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## G = C8.3D8order 128 = 27

### 3rd non-split extension by C8 of D8 acting via D8/C4=C22

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — C8.3D8
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C4○D8 — C8○D8 — C8.3D8
 Lower central C1 — C2 — C4 — C2×C8 — C8.3D8
 Upper central C1 — C2 — C2×C4 — C4×C8 — C8.3D8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C8.3D8

Generators and relations for C8.3D8
G = < a,b,c | a8=c2=1, b8=a4, bab-1=a-1, cac=a3, cbc=a4b7 >

Subgroups: 204 in 77 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×3], C22, C22 [×4], C8 [×4], C8, C2×C4, C2×C4 [×3], D4 [×4], Q8 [×3], C23, C16 [×2], C42, C22⋊C4 [×2], C2×C8 [×2], C2×C8, M4(2) [×2], D8, D8 [×2], SD16 [×3], Q16, Q16 [×2], C2×D4, C2×Q8, C4○D4, C4×C8, C4≀C2, C8.C4, M5(2) [×2], D16, SD32 [×2], Q32, C4.4D4, C8○D4, C2×D8, C2×SD16, C2×Q16, C4○D8, M5(2)⋊C2, C8.17D4, C8.C8, C8○D8, C8.12D4, C16⋊C22, Q32⋊C2, C8.3D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×D8, C8⋊C22, C4⋊D8, C8.3D8

Character table of C8.3D8

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 8E 8F 8G 8H 16A 16B 16C 16D size 1 1 2 8 16 2 2 4 4 8 16 2 2 2 2 4 4 8 8 8 8 8 8 ρ1 1 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 -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 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 -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 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 -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 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 -1 linear of order 2 ρ9 2 2 -2 2 0 -2 2 0 0 -2 0 2 0 0 2 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 -2 0 -2 2 0 0 2 0 2 0 0 2 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 0 0 2 2 -2 -2 0 0 -2 2 2 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 0 0 2 2 2 2 0 0 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 -2 0 0 2 -2 0 0 0 0 0 -2 -2 0 2 0 0 0 -√2 -√2 √2 √2 orthogonal lifted from D8 ρ14 2 2 -2 0 0 2 -2 0 0 0 0 0 2 2 0 -2 0 0 0 -√2 √2 √2 -√2 orthogonal lifted from D8 ρ15 2 2 -2 0 0 2 -2 0 0 0 0 0 2 2 0 -2 0 0 0 √2 -√2 -√2 √2 orthogonal lifted from D8 ρ16 2 2 -2 0 0 2 -2 0 0 0 0 0 -2 -2 0 2 0 0 0 √2 √2 -√2 -√2 orthogonal lifted from D8 ρ17 2 2 -2 0 0 -2 2 0 0 0 0 -2 0 0 -2 0 2 2i -2i 0 0 0 0 complex lifted from C4○D4 ρ18 2 2 -2 0 0 -2 2 0 0 0 0 -2 0 0 -2 0 2 -2i 2i 0 0 0 0 complex lifted from C4○D4 ρ19 4 4 4 0 0 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ20 4 -4 0 0 0 0 0 2i -2i 0 0 -2√2 2√-2 -2√-2 2√2 0 0 0 0 0 0 0 0 complex faithful ρ21 4 -4 0 0 0 0 0 2i -2i 0 0 2√2 -2√-2 2√-2 -2√2 0 0 0 0 0 0 0 0 complex faithful ρ22 4 -4 0 0 0 0 0 -2i 2i 0 0 -2√2 -2√-2 2√-2 2√2 0 0 0 0 0 0 0 0 complex faithful ρ23 4 -4 0 0 0 0 0 -2i 2i 0 0 2√2 2√-2 -2√-2 -2√2 0 0 0 0 0 0 0 0 complex faithful

Smallest permutation representation of C8.3D8
On 32 points
Generators in S32
```(1 29 13 25 9 21 5 17)(2 18 6 22 10 26 14 30)(3 31 15 27 11 23 7 19)(4 20 8 24 12 28 16 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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 21)(18 20)(22 32)(23 31)(24 30)(25 29)(26 28)```

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

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

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

Matrix representation of C8.3D8 in GL4(𝔽17) generated by

 10 7 0 0 5 0 0 0 0 0 7 10 0 0 12 0
,
 0 0 0 6 0 0 3 0 1 0 0 0 1 16 0 0
,
 1 0 0 0 1 16 0 0 0 0 0 6 0 0 3 0
`G:=sub<GL(4,GF(17))| [10,5,0,0,7,0,0,0,0,0,7,12,0,0,10,0],[0,0,1,1,0,0,0,16,0,3,0,0,6,0,0,0],[1,1,0,0,0,16,0,0,0,0,0,3,0,0,6,0] >;`

C8.3D8 in GAP, Magma, Sage, TeX

`C_8._3D_8`
`% in TeX`

`G:=Group("C8.3D8");`
`// GroupNames label`

`G:=SmallGroup(128,944);`
`// by ID`

`G=gap.SmallGroup(128,944);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,64,422,2019,248,1684,438,242,4037,1027,124]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=c^2=1,b^8=a^4,b*a*b^-1=a^-1,c*a*c=a^3,c*b*c=a^4*b^7>;`
`// generators/relations`

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