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## G = M4(2).27D4order 128 = 27

### 8th non-split extension by M4(2) of D4 acting via D4/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — M4(2).27D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C8○2M4(2) — M4(2).27D4
 Lower central C1 — C2 — C2×C4 — M4(2).27D4
 Upper central C1 — C4 — C22×C4 — M4(2).27D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).27D4

Generators and relations for M4(2).27D4
G = < a,b,c,d | a8=b2=1, c4=a4, d2=a2, bab=a5, ac=ca, dad-1=a-1b, bc=cb, dbd-1=a4b, dcd-1=c3 >

Subgroups: 140 in 92 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C4×C8, C8⋊C4, C8.C4, C8.C4, C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), M4(2)⋊4C4, C82M4(2), C2×C8.C4, M4(2).C4, M4(2).27D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C23.65C23, M4(2).27D4

Smallest permutation representation of M4(2).27D4
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 17)(2 22)(3 19)(4 24)(5 21)(6 18)(7 23)(8 20)(9 25)(10 30)(11 27)(12 32)(13 29)(14 26)(15 31)(16 28)
(1 24 3 18 5 20 7 22)(2 17 4 19 6 21 8 23)(9 12 15 10 13 16 11 14)(25 32 31 30 29 28 27 26)
(1 31 3 25 5 27 7 29)(2 10 4 12 6 14 8 16)(9 17 11 19 13 21 15 23)(18 30 20 32 22 26 24 28)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D 8E ··· 8J 8K ··· 8R order 1 2 2 2 2 4 4 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 4 4 2 2 2 2 4 ··· 4 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + - + - + image C1 C2 C2 C2 C2 C4 Q8 D4 Q8 D4 C4○D4 M4(2).27D4 kernel M4(2).27D4 M4(2)⋊4C4 C8○2M4(2) C2×C8.C4 M4(2).C4 C8.C4 C22⋊C4 C2×C8 C2×C8 M4(2) C2×C4 C1 # reps 1 2 1 2 2 8 2 2 2 2 4 4

Matrix representation of M4(2).27D4 in GL4(𝔽17) generated by

 9 0 0 0 16 8 0 0 0 0 8 8 0 0 0 9
,
 13 13 0 0 8 4 0 0 0 0 4 4 0 0 9 13
,
 9 0 0 0 0 9 0 0 0 0 15 0 0 0 0 15
,
 0 0 15 0 0 0 0 15 2 0 0 0 0 2 0 0
`G:=sub<GL(4,GF(17))| [9,16,0,0,0,8,0,0,0,0,8,0,0,0,8,9],[13,8,0,0,13,4,0,0,0,0,4,9,0,0,4,13],[9,0,0,0,0,9,0,0,0,0,15,0,0,0,0,15],[0,0,2,0,0,0,0,2,15,0,0,0,0,15,0,0] >;`

M4(2).27D4 in GAP, Magma, Sage, TeX

`M_4(2)._{27}D_4`
`% in TeX`

`G:=Group("M4(2).27D4");`
`// GroupNames label`

`G:=SmallGroup(128,685);`
`// by ID`

`G=gap.SmallGroup(128,685);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,100,2019,1018,248,2804,172,124]);`
`// Polycyclic`

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

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