,12,5,0,0,4,4,12,12],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;`

C83D4 in GAP, Magma, Sage, TeX

`C_8\rtimes_3D_4`
`% in TeX`

`G:=Group("C8:3D4");`
`// GroupNames label`

`G:=SmallGroup(64,177);`
`// by ID`

`G=gap.SmallGroup(64,177);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,121,55,362,332,86,963,117]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=b^4=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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