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## G = C22.C42order 64 = 26

### 2nd non-split extension by C22 of C42 acting via C42/C2×C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C22.C42
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4⋊C4 — C22.C42
 Lower central C1 — C2 — C22 — C22.C42
 Upper central C1 — C22 — C22×C4 — C22.C42
 Jennings C1 — C2 — C2 — C22×C4 — C22.C42

Generators and relations for C22.C42
G = < a,b,c,d | a2=b2=d4=1, c4=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=abc >

Character table of C22.C42

 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 i -i i -i -1 1 -1 -i i -i 1 i linear of order 4 ρ9 1 -1 1 -1 -1 1 -1 1 -1 1 -i i -i i -1 1 -1 i -i i 1 -i linear of order 4 ρ10 1 -1 1 -1 -1 1 1 -1 1 -1 i -i -i i -i -i i 1 1 -1 i -1 linear of order 4 ρ11 1 -1 1 -1 -1 1 1 -1 1 -1 -i i i -i -i -i i -1 -1 1 i 1 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 1 -1 1 -i i -i -1 i linear of order 4 ρ14 1 -1 1 -1 -1 1 -1 1 -1 1 i -i i -i 1 -1 1 i -i i -1 -i linear of order 4 ρ15 1 -1 1 -1 -1 1 1 -1 1 -1 -i i i -i i i -i 1 1 -1 -i -1 linear of order 4 ρ16 1 -1 1 -1 -1 1 1 -1 1 -1 i -i -i i i i -i -1 -1 1 -i 1 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 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4.D4 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.C42
On 32 points
Generators in S32
```(1 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 20)(10 17)(11 22)(12 19)(13 24)(14 21)(15 18)(16 23)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)
(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 17 31 10)(2 15 32 22)(3 23 25 16)(4 13 26 20)(5 21 27 14)(6 11 28 18)(7 19 29 12)(8 9 30 24)```

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

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

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

Matrix representation of C22.C42 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 11 3 16 0 0 0 6 14 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 1 0 0 0 0 1 0 0 0 0 0 0 0 11 3 15 0 0 0 0 0 1 1 0 0 10 9 16 10 0 0 8 9 1 7
,
 4 0 0 0 0 0 0 13 0 0 0 0 0 0 10 12 0 0 0 0 13 7 0 0 0 0 11 2 4 11 0 0 13 3 11 13

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,11,6,0,0,0,1,3,14,0,0,0,0,16,0,0,0,0,0,0,16],[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],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,11,0,10,8,0,0,3,0,9,9,0,0,15,1,16,1,0,0,0,1,10,7],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,10,13,11,13,0,0,12,7,2,3,0,0,0,0,4,11,0,0,0,0,11,13] >;`

C22.C42 in GAP, Magma, Sage, TeX

`C_2^2.C_4^2`
`% in TeX`

`G:=Group("C2^2.C4^2");`
`// GroupNames label`

`G:=SmallGroup(64,24);`
`// by ID`

`G=gap.SmallGroup(64,24);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,2,-2,48,73,103,650,489,117]);`
`// Polycyclic`

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

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