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## G = C4⋊2Q16order 64 = 26

### The semidirect product of C4 and Q16 acting via Q16/Q8=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C4⋊2Q16
 Chief series C1 — C2 — C4 — C2×C4 — C2×Q8 — C4×Q8 — C4⋊2Q16
 Lower central C1 — C2 — C2×C4 — C4⋊2Q16
 Upper central C1 — C22 — C42 — C4⋊2Q16
 Jennings C1 — C2 — C2 — C2×C4 — C4⋊2Q16

Generators and relations for C42Q16
G = < a,b,c | a4=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=b-1 >

Character table of C42Q16

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 8A 8B 8C 8D size 1 1 1 1 2 2 2 2 4 4 4 4 4 8 8 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 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 1 1 -1 linear of order 2 ρ7 1 1 1 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 -1 -1 -1 linear of order 2 ρ9 2 -2 2 -2 0 2 -2 0 0 0 0 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 -2 -2 2 0 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 0 2 -2 0 0 0 0 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 -2 -2 -2 -2 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 -2 -2 2 2 0 0 -2 0 0 0 0 0 0 0 -√2 -√2 √2 √2 symplectic lifted from Q16, Schur index 2 ρ14 2 -2 -2 2 2 0 0 -2 0 0 0 0 0 0 0 √2 √2 -√2 -√2 symplectic lifted from Q16, Schur index 2 ρ15 2 -2 -2 2 -2 0 0 2 0 0 0 0 0 0 0 √2 -√2 √2 -√2 symplectic lifted from Q16, Schur index 2 ρ16 2 -2 -2 2 -2 0 0 2 0 0 0 0 0 0 0 -√2 √2 -√2 √2 symplectic lifted from Q16, Schur index 2 ρ17 2 -2 2 -2 0 -2 2 0 2i 0 -2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ18 2 -2 2 -2 0 -2 2 0 -2i 0 2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ19 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C42Q16
Regular action on 64 points
Generators in S64
```(1 42 19 35)(2 36 20 43)(3 44 21 37)(4 38 22 45)(5 46 23 39)(6 40 24 47)(7 48 17 33)(8 34 18 41)(9 63 51 30)(10 31 52 64)(11 57 53 32)(12 25 54 58)(13 59 55 26)(14 27 56 60)(15 61 49 28)(16 29 50 62)
(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 13 5 9)(2 12 6 16)(3 11 7 15)(4 10 8 14)(17 49 21 53)(18 56 22 52)(19 55 23 51)(20 54 24 50)(25 40 29 36)(26 39 30 35)(27 38 31 34)(28 37 32 33)(41 60 45 64)(42 59 46 63)(43 58 47 62)(44 57 48 61)```

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

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

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

C42Q16 is a maximal subgroup of
C42.443D4  C42.212D4  C42.445D4  C42.224D4  C42.451D4  C42.231D4  C42.234D4  C42.267D4  C42.270D4  C42.274D4  C42.276D4  C42.296D4  C42.297D4  C42.300D4  C42.303D4  Q8.D12
C4p⋊Q16: C88Q16  C87Q16  C8⋊Q16  C82Q16  C4⋊Dic12  C127Q16  C12⋊Q16  C4⋊Dic20 ...
(Cp×Q8).D4: C42.201C23  Q8.D8  D43Q16  Q83Q16  Q84Q16  D44Q16  C42.213C23  Q8.SD16 ...
C4⋊C4.D2p: C42.19C23  C42.354C23  C42.361C23  C42.409C23  C42.411C23  C42.25C23  C42.28C23  SD168D4 ...
C42Q16 is a maximal quotient of
C2.(C4×Q16)  C42.29Q8  C42.117D4  (C2×C8).1Q8
C4p⋊Q16: C88Q16  C87Q16  C8⋊Q16  C82Q16  C4⋊Dic12  C127Q16  C12⋊Q16  C4⋊Dic20 ...
(Cp×Q8).D4: Q8.1Q16  C8.3Q16  C42.99D4  (C2×C4)⋊9Q16  (C2×C4)⋊2Q16  C4⋊C4.95D4  (C2×C4)⋊3Q16  (C2×C4).19Q16 ...
(C2×C8).D2p: (C2×C8).52D4  Dic3⋊Q16  Dic5⋊Q16  Dic7⋊Q16 ...

Matrix representation of C42Q16 in GL4(𝔽17) generated by

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

C42Q16 in GAP, Magma, Sage, TeX

`C_4\rtimes_2Q_{16}`
`% in TeX`

`G:=Group("C4:2Q16");`
`// GroupNames label`

`G:=SmallGroup(64,143);`
`// by ID`

`G=gap.SmallGroup(64,143);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,55,362,158,1444,376,88]);`
`// Polycyclic`

`G:=Group<a,b,c|a^4=b^8=1,c^2=b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;`
`// generators/relations`

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