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## G = C8.C42order 128 = 27

### 1st non-split extension by C8 of C42 acting via C42/C22=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C8.C42
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C22×C8 — C2×M5(2) — C8.C42
 Lower central C1 — C2 — C4 — C8 — C8.C42
 Upper central C1 — C22 — C22×C4 — C22×C8 — C8.C42
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C22×C8 — C8.C42

Generators and relations for C8.C42
G = < a,b,c | a8=b4=1, c4=a6, bab-1=a-1, cac-1=a5, cbc-1=a3b >

Subgroups: 152 in 70 conjugacy classes, 40 normal (32 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×4], C22 [×3], C22 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×6], C23, C16 [×2], C42, C22⋊C4, C4⋊C4 [×5], C2×C8 [×6], C22×C4, C22×C4, C4.Q8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C16, M5(2) [×2], M5(2), C2×C4⋊C4, C42⋊C2, C22×C8, C2×C4.Q8, C23.25D4, C2×M5(2), C8.C42
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], D8, SD16 [×2], Q16, C2.C42, D4⋊C4 [×2], Q8⋊C4 [×2], C4.Q8, C2.D8, C22.4Q16, D82C4 [×2], C8.C42

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4L 8A 8B 8C 8D 8E 8F 16A ··· 16H order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 8 8 8 8 8 8 16 ··· 16 size 1 1 1 1 2 2 2 2 2 2 8 ··· 8 2 2 2 2 4 4 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + - + - + image C1 C2 C2 C2 C4 C4 C4 D4 Q8 D4 SD16 Q16 D8 D8⋊2C4 kernel C8.C42 C2×C4.Q8 C23.25D4 C2×M5(2) C4.Q8 C2.D8 M5(2) C2×C8 C2×C8 C22×C4 C2×C4 C2×C4 C23 C2 # reps 1 1 1 1 4 4 4 2 1 1 4 2 2 4

Matrix representation of C8.C42 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 5 5 0 0 0 0 12 5 0 0 0 0 0 0 12 12 0 0 0 0 5 12
,
 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 5 0 0 0 0 5 5 0 0 5 12 0 0 0 0 12 12 0 0
,
 0 13 0 0 0 0 13 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 12 5 0 0 0 0 12 12 0 0

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,5,12,0,0,0,0,5,5,0,0,0,0,0,0,12,5,0,0,0,0,12,12],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,5,12,0,0,0,0,12,12,0,0,12,5,0,0,0,0,5,5,0,0],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C8.C42 in GAP, Magma, Sage, TeX

`C_8.C_4^2`
`% in TeX`

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

`G:=SmallGroup(128,118);`
`// by ID`

`G=gap.SmallGroup(128,118);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,723,520,1684,102,4037,124]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=b^4=1,c^4=a^6,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^3*b>;`
`// generators/relations`

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