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## G = M4(2).C4order 64 = 26

### 1st non-split extension by M4(2) of C4 acting via C4/C2=C2

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for M4(2).C4
G = < a,b,c | a8=b2=1, c4=a4, bab=a5, cac-1=a-1, cbc-1=a4b >

Character table of M4(2).C4

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 8K 8L 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 -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 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 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 linear of order 2 ρ9 1 1 -1 -1 1 -1 -1 1 1 -1 -i -1 -1 i 1 1 -i -i i i i -i linear of order 4 ρ10 1 1 -1 1 -1 -1 -1 -1 1 1 i 1 -1 -i -1 1 i -i i -i i -i linear of order 4 ρ11 1 1 -1 -1 1 -1 -1 1 1 -1 i -1 -1 -i 1 1 i i -i -i -i i linear of order 4 ρ12 1 1 -1 1 -1 -1 -1 -1 1 1 -i 1 -1 i -1 1 -i i -i i -i i linear of order 4 ρ13 1 1 -1 1 -1 -1 -1 -1 1 1 i -1 1 i 1 -1 -i i i -i -i -i linear of order 4 ρ14 1 1 -1 -1 1 -1 -1 1 1 -1 -i 1 1 -i -1 -1 i i i i -i -i linear of order 4 ρ15 1 1 -1 1 -1 -1 -1 -1 1 1 -i -1 1 -i 1 -1 i -i -i i i i linear of order 4 ρ16 1 1 -1 -1 1 -1 -1 1 1 -1 i 1 1 i -1 -1 -i -i -i -i 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 symplectic lifted from Q8, Schur index 2 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).C4
On 16 points - transitive group 16T103
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 9 7 11 5 13 3 15)(2 16 8 10 6 12 4 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), (1,9,7,11,5,13,3,15)(2,16,8,10,6,12,4,14)>;`

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

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

`G:=TransitiveGroup(16,103);`

M4(2).C4 is a maximal subgroup of
M4(2).29C23  M4(2).1F5
M4(2).D2p: M4(2).40D4  M4(2).41D4  C42.427D4  M4(2).27D4  M4(2).8D4  M4(2).9D4  C24.Q8  M4(2).15D4 ...
(C2×C8).D2p: (C2×C8).D4  (C2×C8).6D4  C24.11Q8  C42.10D4  C42.32Q8  C22⋊C4.Q8  M5(2).C22  C23.10SD16 ...
M4(2).C4 is a maximal quotient of
M4(2)⋊1C8  C81M4(2)  C42.90D4  C42.91D4  C42.Q8  C24.7Q8  C8.6C42  C42.104D4  M4(2).1F5
M4(2).D2p: C24.10Q8  C42.430D4  M4(2).25D6  C23.8Dic6  M4(2).25D10  C23.Dic10  M4(2).25D14  C23.Dic14 ...
(C2×C8).D2p: C24.9Q8  C42.106D4  C23.9Dic6  M4(2).Dic5  M4(2).Dic7 ...

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

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

M4(2).C4 in GAP, Magma, Sage, TeX

`M_4(2).C_4`
`% in TeX`

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

`G:=SmallGroup(64,111);`
`// by ID`

`G=gap.SmallGroup(64,111);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,55,332,963,117,88]);`
`// Polycyclic`

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

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