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## G = C42.C22order 64 = 26

### 1st non-split extension by C42 of C22 acting faithfully

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

Aliases: C42.1C22, C2.6C4≀C2, C8⋊C46C2, (C2×D4).1C4, (C2×C4).96D4, (C2×Q8).1C4, C4.4D4.1C2, C2.3(C4.D4), C22.37(C22⋊C4), (C2×C4).10(C2×C4), SmallGroup(64,10)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.C22
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4.4D4 — C42.C22
 Lower central C1 — C22 — C2×C4 — C42.C22
 Upper central C1 — C22 — C42 — C42.C22
 Jennings C1 — C22 — C22 — C42 — C42.C22

Generators and relations for C42.C22
G = < a,b,c,d | a4=b4=d2=1, c2=b, ab=ba, cac-1=ab2, dad=a-1, bc=cb, dbd=a2b-1, dcd=a-1b2c >

Character table of C42.C22

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 8 2 2 2 2 4 8 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 trivial ρ2 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 linear of order 2 ρ4 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 i i -i -i i -i i -i linear of order 4 ρ6 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 -i -i i i -i i -i i linear of order 4 ρ8 1 1 1 1 -1 -1 -1 -1 -1 1 1 -i i -i i -i -i i i linear of order 4 ρ9 2 2 2 2 0 -2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 0 2i 0 0 -2i 0 0 0 1-i -1-i 0 0 1+i -1+i 0 complex lifted from C4≀C2 ρ12 2 -2 2 -2 0 -2i 0 0 2i 0 0 0 1+i -1+i 0 0 1-i -1-i 0 complex lifted from C4≀C2 ρ13 2 -2 -2 2 0 0 -2i 2i 0 0 0 1+i 0 0 -1+i -1-i 0 0 1-i complex lifted from C4≀C2 ρ14 2 -2 -2 2 0 0 2i -2i 0 0 0 -1+i 0 0 1+i 1-i 0 0 -1-i complex lifted from C4≀C2 ρ15 2 -2 -2 2 0 0 2i -2i 0 0 0 1-i 0 0 -1-i -1+i 0 0 1+i complex lifted from C4≀C2 ρ16 2 -2 2 -2 0 2i 0 0 -2i 0 0 0 -1+i 1+i 0 0 -1-i 1-i 0 complex lifted from C4≀C2 ρ17 2 -2 -2 2 0 0 -2i 2i 0 0 0 -1-i 0 0 1-i 1+i 0 0 -1+i complex lifted from C4≀C2 ρ18 2 -2 2 -2 0 -2i 0 0 2i 0 0 0 -1-i 1-i 0 0 -1+i 1+i 0 complex lifted from C4≀C2 ρ19 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4.D4

Smallest permutation representation of C42.C22
On 32 points
Generators in S32
```(1 22 31 15)(2 19 32 12)(3 24 25 9)(4 21 26 14)(5 18 27 11)(6 23 28 16)(7 20 29 13)(8 17 30 10)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 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)
(2 12)(3 29)(4 17)(6 16)(7 25)(8 21)(9 13)(10 26)(11 18)(14 30)(15 22)(19 32)(20 24)(23 28)```

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

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

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

C42.C22 is a maximal subgroup of
C42.C23  C42.2C23  C42.3C23  C42.5C23  C42.6C23  C42.7C23  C42.8C23  C42.10C23  (C2×D4).F5  (C2×Q8).F5
C42.D2p: C42.2D4  C42.3D4  C42.66D4  C42.405D4  C42.407D4  C42.376D4  C42.67D4  C42.69D4 ...
C42.C22 is a maximal quotient of
(C2×C4).98D8  (C2×Q8)⋊C8  C4.C4≀C2  C42.(C2×C4)  (C2×D4).F5  (C2×Q8).F5
C42.D2p: C42.7Q8  C42.D6  C42.7D6  C42.D10  C42.7D10  C42.D14  C42.7D14 ...

Matrix representation of C42.C22 in GL4(𝔽17) generated by

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

C42.C22 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(64,10);`
`// by ID`

`G=gap.SmallGroup(64,10);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,362,332,158,681,69]);`
`// Polycyclic`

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

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