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## G = C4.Q16order 64 = 26

### 3rd non-split extension by C4 of 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.Q16
 Chief series C1 — C2 — C4 — C2×C4 — C2×Q8 — C4×Q8 — C4.Q16
 Lower central C1 — C2 — C2×C4 — C4.Q16
 Upper central C1 — C22 — C42 — C4.Q16
 Jennings C1 — C2 — C2 — C2×C4 — C4.Q16

Generators and relations for C4.Q16
G = < a,b,c | a4=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=a2b-1 >

Character table of C4.Q16

 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 2 -2 -2 2 0 -2 0 0 0 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 symplectic lifted from Q8, Schur index 2 ρ12 2 -2 2 -2 0 2 -2 0 0 0 0 -2 2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 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 orthogonal lifted from C8⋊C22

Smallest permutation representation of C4.Q16
Regular action on 64 points
Generators in S64
```(1 17 28 46)(2 47 29 18)(3 19 30 48)(4 41 31 20)(5 21 32 42)(6 43 25 22)(7 23 26 44)(8 45 27 24)(9 38 50 62)(10 63 51 39)(11 40 52 64)(12 57 53 33)(13 34 54 58)(14 59 55 35)(15 36 56 60)(16 61 49 37)
(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 55 5 51)(2 13 6 9)(3 53 7 49)(4 11 8 15)(10 28 14 32)(12 26 16 30)(17 35 21 39)(18 58 22 62)(19 33 23 37)(20 64 24 60)(25 50 29 54)(27 56 31 52)(34 43 38 47)(36 41 40 45)(42 63 46 59)(44 61 48 57)```

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

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

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

Matrix representation of C4.Q16 in GL4(𝔽17) generated by

 4 2 0 0 0 13 0 0 0 0 16 0 0 0 0 16
,
 5 3 0 0 14 12 0 0 0 0 15 0 0 0 0 8
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 16 0
`G:=sub<GL(4,GF(17))| [4,0,0,0,2,13,0,0,0,0,16,0,0,0,0,16],[5,14,0,0,3,12,0,0,0,0,15,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0] >;`

C4.Q16 in GAP, Magma, Sage, TeX

`C_4.Q_{16}`
`% in TeX`

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

`G:=SmallGroup(64,158);`
`// by ID`

`G=gap.SmallGroup(64,158);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,240,121,55,362,158,1444,376,88]);`
`// 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=a^2*b^-1>;`
`// generators/relations`

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