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## G = M4(2).50D4order 128 = 27

### 14th non-split extension by M4(2) of D4 acting via D4/C22=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — M4(2).50D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — Q8○M4(2) — M4(2).50D4
 Lower central C1 — C2 — C23 — M4(2).50D4
 Upper central C1 — C2 — C22×C4 — M4(2).50D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).50D4

Generators and relations for M4(2).50D4
G = < a,b,c,d | a8=b2=1, c4=a4, d2=a2b, bab=a5, cac-1=a5b, dad-1=ab, bc=cb, dbd-1=a4b, dcd-1=a6bc3 >

Subgroups: 284 in 154 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4.D4, C4.10D4, C42⋊C2, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C2×M4(2), C2×M4(2), C8○D4, C22×Q8, C2×C4○D4, M4(2)⋊4C4, C2×C4.10D4, M4(2).8C22, C23.38C23, Q8○M4(2), M4(2).50D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C23.23D4, M4(2).50D4

Character table of M4(2).50D4

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L size 1 1 2 2 2 4 4 2 2 2 2 4 4 8 8 8 8 4 4 4 4 4 4 4 4 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 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 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 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 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 -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 -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 -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 -1 -1 1 1 1 1 linear of order 2 ρ9 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 -1 -1 1 1 -i i i -i i -i i -i i -i -i i linear of order 4 ρ10 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 -1 -1 1 1 i -i -i i -i i -i i -i i i -i linear of order 4 ρ11 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 -1 -1 i -i -i i -i i -i i i -i -i i linear of order 4 ρ12 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 -1 -1 -i i i -i i -i i -i -i i i -i linear of order 4 ρ13 1 1 1 1 1 -1 1 -1 -1 -1 -1 1 -1 1 -1 1 -1 -i i -i i -i i i -i -i -i i i linear of order 4 ρ14 1 1 1 1 1 -1 1 -1 -1 -1 -1 1 -1 1 -1 1 -1 i -i i -i i -i -i i i i -i -i linear of order 4 ρ15 1 1 1 1 1 -1 1 -1 -1 -1 -1 1 -1 -1 1 -1 1 i -i i -i i -i -i i -i -i i i linear of order 4 ρ16 1 1 1 1 1 -1 1 -1 -1 -1 -1 1 -1 -1 1 -1 1 -i i -i i -i i i -i i i -i -i linear of order 4 ρ17 2 2 -2 2 -2 -2 -2 -2 2 2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 -2 2 0 0 -2 2 -2 2 0 0 0 0 0 0 2 -2 0 0 0 0 2 -2 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 2 -2 -2 2 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 -2 2 -2 2 -2 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 2 -2 -2 0 0 -2 -2 2 2 0 0 0 0 0 0 0 0 -2 -2 2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ22 2 2 -2 -2 2 0 0 -2 2 -2 2 0 0 0 0 0 0 -2 2 0 0 0 0 -2 2 0 0 0 0 orthogonal lifted from D4 ρ23 2 2 -2 2 -2 2 2 -2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ24 2 2 2 -2 -2 0 0 -2 -2 2 2 0 0 0 0 0 0 0 0 2 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ25 2 2 2 -2 -2 0 0 2 2 -2 -2 0 0 0 0 0 0 0 0 2i -2i -2i 2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ26 2 2 -2 -2 2 0 0 2 -2 2 -2 0 0 0 0 0 0 -2i -2i 0 0 0 0 2i 2i 0 0 0 0 complex lifted from C4○D4 ρ27 2 2 2 -2 -2 0 0 2 2 -2 -2 0 0 0 0 0 0 0 0 -2i 2i 2i -2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ28 2 2 -2 -2 2 0 0 2 -2 2 -2 0 0 0 0 0 0 2i 2i 0 0 0 0 -2i -2i 0 0 0 0 complex lifted from C4○D4 ρ29 8 -8 0 0 0 0 0 0 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).50D4
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 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 19)(10 24)(11 21)(12 18)(13 23)(14 20)(15 17)(16 22)
(1 30 25 2 5 26 29 6)(3 28 27 8 7 32 31 4)(9 22 21 14 13 18 17 10)(11 20 23 12 15 24 19 16)
(1 18 25 14 5 22 29 10)(2 13 30 17 6 9 26 21)(3 24 27 12 7 20 31 16)(4 11 32 23 8 15 28 19)```

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

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

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

Matrix representation of M4(2).50D4 in GL8(𝔽17)

 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0
,
 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0
,
 0 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 0 0 0 13 0 0 0 0 4 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0

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

M4(2).50D4 in GAP, Magma, Sage, TeX

`M_4(2)._{50}D_4`
`% in TeX`

`G:=Group("M4(2).50D4");`
`// GroupNames label`

`G:=SmallGroup(128,647);`
`// by ID`

`G=gap.SmallGroup(128,647);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,2019,1018,521,2804,1411,2028,1027,124]);`
`// Polycyclic`

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

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