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## G = C8.5M4(2)  order 128 = 27

### 5th non-split extension by C8 of M4(2) acting via M4(2)/C4=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C8.5M4(2)
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C8⋊C4 — C4×M4(2) — C8.5M4(2)
 Lower central C1 — C2 — C2×C4 — C8.5M4(2)
 Upper central C1 — C4 — C8⋊C4 — C8.5M4(2)
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C8.5M4(2)

Generators and relations for C8.5M4(2)
G = < a,b,c | a8=1, b8=c2=a4, bab-1=a3, cac-1=a5, cbc-1=a6b5 >

Subgroups: 100 in 64 conjugacy classes, 38 normal (24 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C23, C16 [×4], C42 [×2], C42, C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4, C4×C8 [×2], C8⋊C4 [×2], M5(2) [×4], C2×C42, C2×M4(2) [×2], C16⋊C4 [×2], C23.C8 [×2], C8.C8 [×2], C4×M4(2), C8.5M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, M4(2) [×4], C22×C4, C4○D4 [×2], C42⋊C2, C2×M4(2) [×2], C42.6C4, C8.5M4(2)

Permutation representations of C8.5M4(2)
On 16 points - transitive group 16T220
Generators in S16
```(1 7 13 3 9 15 5 11)(2 4 6 8 10 12 14 16)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)
(1 13 9 5)(2 6 10 14)(3 7 11 15)(4 16 12 8)```

`G:=sub<Sym(16)| (1,7,13,3,9,15,5,11)(2,4,6,8,10,12,14,16), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (1,13,9,5)(2,6,10,14)(3,7,11,15)(4,16,12,8)>;`

`G:=Group( (1,7,13,3,9,15,5,11)(2,4,6,8,10,12,14,16), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (1,13,9,5)(2,6,10,14)(3,7,11,15)(4,16,12,8) );`

`G=PermutationGroup([(1,7,13,3,9,15,5,11),(2,4,6,8,10,12,14,16)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)], [(1,13,9,5),(2,6,10,14),(3,7,11,15),(4,16,12,8)])`

`G:=TransitiveGroup(16,220);`

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C4 M4(2) C4○D4 M4(2) M4(2) C8.5M4(2) kernel C8.5M4(2) C16⋊C4 C23.C8 C8.C8 C4×M4(2) C4×C8 C8⋊C4 C2×C42 C2×M4(2) C8 C8 C2×C4 C23 C1 # reps 1 2 2 2 1 2 2 2 2 4 4 2 2 4

Matrix representation of C8.5M4(2) in GL4(𝔽5) generated by

 3 4 1 0 3 4 0 2 3 0 1 4 0 4 3 2
,
 3 4 3 1 0 4 1 1 3 4 2 3 4 0 1 1
,
 3 3 1 0 0 1 1 0 0 3 4 0 0 4 1 2
`G:=sub<GL(4,GF(5))| [3,3,3,0,4,4,0,4,1,0,1,3,0,2,4,2],[3,0,3,4,4,4,4,0,3,1,2,1,1,1,3,1],[3,0,0,0,3,1,3,4,1,1,4,1,0,0,0,2] >;`

C8.5M4(2) in GAP, Magma, Sage, TeX

`C_8._5M_4(2)`
`% in TeX`

`G:=Group("C8.5M4(2)");`
`// GroupNames label`

`G:=SmallGroup(128,897);`
`// by ID`

`G=gap.SmallGroup(128,897);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,758,58,2019,1411,718,102,124]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=1,b^8=c^2=a^4,b*a*b^-1=a^3,c*a*c^-1=a^5,c*b*c^-1=a^6*b^5>;`
`// generators/relations`

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