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## G = Q8○M4(2)  order 64 = 26

### Central product of Q8 and M4(2)

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for Q8○M4(2)
G = < a,b,c,d | a4=d2=1, b2=c4=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c >

Subgroups: 145 in 129 conjugacy classes, 119 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C2×M4(2), C8○D4, C2×C4○D4, Q8○M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C23×C4, Q8○M4(2)

Permutation representations of Q8○M4(2)
On 16 points - transitive group 16T70
Generators in S16
```(1 16 5 12)(2 9 6 13)(3 10 7 14)(4 11 8 15)
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 6)(4 8)(9 13)(11 15)```

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

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

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

`G:=TransitiveGroup(16,70);`

34 conjugacy classes

 class 1 2A 2B ··· 2H 4A 4B 4C ··· 4I 8A ··· 8P order 1 2 2 ··· 2 4 4 4 ··· 4 8 ··· 8 size 1 1 2 ··· 2 1 1 2 ··· 2 2 ··· 2

34 irreducible representations

 dim 1 1 1 1 1 1 1 4 type + + + + image C1 C2 C2 C2 C4 C4 C4 Q8○M4(2) kernel Q8○M4(2) C2×M4(2) C8○D4 C2×C4○D4 C2×D4 C2×Q8 C4○D4 C1 # reps 1 6 8 1 6 2 8 2

Matrix representation of Q8○M4(2) in GL4(𝔽5) generated by

 2 0 0 0 0 2 0 0 0 0 3 0 0 0 0 3
,
 0 0 1 0 0 0 0 4 4 0 0 0 0 1 0 0
,
 0 3 0 0 1 0 0 0 0 0 0 2 0 0 4 0
,
 4 0 0 0 0 1 0 0 0 0 4 0 0 0 0 1
`G:=sub<GL(4,GF(5))| [2,0,0,0,0,2,0,0,0,0,3,0,0,0,0,3],[0,0,4,0,0,0,0,1,1,0,0,0,0,4,0,0],[0,1,0,0,3,0,0,0,0,0,0,4,0,0,2,0],[4,0,0,0,0,1,0,0,0,0,4,0,0,0,0,1] >;`

Q8○M4(2) in GAP, Magma, Sage, TeX

`Q_8\circ M_4(2)`
`% in TeX`

`G:=Group("Q8oM4(2)");`
`// GroupNames label`

`G:=SmallGroup(64,249);`
`// by ID`

`G=gap.SmallGroup(64,249);`
`# by ID`

`G:=PCGroup([6,-2,2,2,2,-2,-2,96,332,963,88]);`
`// Polycyclic`

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

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