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## G = C8.C16order 128 = 27

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

p-group, metacyclic, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C8.C16
G = < a,b | a8=1, b16=a4, bab-1=a-1 >

Smallest permutation representation of C8.C16
On 32 points
Generators in S32
```(1 5 9 13 17 21 25 29)(2 30 26 22 18 14 10 6)(3 7 11 15 19 23 27 31)(4 32 28 24 20 16 12 8)
(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)```

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

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

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

56 conjugacy classes

 class 1 2A 2B 4A 4B 4C ··· 4G 8A 8B 8C 8D 8E ··· 8J 16A ··· 16H 16I ··· 16T 32A ··· 32P order 1 2 2 4 4 4 ··· 4 8 8 8 8 8 ··· 8 16 ··· 16 16 ··· 16 32 ··· 32 size 1 1 2 1 1 2 ··· 2 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + - image C1 C2 C2 C4 C4 C8 C8 C16 D4 Q8 M4(2) M5(2) C8.C16 kernel C8.C16 C4×C16 M6(2) C4×C8 C2×C16 C42 C2×C8 C8 C16 C16 C8 C22 C1 # reps 1 1 2 2 2 4 4 16 1 1 2 4 16

Matrix representation of C8.C16 in GL2(𝔽17) generated by

 15 0 0 8
,
 0 12 1 0
`G:=sub<GL(2,GF(17))| [15,0,0,8],[0,1,12,0] >;`

C8.C16 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(128,154);`
`// by ID`

`G=gap.SmallGroup(128,154);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,36,1430,352,80,102,124]);`
`// Polycyclic`

`G:=Group<a,b|a^8=1,b^16=a^4,b*a*b^-1=a^-1>;`
`// generators/relations`

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