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## G = M4(2).1C8order 128 = 27

### 1st non-split extension by M4(2) of C8 acting via C8/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — M4(2).1C8
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C22×C8 — C8○2M4(2) — M4(2).1C8
 Lower central C1 — C2 — C4 — M4(2).1C8
 Upper central C1 — C8 — C22×C8 — M4(2).1C8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — M4(2).1C8

Generators and relations for M4(2).1C8
G = < a,b,c | a8=b2=1, c8=a4, bab=a5, cac-1=a-1, cbc-1=a4b >

Subgroups: 100 in 76 conjugacy classes, 58 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, 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), M5(2), C42⋊C2, C22×C8, C2×M4(2), C8.C8, C82M4(2), C2×M5(2), M4(2).1C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4⋊C8, C2×C4⋊C4, C22×C8, C2×M4(2), C2×C4⋊C8, M4(2).1C8

Smallest permutation representation of M4(2).1C8
On 32 points
Generators in S32
```(1 27 13 23 9 19 5 31)(2 32 6 20 10 24 14 28)(3 29 15 25 11 21 7 17)(4 18 8 22 12 26 16 30)
(1 9)(3 11)(5 13)(7 15)(18 26)(20 28)(22 30)(24 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)```

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

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

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

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 2 2 2 2 4 type + + + + + - image C1 C2 C2 C2 C4 C4 C4 C8 D4 Q8 M4(2) M4(2) M4(2).1C8 kernel M4(2).1C8 C8.C8 C8○2M4(2) C2×M5(2) C8⋊C4 C42⋊C2 C2×M4(2) M4(2) C2×C8 C2×C8 C2×C4 C23 C1 # reps 1 4 1 2 4 2 2 16 2 2 2 2 4

Matrix representation of M4(2).1C8 in GL4(𝔽17) generated by

 0 4 0 0 1 0 0 0 0 0 0 1 0 0 13 0
,
 16 0 0 0 0 1 0 0 0 0 1 0 0 0 0 16
,
 0 0 1 0 0 0 0 1 8 0 0 0 0 8 0 0
`G:=sub<GL(4,GF(17))| [0,1,0,0,4,0,0,0,0,0,0,13,0,0,1,0],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16],[0,0,8,0,0,0,0,8,1,0,0,0,0,1,0,0] >;`

M4(2).1C8 in GAP, Magma, Sage, TeX

`M_4(2)._1C_8`
`% in TeX`

`G:=Group("M4(2).1C8");`
`// GroupNames label`

`G:=SmallGroup(128,885);`
`// by ID`

`G=gap.SmallGroup(128,885);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,64,723,2019,248,102,124]);`
`// Polycyclic`

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

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