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## G = C23.D8order 128 = 27

### 1st non-split extension by C23 of D8 acting via D8/C2=D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — C23.D8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×M4(2) — C8⋊2D4 — C23.D8
 Lower central C1 — C2 — C2×C4 — C2×C8 — C23.D8
 Upper central C1 — C2 — C2×C4 — C2×M4(2) — C23.D8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C2×C4 — C2×M4(2) — C23.D8

Generators and relations for C23.D8
G = < a,b,c,d,e | a2=b2=c2=1, d8=c, e2=a, dad-1=ab=ba, ac=ca, ae=ea, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=acd7 >

Character table of C23.D8

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 16A 16B 16C 16D size 1 1 2 4 16 2 2 4 8 8 8 8 16 4 4 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 trivial ρ2 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 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 linear of order 2 ρ5 1 1 1 -1 1 1 1 -1 i -i -i i -1 -1 -1 1 -i i i -i linear of order 4 ρ6 1 1 1 -1 -1 1 1 -1 i -i -i i 1 -1 -1 1 i -i -i i linear of order 4 ρ7 1 1 1 -1 -1 1 1 -1 -i i i -i 1 -1 -1 1 -i i i -i linear of order 4 ρ8 1 1 1 -1 1 1 1 -1 -i i i -i -1 -1 -1 1 i -i -i i linear of order 4 ρ9 2 2 2 2 0 2 2 2 0 0 0 0 0 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 -2 0 2 2 -2 0 0 0 0 0 2 2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 -2 -2 -2 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ12 2 2 2 2 0 -2 -2 -2 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ13 2 2 -2 0 0 2 -2 0 -1-i -1+i 1-i 1+i 0 2i -2i 0 0 0 0 0 complex lifted from C4≀C2 ρ14 2 2 -2 0 0 2 -2 0 1+i 1-i -1+i -1-i 0 2i -2i 0 0 0 0 0 complex lifted from C4≀C2 ρ15 2 2 -2 0 0 2 -2 0 1-i 1+i -1-i -1+i 0 -2i 2i 0 0 0 0 0 complex lifted from C4≀C2 ρ16 2 2 2 -2 0 -2 -2 2 0 0 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 complex lifted from SD16 ρ17 2 2 -2 0 0 2 -2 0 -1+i -1-i 1+i 1-i 0 -2i 2i 0 0 0 0 0 complex lifted from C4≀C2 ρ18 2 2 2 -2 0 -2 -2 2 0 0 0 0 0 0 0 0 √-2 √-2 -√-2 -√-2 complex lifted from SD16 ρ19 4 4 -4 0 0 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ20 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C23.D8
On 16 points - transitive group 16T371
Generators in S16
```(1 9)(4 12)(5 13)(8 16)
(1 9)(3 11)(5 13)(7 15)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)
(1 8 9 16)(2 7)(3 6)(4 13 12 5)(10 15)(11 14)```

`G:=sub<Sym(16)| (1,9)(4,12)(5,13)(8,16), (1,9)(3,11)(5,13)(7,15), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (1,8,9,16)(2,7)(3,6)(4,13,12,5)(10,15)(11,14)>;`

`G:=Group( (1,9)(4,12)(5,13)(8,16), (1,9)(3,11)(5,13)(7,15), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (1,8,9,16)(2,7)(3,6)(4,13,12,5)(10,15)(11,14) );`

`G=PermutationGroup([[(1,9),(4,12),(5,13),(8,16)], [(1,9),(3,11),(5,13),(7,15)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)], [(1,8,9,16),(2,7),(3,6),(4,13,12,5),(10,15),(11,14)]])`

`G:=TransitiveGroup(16,371);`

Matrix representation of C23.D8 in GL8(ℤ)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1
,
 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1
,
 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0
,
 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0

`G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;`

C23.D8 in GAP, Magma, Sage, TeX

`C_2^3.D_8`
`% in TeX`

`G:=Group("C2^3.D8");`
`// GroupNames label`

`G:=SmallGroup(128,71);`
`// by ID`

`G=gap.SmallGroup(128,71);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,387,520,1690,521,1411,172,4037,2028,124]);`
`// Polycyclic`

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

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