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## G = C42.30C22order 64 = 26

### 30th non-split extension by C42 of C22 acting faithfully

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

Aliases: C42.30C22, C4⋊Q8.8C2, (C2×C4).43D4, C8⋊C4.6C2, C4.18(C4○D4), C4⋊C4.21C22, (C2×C8).56C22, Q8⋊C4.7C2, C42.C2.3C2, (C2×C4).116C23, C22.112(C2×D4), (C2×Q8).23C22, C2.14(C4.4D4), C2.21(C8.C22), SmallGroup(64,172)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.30C22
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C8⋊C4 — C42.30C22
 Lower central C1 — C2 — C2×C4 — C42.30C22
 Upper central C1 — C22 — C42 — C42.30C22
 Jennings C1 — C2 — C2 — C2×C4 — C42.30C22

Generators and relations for C42.30C22
G = < a,b,c,d | a4=b4=1, c2=b2, d2=b, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=a2b-1c >

Character table of C42.30C22

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D size 1 1 1 1 2 2 4 4 8 8 8 8 4 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 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 linear of order 2 ρ7 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 linear of order 2 ρ9 2 2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 -2 2 0 0 0 0 0 0 -2i 0 2i 0 complex lifted from C4○D4 ρ12 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 2i 0 -2i complex lifted from C4○D4 ρ13 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 -2i 0 2i complex lifted from C4○D4 ρ14 2 -2 2 -2 -2 2 0 0 0 0 0 0 2i 0 -2i 0 complex lifted from C4○D4 ρ15 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ16 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2

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

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

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

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

C42.30C22 is a maximal subgroup of
C42.(C2×C4)  C8⋊C4.C4
C42.D2p: C42.239D4  C42.241D4  C42.244D4  C42.256D4  C42.258D4  C42.276D4  C42.277D4  C42.288D4 ...
C4⋊C4.D2p: C42.8C23  C42.9C23  C42.10C23  C42.367C23  C42.386C23  C42.389C23  C42.407C23  C42.409C23 ...
C42.30C22 is a maximal quotient of
C42.24Q8  C2.(C8⋊D4)  (C2×Q8).109D4  (C2×C4).28D8
C42.D2p: C42.111D4  C42.124D4  C42.14D6  C42.71D6  C42.77D6  C42.14D10  C42.71D10  C42.77D10 ...
C4⋊C4.D2p: C4⋊C4.85D4  C4⋊C4.95D4  (C2×Q8).36D6  C408C4.C2  C56⋊C4.C2 ...

Matrix representation of C42.30C22 in GL6(𝔽17)

 0 13 0 0 0 0 13 0 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 16 0 0 0 0 0 0 7 16 0 0 0 0 16 10 0 0 0 0 0 0 1 7 0 0 0 0 7 16
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 16 0 0 0 0 0 0 16 0 0

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

C42.30C22 in GAP, Magma, Sage, TeX

`C_4^2._{30}C_2^2`
`% in TeX`

`G:=Group("C4^2.30C2^2");`
`// GroupNames label`

`G:=SmallGroup(64,172);`
`// by ID`

`G=gap.SmallGroup(64,172);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,295,362,332,50,963,117,1444,88]);`
`// Polycyclic`

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

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