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## G = M4(2)⋊4C4order 64 = 26

### 4th semidirect product of M4(2) and C4 acting via C4/C2=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — M4(2)⋊4C4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×M4(2) — M4(2)⋊4C4
 Lower central C1 — C2 — C22 — M4(2)⋊4C4
 Upper central C1 — C4 — C22×C4 — M4(2)⋊4C4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2)⋊4C4

Generators and relations for M4(2)⋊4C4
G = < a,b,c | a8=b2=c4=1, bab=a5, cac-1=ab, cbc-1=a4b >

Character table of M4(2)⋊4C4

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 2 2 2 1 1 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 ρ1 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 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 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 linear of order 2 ρ5 1 1 1 -1 -1 1 1 -1 1 -1 -i i -i i -i 1 1 -i i i -1 -1 linear of order 4 ρ6 1 1 1 -1 -1 -1 -1 1 -1 1 i -i -i i 1 i -i -1 -1 1 -i i linear of order 4 ρ7 1 1 1 -1 -1 1 1 -1 1 -1 -i i -i i i -1 -1 i -i -i 1 1 linear of order 4 ρ8 1 1 1 -1 -1 -1 -1 1 -1 1 i -i -i i -1 -i i 1 1 -1 i -i linear of order 4 ρ9 1 1 1 -1 -1 1 1 -1 1 -1 i -i i -i -i -1 -1 -i i i 1 1 linear of order 4 ρ10 1 1 1 -1 -1 -1 -1 1 -1 1 -i i i -i 1 -i i -1 -1 1 i -i linear of order 4 ρ11 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 -1 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 i i -i -i i -i i -i linear of order 4 ρ13 1 1 1 -1 -1 1 1 -1 1 -1 i -i i -i i 1 1 i -i -i -1 -1 linear of order 4 ρ14 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 -1 -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 -i -i i i -i i -i i linear of order 4 ρ16 1 1 1 -1 -1 -1 -1 1 -1 1 -i i i -i -1 i -i 1 1 -1 -i i linear of order 4 ρ17 2 2 -2 -2 2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 -2 2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 2 -2 2 2 -2 -2 2 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 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ21 4 -4 0 0 0 4i -4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ22 4 -4 0 0 0 -4i 4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of M4(2)⋊4C4
On 16 points - transitive group 16T90
Generators in S16
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 6)(4 8)(9 13)(11 15)
(1 9)(2 14 6 10)(3 15)(4 12 8 16)(5 13)(7 11)```

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

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

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

`G:=TransitiveGroup(16,90);`

On 16 points - transitive group 16T110
Generators in S16
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 15)(2 12)(3 9)(4 14)(5 11)(6 16)(7 13)(8 10)
(1 9 5 13)(2 4)(3 15 7 11)(6 8)(10 12)(14 16)```

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

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

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

`G:=TransitiveGroup(16,110);`

Matrix representation of M4(2)⋊4C4 in GL4(𝔽5) generated by

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

M4(2)⋊4C4 in GAP, Magma, Sage, TeX

`M_4(2)\rtimes_4C_4`
`% in TeX`

`G:=Group("M4(2):4C4");`
`// GroupNames label`

`G:=SmallGroup(64,25);`
`// by ID`

`G=gap.SmallGroup(64,25);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,2,-2,48,73,103,650,489,88]);`
`// Polycyclic`

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

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