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## G = M4(2).37D4order 128 = 27

### 1st 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).37D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — Q8○M4(2) — M4(2).37D4
 Lower central C1 — C2 — C2×C4 — M4(2).37D4
 Upper central C1 — C2 — C22×C4 — M4(2).37D4
 Jennings C1 — C2 — C2 — C2×C4 — M4(2).37D4

Generators and relations for M4(2).37D4
G = < a,b,c,d | a8=b2=d2=1, c4=a4, bab=a5, cac-1=dad=a-1, bc=cb, bd=db, dcd=a4c3 >

Subgroups: 476 in 233 conjugacy classes, 98 normal (36 characteristic)
C1, C2, C2 [×8], C4 [×4], C4 [×3], C22 [×3], C22 [×14], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×8], D4 [×2], D4 [×15], Q8 [×2], Q8 [×3], C23, C23 [×8], C2×C8 [×2], C2×C8 [×7], M4(2) [×10], M4(2) [×7], D8 [×10], SD16 [×12], Q16 [×2], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×D4 [×7], C2×Q8 [×2], C4○D4 [×4], C4○D4 [×6], C24, C4.D4 [×6], C4.10D4 [×2], C8.C4 [×4], C2×M4(2) [×4], C2×M4(2) [×2], C8○D4 [×4], C8○D4 [×2], C2×D8 [×2], C2×D8, C2×SD16 [×2], C2×SD16, C4○D8 [×2], C8⋊C22 [×6], C8⋊C22 [×7], C8.C22 [×2], C8.C22, C22×D4, C2×C4○D4 [×2], C2×C4.D4, M4(2).8C22, M4(2).C4, D4.3D4 [×4], D4.4D4 [×4], Q8○M4(2), C2×C8⋊C22 [×2], D8⋊C22, M4(2).37D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, M4(2).37D4

Character table of M4(2).37D4

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

Permutation representations of M4(2).37D4
On 16 points - transitive group 16T312
Generators in S16
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 6)(4 8)(10 14)(12 16)
(1 11 7 13 5 15 3 9)(2 10 8 12 6 14 4 16)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)```

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

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

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

`G:=TransitiveGroup(16,312);`

Matrix representation of M4(2).37D4 in GL8(ℤ)

 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 -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
,
 -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 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 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
,
 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())| [0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,-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,-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],[0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-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],[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).37D4 in GAP, Magma, Sage, TeX

`M_4(2)._{37}D_4`
`% in TeX`

`G:=Group("M4(2).37D4");`
`// GroupNames label`

`G:=SmallGroup(128,1800);`
`// by ID`

`G=gap.SmallGroup(128,1800);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,1018,2804,172,4037,1027,124]);`
`// Polycyclic`

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

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