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## G = C16.C8order 128 = 27

### 1st non-split extension by C16 of C8 acting via C8/C2=C4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C16.C8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C4×C8 — C16⋊5C4 — C16.C8
 Lower central C1 — C2 — C4 — C8 — C16.C8
 Upper central C1 — C4 — C2×C8 — C4×C8 — C16.C8
 Jennings C1 — C2 — C2 — C2 — C2 — C2×C4 — C2×C4 — C4×C8 — C16.C8

Generators and relations for C16.C8
G = < a,b | a16=1, b8=a8, bab-1=a3 >

Smallest permutation representation of C16.C8
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 31 7 17 13 19 3 21 9 23 15 25 5 27 11 29)(2 26 16 20 14 30 12 24 10 18 8 28 6 22 4 32)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 4A 4B 4C 4D 4E 8A ··· 8H 16A ··· 16H 16I ··· 16P order 1 2 2 4 4 4 4 4 8 ··· 8 16 ··· 16 16 ··· 16 size 1 1 2 1 1 2 4 4 2 ··· 2 4 ··· 4 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 type + + + - + image C1 C2 C2 C4 C8 Q8 D4 M4(2) SD16 C8.C4 C16.C8 kernel C16.C8 C16⋊5C4 C8.C8 C2×C16 C16 C42 C2×C8 C8 C8 C22 C1 # reps 1 1 2 4 8 1 1 2 4 4 4

Matrix representation of C16.C8 in GL4(𝔽17) generated by

 0 13 0 0 9 0 0 0 0 0 0 9 0 0 1 0
,
 0 0 1 0 0 0 0 1 9 0 0 0 0 8 0 0
`G:=sub<GL(4,GF(17))| [0,9,0,0,13,0,0,0,0,0,0,1,0,0,9,0],[0,0,9,0,0,0,0,8,1,0,0,0,0,1,0,0] >;`

C16.C8 in GAP, Magma, Sage, TeX

`C_{16}.C_8`
`% in TeX`

`G:=Group("C16.C8");`
`// GroupNames label`

`G:=SmallGroup(128,101);`
`// by ID`

`G=gap.SmallGroup(128,101);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,36,422,100,1018,136,2804,172,124]);`
`// Polycyclic`

`G:=Group<a,b|a^16=1,b^8=a^8,b*a*b^-1=a^3>;`
`// generators/relations`

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