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## G = D8⋊C23order 128 = 27

### 6th semidirect product of D8 and C23 acting via C23/C22=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — D8⋊C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×2+ 1+4 — D8⋊C23
 Lower central C1 — C2 — C4 — D8⋊C23
 Upper central C1 — C2 — C2×C4○D4 — D8⋊C23
 Jennings C1 — C2 — C2 — C4 — D8⋊C23

Generators and relations for D8⋊C23
G = < a,b,c,d,e | a8=b2=c2=d2=e2=1, bab=a-1, cac=dad=a3, eae=a5, cbc=a2b, dbd=a6b, ebe=a4b, cd=dc, ce=ec, de=ed >

Subgroups: 1220 in 732 conjugacy classes, 426 normal (13 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, C2×M4(2), C8○D4, C2×D8, C2×SD16, C4○D8, C8⋊C22, C8.C22, C22×D4, C22×D4, C2×C4○D4, C2×C4○D4, C2×C4○D4, 2+ 1+4, 2+ 1+4, 2- 1+4, 2- 1+4, Q8○M4(2), C2×C8⋊C22, D8⋊C22, D4○D8, D4○SD16, C2×2+ 1+4, C2.C25, D8⋊C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, C25, D4×C23, D8⋊C23

Permutation representations of D8⋊C23
On 16 points - transitive group 16T204
Generators in S16
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)
(1 7)(3 5)(4 8)(9 11)(10 14)(13 15)
(1 7)(3 5)(4 8)(9 15)(11 13)(12 16)
(1 5)(3 7)(9 13)(11 15)```

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

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

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

`G:=TransitiveGroup(16,204);`

41 conjugacy classes

 class 1 2A 2B ··· 2H 2I ··· 2R 4A ··· 4H 4I ··· 4N 8A ··· 8H order 1 2 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

41 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 8 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D8⋊C23 kernel D8⋊C23 Q8○M4(2) C2×C8⋊C22 D8⋊C22 D4○D8 D4○SD16 C2×2+ 1+4 C2.C25 C2×D4 C2×Q8 C4○D4 C1 # reps 1 1 6 6 8 8 1 1 3 1 4 1

Matrix representation of D8⋊C23 in GL8(ℤ)

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

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

D8⋊C23 in GAP, Magma, Sage, TeX

`D_8\rtimes C_2^3`
`% in TeX`

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

`G:=SmallGroup(128,2317);`
`// by ID`

`G=gap.SmallGroup(128,2317);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,2,-2,-2,477,521,2804,4037,2028,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^8=b^2=c^2=d^2=e^2=1,b*a*b=a^-1,c*a*c=d*a*d=a^3,e*a*e=a^5,c*b*c=a^2*b,d*b*d=a^6*b,e*b*e=a^4*b,c*d=d*c,c*e=e*c,d*e=e*d>;`
`// generators/relations`

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