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## G = M4(2).9D4order 128 = 27

### 9th non-split extension by M4(2) of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — M4(2).9D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×M4(2) — M4(2).8C22 — M4(2).9D4
 Lower central C1 — C2 — C22×C4 — M4(2).9D4
 Upper central C1 — C2 — C22×C4 — M4(2).9D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).9D4

Generators and relations for M4(2).9D4
G = < a,b,c,d | a8=b2=1, c4=d2=a4, bab=a5, cac-1=a-1, dad-1=a5b, cbc-1=a4b, bd=db, dcd-1=c3 >

Subgroups: 264 in 127 conjugacy classes, 40 normal (34 characteristic)
C1, C2, C2 [×4], C4 [×4], C4 [×4], C22 [×3], C22 [×3], C8 [×6], C2×C4 [×6], C2×C4 [×8], D4 [×4], Q8 [×8], C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8 [×4], M4(2) [×4], M4(2) [×6], SD16 [×4], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×4], C4.D4, C4.10D4 [×3], Q8⋊C4 [×2], C4≀C2 [×2], C8.C4 [×2], C42⋊C2, C2×M4(2) [×4], C2×SD16, C2×Q16, C8.C22 [×4], C22×Q8, C2×C4○D4, M4(2)⋊4C4, C2×C4.10D4, M4(2).8C22, C23.38D4, C42⋊C22, M4(2).C4, C2×C8.C22, M4(2).9D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×8], C23, C2×D4 [×4], C4○D4 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C23.10D4, M4(2).9D4

Character table of M4(2).9D4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 2 2 2 8 2 2 2 2 8 8 8 8 8 8 8 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 1 1 1 1 1 1 -1 1 1 1 -1 -1 -1 -1 -1 -1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 -1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 -1 1 1 1 1 -1 1 -1 1 -1 -1 1 1 1 -1 -1 1 -1 linear of order 2 ρ5 1 1 1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 1 -1 1 1 -1 -1 -1 1 linear of order 2 ρ6 1 1 1 1 1 -1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 linear of order 2 ρ7 1 1 1 1 1 1 1 1 1 1 1 -1 1 -1 1 -1 1 -1 -1 -1 -1 1 -1 linear of order 2 ρ8 1 1 1 1 1 -1 1 1 1 1 1 -1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 linear of order 2 ρ9 2 2 -2 -2 2 0 2 -2 2 -2 0 0 0 0 0 0 2 0 0 0 0 -2 0 orthogonal lifted from D4 ρ10 2 2 -2 -2 2 0 -2 2 -2 2 0 2 0 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 2 -2 2 2 -2 -2 2 0 0 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 -2 2 0 -2 2 -2 2 0 -2 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 -2 -2 0 -2 -2 2 2 0 0 0 0 0 0 0 0 0 2 -2 0 0 orthogonal lifted from D4 ρ14 2 2 -2 -2 2 0 2 -2 2 -2 0 0 0 0 0 0 -2 0 0 0 0 2 0 orthogonal lifted from D4 ρ15 2 2 2 -2 -2 0 -2 -2 2 2 0 0 0 0 0 0 0 0 0 -2 2 0 0 orthogonal lifted from D4 ρ16 2 2 -2 2 -2 -2 2 -2 -2 2 0 0 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ17 2 2 -2 2 -2 0 -2 2 2 -2 0 0 0 0 0 2i 0 0 0 0 0 0 -2i complex lifted from C4○D4 ρ18 2 2 2 -2 -2 0 2 2 -2 -2 -2i 0 0 0 2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ19 2 2 -2 2 -2 0 -2 2 2 -2 0 0 0 0 0 -2i 0 0 0 0 0 0 2i complex lifted from C4○D4 ρ20 2 2 2 2 2 0 -2 -2 -2 -2 0 0 0 0 0 0 0 -2i 2i 0 0 0 0 complex lifted from C4○D4 ρ21 2 2 2 2 2 0 -2 -2 -2 -2 0 0 0 0 0 0 0 2i -2i 0 0 0 0 complex lifted from C4○D4 ρ22 2 2 2 -2 -2 0 2 2 -2 -2 2i 0 0 0 -2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of M4(2).9D4 in GL8(𝔽17)

 7 0 7 0 6 8 6 8 3 0 3 0 16 7 16 7 11 0 6 0 11 9 6 8 12 0 5 0 1 10 16 7 9 3 10 14 12 16 14 13 5 4 14 13 12 16 14 13 0 3 9 3 5 1 14 13 13 4 5 4 5 1 14 13
,
 0 0 16 2 0 0 0 0 0 0 16 1 0 0 0 0 1 15 0 0 0 0 0 0 1 16 0 0 0 0 0 0 6 11 0 10 0 0 0 16 0 11 7 10 0 0 1 0 0 7 11 6 0 1 0 0 10 7 0 6 16 0 0 0
,
 7 9 10 8 0 0 0 0 3 10 14 7 0 0 0 0 7 9 7 9 0 0 0 0 3 10 3 10 0 0 0 0 7 4 1 16 13 3 4 14 8 4 0 16 3 4 14 13 13 4 5 1 13 3 13 3 0 4 9 1 3 4 3 4
,
 7 9 10 8 0 0 0 0 3 10 14 7 0 0 0 0 10 8 10 8 0 0 0 0 14 7 14 7 0 0 0 0 16 1 10 13 13 3 4 14 0 1 9 13 3 4 14 13 12 16 4 13 4 14 4 14 8 16 0 13 14 13 14 13

`G:=sub<GL(8,GF(17))| [7,3,11,12,9,5,0,13,0,0,0,0,3,4,3,4,7,3,6,5,10,14,9,5,0,0,0,0,14,13,3,4,6,16,11,1,12,12,5,5,8,7,9,10,16,16,1,1,6,16,6,16,14,14,14,14,8,7,8,7,13,13,13,13],[0,0,1,1,6,0,0,10,0,0,15,16,11,11,7,7,16,16,0,0,0,7,11,0,2,1,0,0,10,10,6,6,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0],[7,3,7,3,7,8,13,0,9,10,9,10,4,4,4,4,10,14,7,3,1,0,5,9,8,7,9,10,16,16,1,1,0,0,0,0,13,3,13,3,0,0,0,0,3,4,3,4,0,0,0,0,4,14,13,3,0,0,0,0,14,13,3,4],[7,3,10,14,16,0,12,8,9,10,8,7,1,1,16,16,10,14,10,14,10,9,4,0,8,7,8,7,13,13,13,13,0,0,0,0,13,3,4,14,0,0,0,0,3,4,14,13,0,0,0,0,4,14,4,14,0,0,0,0,14,13,14,13] >;`

M4(2).9D4 in GAP, Magma, Sage, TeX

`M_4(2)._9D_4`
`% in TeX`

`G:=Group("M4(2).9D4");`
`// GroupNames label`

`G:=SmallGroup(128,781);`
`// by ID`

`G=gap.SmallGroup(128,781);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,422,387,58,2019,1018,248,2804,172,1027]);`
`// Polycyclic`

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

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