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## G = C8.23C42order 128 = 27

### 4th central extension by C8 of C42

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C8.23C42
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C22×C8 — C8○2M4(2) — C8.23C42
 Lower central C1 — C4 — C8.23C42
 Upper central C1 — C8 — C8.23C42
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C8.23C42

Generators and relations for C8.23C42
G = < a,b,c | a8=c4=1, b4=a2, ab=ba, ac=ca, cbc-1=a2b >

Subgroups: 100 in 80 conjugacy classes, 66 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C8, C2×C4, C2×C4, C2×C4, C23, C16, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C4×C8, C8⋊C4, C2×C16, M5(2), C42⋊C2, C22×C8, C2×M4(2), C16⋊C4, C82M4(2), C2×M5(2), C8.23C42
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, M4(2), C22×C4, C8⋊C4, C2×C42, C2×M4(2), C2×C8⋊C4, C8.23C42

Smallest permutation representation of C8.23C42
On 32 points
Generators in S32
```(1 3 5 7 9 11 13 15)(2 4 6 8 10 12 14 16)(17 27 21 31 25 19 29 23)(18 28 22 32 26 20 30 24)
(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 26)(2 23 10 31)(3 20)(4 17 12 25)(5 30)(6 27 14 19)(7 24)(8 21 16 29)(9 18)(11 28)(13 22)(15 32)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D 8E ··· 8J 8K 8L 8M 8N 16A ··· 16P order 1 2 2 2 2 4 4 4 4 4 4 4 4 4 8 8 8 8 8 ··· 8 8 8 8 8 16 ··· 16 size 1 1 2 2 2 1 1 2 2 2 4 4 4 4 1 1 1 1 2 ··· 2 4 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 4 type + + + + image C1 C2 C2 C2 C4 C4 C4 C4 C4 M4(2) M4(2) C8.23C42 kernel C8.23C42 C16⋊C4 C8○2M4(2) C2×M5(2) C4×C8 C2×C16 M5(2) C42⋊C2 C2×M4(2) C2×C4 C23 C1 # reps 1 4 1 2 4 8 8 2 2 6 2 4

Matrix representation of C8.23C42 in GL4(𝔽17) generated by

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

C8.23C42 in GAP, Magma, Sage, TeX

`C_8._{23}C_4^2`
`% in TeX`

`G:=Group("C8.23C4^2");`
`// GroupNames label`

`G:=SmallGroup(128,842);`
`// by ID`

`G=gap.SmallGroup(128,842);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,477,120,723,1018,136,2804,124]);`
`// Polycyclic`

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

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