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## G = C42.2C22order 64 = 26

### 2nd non-split extension by C42 of C22 acting faithfully

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

Aliases: C42.2C22, C2.7C4≀C2, C4⋊C4.1C4, (C2×C4).97D4, C8⋊C4.3C2, C42.C2.1C2, C2.3(C4.10D4), C22.38(C22⋊C4), (C2×C4).11(C2×C4), SmallGroup(64,11)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.2C22
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C42.C2 — C42.2C22
 Lower central C1 — C22 — C2×C4 — C42.2C22
 Upper central C1 — C22 — C42 — C42.2C22
 Jennings C1 — C22 — C22 — C42 — C42.2C22

Generators and relations for C42.2C22
G = < a,b,c,d | a4=b4=1, c2=b, d2=b2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=a2b-1, dcd-1=a-1b2c >

Character table of C42.2C22

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 2 2 2 2 4 8 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 -2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 2i 0 0 -2i 0 0 0 0 -1+i 1+i 0 0 -1-i 1-i 0 complex lifted from C4≀C2 ρ12 2 -2 2 -2 -2i 0 0 2i 0 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 2i -2i 0 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 -2i 2i 0 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 -2i 0 0 2i 0 0 0 0 -1-i 1-i 0 0 -1+i 1+i 0 complex lifted from C4≀C2 ρ16 2 -2 -2 2 0 -2i 2i 0 0 0 0 1-i 0 0 1+i -1+i 0 0 -1-i complex lifted from C4≀C2 ρ17 2 -2 -2 2 0 2i -2i 0 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 2i 0 0 -2i 0 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 symplectic lifted from C4.10D4, Schur index 2

Smallest permutation representation of C42.2C22
Regular action on 64 points
Generators in S64
```(1 31 55 47)(2 28 56 44)(3 25 49 41)(4 30 50 46)(5 27 51 43)(6 32 52 48)(7 29 53 45)(8 26 54 42)(9 57 40 17)(10 62 33 22)(11 59 34 19)(12 64 35 24)(13 61 36 21)(14 58 37 18)(15 63 38 23)(16 60 39 20)
(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)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 63 5 59)(2 12 6 16)(3 21 7 17)(4 33 8 37)(9 41 13 45)(10 54 14 50)(11 31 15 27)(18 46 22 42)(19 55 23 51)(20 28 24 32)(25 36 29 40)(26 58 30 62)(34 47 38 43)(35 52 39 56)(44 64 48 60)(49 61 53 57)```

`G:=sub<Sym(64)| (1,31,55,47)(2,28,56,44)(3,25,49,41)(4,30,50,46)(5,27,51,43)(6,32,52,48)(7,29,53,45)(8,26,54,42)(9,57,40,17)(10,62,33,22)(11,59,34,19)(12,64,35,24)(13,61,36,21)(14,58,37,18)(15,63,38,23)(16,60,39,20), (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)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,63,5,59)(2,12,6,16)(3,21,7,17)(4,33,8,37)(9,41,13,45)(10,54,14,50)(11,31,15,27)(18,46,22,42)(19,55,23,51)(20,28,24,32)(25,36,29,40)(26,58,30,62)(34,47,38,43)(35,52,39,56)(44,64,48,60)(49,61,53,57)>;`

`G:=Group( (1,31,55,47)(2,28,56,44)(3,25,49,41)(4,30,50,46)(5,27,51,43)(6,32,52,48)(7,29,53,45)(8,26,54,42)(9,57,40,17)(10,62,33,22)(11,59,34,19)(12,64,35,24)(13,61,36,21)(14,58,37,18)(15,63,38,23)(16,60,39,20), (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)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,63,5,59)(2,12,6,16)(3,21,7,17)(4,33,8,37)(9,41,13,45)(10,54,14,50)(11,31,15,27)(18,46,22,42)(19,55,23,51)(20,28,24,32)(25,36,29,40)(26,58,30,62)(34,47,38,43)(35,52,39,56)(44,64,48,60)(49,61,53,57) );`

`G=PermutationGroup([(1,31,55,47),(2,28,56,44),(3,25,49,41),(4,30,50,46),(5,27,51,43),(6,32,52,48),(7,29,53,45),(8,26,54,42),(9,57,40,17),(10,62,33,22),(11,59,34,19),(12,64,35,24),(13,61,36,21),(14,58,37,18),(15,63,38,23),(16,60,39,20)], [(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),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,63,5,59),(2,12,6,16),(3,21,7,17),(4,33,8,37),(9,41,13,45),(10,54,14,50),(11,31,15,27),(18,46,22,42),(19,55,23,51),(20,28,24,32),(25,36,29,40),(26,58,30,62),(34,47,38,43),(35,52,39,56),(44,64,48,60),(49,61,53,57)])`

C42.2C22 is a maximal subgroup of
C42.2C23  C42.3C23  C42.4C23  C42.6C23  C42.7C23  C42.8C23  C42.9C23  C42.10C23  C10.C4≀C2
C42.D2p: C42.66D4  C42.406D4  C42.408D4  C42.376D4  C42.68D4  C42.69D4  C42.71D4  C42.72D4 ...
C42.2C22 is a maximal quotient of
C4⋊C4⋊C8  C10.C4≀C2
C42.D2p: C42.7Q8  C42.2D6  C42.8D6  C42.2D10  C42.8D10  C42.2D14  C42.8D14 ...

Matrix representation of C42.2C22 in GL4(𝔽17) generated by

 0 13 0 0 13 0 0 0 0 0 0 1 0 0 16 0
,
 0 1 0 0 1 0 0 0 0 0 4 0 0 0 0 4
,
 6 10 0 0 10 6 0 0 0 0 0 2 0 0 2 0
,
 4 10 0 0 7 13 0 0 0 0 5 12 0 0 12 12
`G:=sub<GL(4,GF(17))| [0,13,0,0,13,0,0,0,0,0,0,16,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,4],[6,10,0,0,10,6,0,0,0,0,0,2,0,0,2,0],[4,7,0,0,10,13,0,0,0,0,5,12,0,0,12,12] >;`

C42.2C22 in GAP, Magma, Sage, TeX

`C_4^2._2C_2^2`
`% in TeX`

`G:=Group("C4^2.2C2^2");`
`// GroupNames label`

`G:=SmallGroup(64,11);`
`// by ID`

`G=gap.SmallGroup(64,11);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,103,362,332,158,681,69]);`
`// Polycyclic`

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

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