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## G = C8.16C42order 128 = 27

### 10th non-split extension by C8 of C42 acting via C42/C2×C4=C2

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C8.16C42
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C22×C8 — C8○2M4(2) — C8.16C42
 Lower central C1 — C4 — C8.16C42
 Upper central C1 — C8 — C8.16C42
 Jennings C1 — C2 — C2 — C22×C4 — C8.16C42

Generators and relations for C8.16C42
G = < a,b,c | a8=b4=c4=1, bab-1=cac-1=a5, cbc-1=a2b >

Subgroups: 148 in 100 conjugacy classes, 66 normal (9 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C8, C2×C4, C2×C4, C23, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C4×C8, C8⋊C4, C42⋊C2, C22×C8, C2×M4(2), C4.9C42, C4.10C42, C82M4(2), C8.16C42
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C4○D4, C2×C42, C42⋊C2, C424C4, C8.16C42

Smallest permutation representation of C8.16C42
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 27 23 11)(2 32 24 16)(3 29 17 13)(4 26 18 10)(5 31 19 15)(6 28 20 12)(7 25 21 9)(8 30 22 14)
(1 12 19 28)(2 9 20 25)(3 14 21 30)(4 11 22 27)(5 16 23 32)(6 13 24 29)(7 10 17 26)(8 15 18 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), (1,27,23,11)(2,32,24,16)(3,29,17,13)(4,26,18,10)(5,31,19,15)(6,28,20,12)(7,25,21,9)(8,30,22,14), (1,12,19,28)(2,9,20,25)(3,14,21,30)(4,11,22,27)(5,16,23,32)(6,13,24,29)(7,10,17,26)(8,15,18,31)>;`

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F ··· 4Q 8A 8B 8C 8D 8E ··· 8J 8K ··· 8V order 1 2 2 2 2 4 4 4 4 4 4 ··· 4 8 8 8 8 8 ··· 8 8 ··· 8 size 1 1 2 2 2 1 1 2 2 2 4 ··· 4 1 1 1 1 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 type + + + + image C1 C2 C2 C2 C4 C4 C4○D4 C4○D4 C8.16C42 kernel C8.16C42 C4.9C42 C4.10C42 C8○2M4(2) C4×C8 C8⋊C4 C2×C4 C23 C1 # reps 1 3 1 3 12 12 6 2 4

Matrix representation of C8.16C42 in GL4(𝔽17) generated by

 2 0 0 0 0 2 0 0 0 0 15 0 0 0 0 15
,
 0 0 2 4 0 0 15 15 8 0 0 0 9 9 0 0
,
 0 0 1 0 0 0 0 1 16 15 0 0 0 1 0 0
`G:=sub<GL(4,GF(17))| [2,0,0,0,0,2,0,0,0,0,15,0,0,0,0,15],[0,0,8,9,0,0,0,9,2,15,0,0,4,15,0,0],[0,0,16,0,0,0,15,1,1,0,0,0,0,1,0,0] >;`

C8.16C42 in GAP, Magma, Sage, TeX

`C_8._{16}C_4^2`
`% in TeX`

`G:=Group("C8.16C4^2");`
`// GroupNames label`

`G:=SmallGroup(128,479);`
`// by ID`

`G=gap.SmallGroup(128,479);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,58,248,1411,4037]);`
`// Polycyclic`

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

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