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## G = M4(2).47D4order 128 = 27

### 11st 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 — C2×C4 — M4(2).47D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — Q8○M4(2) — M4(2).47D4
 Lower central C1 — C2 — C2×C4 — M4(2).47D4
 Upper central C1 — C2 — C22×C4 — M4(2).47D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).47D4

Generators and relations for M4(2).47D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=ab, dad=a5b, cbc-1=a4b, bd=db, dcd=a2c-1 >

Subgroups: 364 in 162 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C4.D4, D4⋊C4, C4≀C2, C42⋊C2, C2×M4(2), C2×M4(2), C8○D4, C2×D8, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C4.10C42, M4(2)⋊4C4, C2×C4.D4, C23.37D4, C42⋊C22, Q8○M4(2), C2×C8⋊C22, M4(2).47D4
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).47D4

Character table of M4(2).47D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L size 1 1 2 2 2 4 4 8 8 2 2 2 2 4 4 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 -i i -i -i i i -i i i -i -i i -1 1 linear of order 4 ρ10 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 1 -1 linear of order 4 ρ11 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 -1 1 linear of order 4 ρ12 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 1 -1 linear of order 4 ρ13 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 1 -1 linear of order 4 ρ14 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 -1 1 linear of order 4 ρ15 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 1 -1 linear of order 4 ρ16 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 -1 1 linear of order 4 ρ17 2 2 -2 2 -2 0 0 0 0 -2 2 -2 2 0 0 0 0 0 0 0 -2 -2 2 0 2 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 -2 -2 -2 2 0 0 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 -2 -2 0 0 0 0 -2 -2 2 2 0 0 0 0 -2 2 2 0 0 0 -2 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 -2 -2 2 2 -2 0 0 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 -2 -2 2 -2 2 0 0 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ22 2 2 2 -2 -2 2 -2 0 0 2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ23 2 2 2 -2 -2 0 0 0 0 -2 -2 2 2 0 0 0 0 2 -2 -2 0 0 0 2 0 0 0 0 0 orthogonal lifted from D4 ρ24 2 2 -2 2 -2 0 0 0 0 -2 2 -2 2 0 0 0 0 0 0 0 2 2 -2 0 -2 0 0 0 0 orthogonal lifted from D4 ρ25 2 2 -2 -2 2 0 0 0 0 -2 2 2 -2 0 0 0 0 2i -2i 2i 0 0 0 -2i 0 0 0 0 0 complex lifted from C4○D4 ρ26 2 2 -2 -2 2 0 0 0 0 -2 2 2 -2 0 0 0 0 -2i 2i -2i 0 0 0 2i 0 0 0 0 0 complex lifted from C4○D4 ρ27 2 2 2 2 2 0 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 -2i 2i 2i 0 -2i 0 0 0 0 complex lifted from C4○D4 ρ28 2 2 2 2 2 0 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 2i -2i -2i 0 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 orthogonal faithful

Permutation representations of M4(2).47D4
On 16 points - transitive group 16T267
Generators in S16
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 5)(3 7)(10 14)(12 16)
(1 11 5 15)(2 12)(3 9 7 13)(4 10)(6 16)(8 14)
(2 6)(3 7)(9 15)(10 16)(11 13)(12 14)```

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

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

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

`G:=TransitiveGroup(16,267);`

Matrix representation of M4(2).47D4 in GL8(ℤ)

 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 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 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 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 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 1 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 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0
,
 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1

`G:=sub<GL(8,Integers())| [0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,-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],[-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,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1] >;`

M4(2).47D4 in GAP, Magma, Sage, TeX

`M_4(2)._{47}D_4`
`% in TeX`

`G:=Group("M4(2).47D4");`
`// GroupNames label`

`G:=SmallGroup(128,635);`
`// by ID`

`G=gap.SmallGroup(128,635);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,2019,521,248,1411,718,172,1027]);`
`// Polycyclic`

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

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