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## G = C2.Q32order 64 = 26

### 1st central extension by C2 of Q32

p-group, metabelian, nilpotent (class 4), monomial

Aliases: Q161C4, C8.15D4, C2.1Q32, C4.2SD16, C2.2SD32, C22.9D8, C8.8(C2×C4), (C2×C16).1C2, (C2×C4).61D4, C2.D8.1C2, (C2×Q16).1C2, C4.2(C22⋊C4), (C2×C8).70C22, C2.7(D4⋊C4), SmallGroup(64,39)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C2.Q32
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×Q16 — C2.Q32
 Lower central C1 — C2 — C4 — C8 — C2.Q32
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2.Q32
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C2.Q32

Generators and relations for C2.Q32
G = < a,b,c | a2=b16=1, c2=b8, ab=ba, ac=ca, cbc-1=ab-1 >

Character table of C2.Q32

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 1 1 2 2 8 8 8 8 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 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 -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 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 -1 -1 -1 linear of order 2 ρ5 1 -1 -1 1 -1 1 i -1 -i 1 1 -1 1 -1 i -i -i i i i -i -i linear of order 4 ρ6 1 -1 -1 1 -1 1 -i -1 i 1 1 -1 1 -1 -i i i -i -i -i i i linear of order 4 ρ7 1 -1 -1 1 -1 1 -i 1 i -1 1 -1 1 -1 i -i -i i i i -i -i linear of order 4 ρ8 1 -1 -1 1 -1 1 i 1 -i -1 1 -1 1 -1 -i i i -i -i -i i i linear of order 4 ρ9 2 2 2 2 2 2 0 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 -2 2 -2 2 0 0 0 0 -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 0 0 √2 -√2 √2 -√2 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ12 2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ13 2 2 -2 -2 0 0 0 0 0 0 √2 -√2 -√2 √2 ζ165-ζ163 -ζ1615+ζ169 ζ165-ζ163 -ζ1615+ζ169 -ζ165+ζ163 ζ1615-ζ169 -ζ165+ζ163 ζ1615-ζ169 symplectic lifted from Q32, Schur index 2 ρ14 2 2 -2 -2 0 0 0 0 0 0 -√2 √2 √2 -√2 ζ1615-ζ169 ζ165-ζ163 ζ1615-ζ169 ζ165-ζ163 -ζ1615+ζ169 -ζ165+ζ163 -ζ1615+ζ169 -ζ165+ζ163 symplectic lifted from Q32, Schur index 2 ρ15 2 2 -2 -2 0 0 0 0 0 0 √2 -√2 -√2 √2 -ζ165+ζ163 ζ1615-ζ169 -ζ165+ζ163 ζ1615-ζ169 ζ165-ζ163 -ζ1615+ζ169 ζ165-ζ163 -ζ1615+ζ169 symplectic lifted from Q32, Schur index 2 ρ16 2 2 -2 -2 0 0 0 0 0 0 -√2 √2 √2 -√2 -ζ1615+ζ169 -ζ165+ζ163 -ζ1615+ζ169 -ζ165+ζ163 ζ1615-ζ169 ζ165-ζ163 ζ1615-ζ169 ζ165-ζ163 symplectic lifted from Q32, Schur index 2 ρ17 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 -√-2 √-2 √-2 -√-2 complex lifted from SD16 ρ18 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 √-2 √-2 -√-2 -√-2 √-2 -√-2 -√-2 √-2 complex lifted from SD16 ρ19 2 -2 2 -2 0 0 0 0 0 0 -√2 -√2 √2 √2 ζ1613+ζ1611 ζ1615+ζ169 ζ165+ζ163 ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 ζ167+ζ16 complex lifted from SD32 ρ20 2 -2 2 -2 0 0 0 0 0 0 √2 √2 -√2 -√2 ζ167+ζ16 ζ1613+ζ1611 ζ1615+ζ169 ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 ζ167+ζ16 ζ165+ζ163 complex lifted from SD32 ρ21 2 -2 2 -2 0 0 0 0 0 0 √2 √2 -√2 -√2 ζ1615+ζ169 ζ165+ζ163 ζ167+ζ16 ζ1613+ζ1611 ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 complex lifted from SD32 ρ22 2 -2 2 -2 0 0 0 0 0 0 -√2 -√2 √2 √2 ζ165+ζ163 ζ167+ζ16 ζ1613+ζ1611 ζ1615+ζ169 ζ1613+ζ1611 ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 complex lifted from SD32

Smallest permutation representation of C2.Q32
Regular action on 64 points
Generators in S64
```(1 56)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 64)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 33)
(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 41 9 33)(2 23 10 31)(3 39 11 47)(4 21 12 29)(5 37 13 45)(6 19 14 27)(7 35 15 43)(8 17 16 25)(18 54 26 62)(20 52 28 60)(22 50 30 58)(24 64 32 56)(34 55 42 63)(36 53 44 61)(38 51 46 59)(40 49 48 57)```

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

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

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

Matrix representation of C2.Q32 in GL3(𝔽17) generated by

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

C2.Q32 in GAP, Magma, Sage, TeX

`C_2.Q_{32}`
`% in TeX`

`G:=Group("C2.Q32");`
`// GroupNames label`

`G:=SmallGroup(64,39);`
`// by ID`

`G=gap.SmallGroup(64,39);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,199,362,188,230,1444,730,88]);`
`// Polycyclic`

`G:=Group<a,b,c|a^2=b^16=1,c^2=b^8,a*b=b*a,a*c=c*a,c*b*c^-1=a*b^-1>;`
`// generators/relations`

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