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## G = C22.M4(2)  order 64 = 26

### 2nd non-split extension by C22 of M4(2) acting via M4(2)/C2×C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C22.M4(2)
G = < a,b,c,d | a2=b2=c8=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=abc5 >

Character table of C22.M4(2)

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 2 2 2 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 i -i -i i -i i -i i linear of order 4 ρ6 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 ρ7 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 ρ8 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 ρ9 1 1 -1 -1 -1 1 -i i i -i 1 i -1 -i ζ83 ζ87 ζ85 ζ87 ζ83 ζ8 ζ8 ζ85 linear of order 8 ρ10 1 1 -1 -1 -1 1 i -i -i i 1 -i -1 i ζ85 ζ8 ζ83 ζ8 ζ85 ζ87 ζ87 ζ83 linear of order 8 ρ11 1 1 -1 -1 -1 1 -i i i -i 1 i -1 -i ζ87 ζ83 ζ8 ζ83 ζ87 ζ85 ζ85 ζ8 linear of order 8 ρ12 1 1 -1 -1 -1 1 i -i -i i 1 -i -1 i ζ8 ζ85 ζ87 ζ85 ζ8 ζ83 ζ83 ζ87 linear of order 8 ρ13 1 1 -1 -1 -1 1 i -i -i i -1 i 1 -i ζ8 ζ8 ζ87 ζ85 ζ85 ζ87 ζ83 ζ83 linear of order 8 ρ14 1 1 -1 -1 -1 1 -i i i -i -1 -i 1 i ζ83 ζ83 ζ85 ζ87 ζ87 ζ85 ζ8 ζ8 linear of order 8 ρ15 1 1 -1 -1 -1 1 -i i i -i -1 -i 1 i ζ87 ζ87 ζ8 ζ83 ζ83 ζ8 ζ85 ζ85 linear of order 8 ρ16 1 1 -1 -1 -1 1 i -i -i i -1 i 1 -i ζ85 ζ85 ζ83 ζ8 ζ8 ζ83 ζ87 ζ87 linear of order 8 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 -2i -2i 2i 2i 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ20 2 2 -2 -2 2 -2 2i 2i -2i -2i 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ21 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ22 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C4.10D4, Schur index 2

Smallest permutation representation of C22.M4(2)
On 32 points
Generators in S32
```(1 5)(2 30)(3 7)(4 32)(6 26)(8 28)(9 24)(10 14)(11 18)(12 16)(13 20)(15 22)(17 21)(19 23)(25 29)(27 31)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)
(1 12 25 23)(2 24 26 13)(3 17 27 14)(4 15 28 18)(5 16 29 19)(6 20 30 9)(7 21 31 10)(8 11 32 22)```

`G:=sub<Sym(32)| (1,5)(2,30)(3,7)(4,32)(6,26)(8,28)(9,24)(10,14)(11,18)(12,16)(13,20)(15,22)(17,21)(19,23)(25,29)(27,31), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,12,25,23)(2,24,26,13)(3,17,27,14)(4,15,28,18)(5,16,29,19)(6,20,30,9)(7,21,31,10)(8,11,32,22)>;`

`G:=Group( (1,5)(2,30)(3,7)(4,32)(6,26)(8,28)(9,24)(10,14)(11,18)(12,16)(13,20)(15,22)(17,21)(19,23)(25,29)(27,31), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,12,25,23)(2,24,26,13)(3,17,27,14)(4,15,28,18)(5,16,29,19)(6,20,30,9)(7,21,31,10)(8,11,32,22) );`

`G=PermutationGroup([(1,5),(2,30),(3,7),(4,32),(6,26),(8,28),(9,24),(10,14),(11,18),(12,16),(13,20),(15,22),(17,21),(19,23),(25,29),(27,31)], [(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,20),(10,21),(11,22),(12,23),(13,24),(14,17),(15,18),(16,19)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32)], [(1,12,25,23),(2,24,26,13),(3,17,27,14),(4,15,28,18),(5,16,29,19),(6,20,30,9),(7,21,31,10),(8,11,32,22)])`

C22.M4(2) is a maximal subgroup of
C42.371D4  C42.394D4  C42.42D4  C42.44D4  C42.396D4  C42.372D4  (C2×C20)⋊1C8  (C22×C4).F5
(C2×C4).D4p: C2.C2≀C4  (C2×C4).D8  (C2×C4).5D8  (C2×Dic3)⋊C8  (C2×Dic5)⋊C8  (C2×Dic7)⋊C8 ...
(C22×C4).D2p: (C2×D4)⋊C8  (C2×C42).C4  C42⋊C8  C423C8  (C2×C4).Q16  C2.7C2≀C4  C23.8M4(2)  (C2×C4)⋊M4(2) ...
C22.M4(2) is a maximal quotient of
C4⋊C4⋊C8  C23.19C42  C42⋊C8  C423C8  C23.2M4(2)  (C2×C20)⋊1C8  (C22×C4).F5
(C2×C4p).D4: C22.M5(2)  C23.7M4(2)  C42.C8  C22⋊C4.C8  (C2×Dic3)⋊C8  (C2×C12)⋊C8  (C2×Dic5)⋊C8  (C2×C20)⋊C8 ...

Matrix representation of C22.M4(2) in GL6(𝔽17)

 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 16 0 0 0 4 13 0 16
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 0 8 0 0 0 0 9 0 0 0 0 0 0 0 5 12 9 6 0 0 3 14 8 7 0 0 14 12 0 0 0 0 0 9 2 15
,
 0 16 0 0 0 0 16 0 0 0 0 0 0 0 12 14 0 0 0 0 3 5 0 0 0 0 5 12 9 6 0 0 2 0 9 8

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,4,0,0,0,1,0,13,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,9,0,0,0,0,8,0,0,0,0,0,0,0,5,3,14,0,0,0,12,14,12,9,0,0,9,8,0,2,0,0,6,7,0,15],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,12,3,5,2,0,0,14,5,12,0,0,0,0,0,9,9,0,0,0,0,6,8] >;`

C22.M4(2) in GAP, Magma, Sage, TeX

`C_2^2.M_4(2)`
`% in TeX`

`G:=Group("C2^2.M4(2)");`
`// GroupNames label`

`G:=SmallGroup(64,5);`
`// by ID`

`G=gap.SmallGroup(64,5);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,103,362,297,117]);`
`// Polycyclic`

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

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