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G = C16.3C8order 128 = 27

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

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C16.3C8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C4×C8 — C4×C16 — C16.3C8
 Lower central C1 — C2 — C4 — C8 — C16.3C8
 Upper central C1 — C8 — C2×C8 — C4×C8 — C16.3C8
 Jennings C1 — C2 — C2 — C2 — C2 — C2×C4 — C2×C4 — C4×C8 — C16.3C8

Generators and relations for C16.3C8
G = < a,b | a16=1, b8=a8, bab-1=a7 >

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

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

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

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

44 conjugacy classes

 class 1 2A 2B 4A 4B 4C ··· 4G 8A 8B 8C 8D 8E ··· 8J 16A ··· 16P 16Q ··· 16X order 1 2 2 4 4 4 ··· 4 8 8 8 8 8 ··· 8 16 ··· 16 16 ··· 16 size 1 1 2 1 1 2 ··· 2 1 1 1 1 2 ··· 2 2 ··· 2 8 ··· 8

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 type + + + - + + - image C1 C2 C2 C4 C8 Q8 D4 M4(2) D8 Q16 C8.C4 C16.3C8 kernel C16.3C8 C4×C16 C8.C8 C2×C16 C16 C42 C2×C8 C8 C8 C8 C22 C1 # reps 1 1 2 4 8 1 1 2 2 2 4 16

Matrix representation of C16.3C8 in GL2(𝔽17) generated by

 11 6 0 3
,
 14 10 4 3
`G:=sub<GL(2,GF(17))| [11,0,6,3],[14,4,10,3] >;`

C16.3C8 in GAP, Magma, Sage, TeX

`C_{16}._3C_8`
`% in TeX`

`G:=Group("C16.3C8");`
`// GroupNames label`

`G:=SmallGroup(128,105);`
`// by ID`

`G=gap.SmallGroup(128,105);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,148,422,436,136,2804,172,124]);`
`// Polycyclic`

`G:=Group<a,b|a^16=1,b^8=a^8,b*a*b^-1=a^7>;`
`// generators/relations`

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