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G = Q64  order 64 = 26

Generalised quaternion group

p-group, metacyclic, nilpotent (class 5), monomial

Aliases: Q64, Dic16, C32.C2, C8.7D4, C4.3D8, Q32.C2, C2.5D16, C16.4C22, 2-Sylow(SL(2,31)), SmallGroup(64,54)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C16 — Q64
Chief series
C1 — C2 — C4 — C8 — C16 — Q32 — Q64
Lower central
C1 — C2 — C4 — C8 — C16 — Q64
Upper central
C1 — C2 — C4 — C8 — C16 — Q64
Jennings
C1 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C4 — C4 — C4 — C4 — C8 — C8 — C16 — Q64

Generators and relations for Q64
 G = < a,b | a32=1, b2=a16, bab-1=a-1 >

C1
C2
C4
8C4
8C4
C8
4Q8
4Q8
C16
2Q16
2Q16
Q32
Q32
C32
Q64

Character table of Q64

 class 124A4B4C8A8B16A16B16C16D32A32B32C32D32E32F32G32H
 size 112161622222222222222
ρ11111111111111111111    trivial
ρ2111-1-111111111111111    linear of order 2
ρ31111-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ4111-11111111-1-1-1-1-1-1-1-1    linear of order 2
ρ52220022-2-2-2-200000000    orthogonal lifted from D4
ρ622200-2-20000-√2-√2-√2√2√2-√2√2√2    orthogonal lifted from D8
ρ722200-2-20000√2√2√2-√2-√2√2-√2-√2    orthogonal lifted from D8
ρ822-20000-√2√2√2-√2-ζ167+ζ16-ζ167+ζ16ζ167-ζ16ζ165-ζ163ζ165-ζ163ζ167-ζ16-ζ165+ζ163-ζ165+ζ163    orthogonal lifted from D16
ρ922-20000√2-√2-√2√2ζ165-ζ163ζ165-ζ163-ζ165+ζ163ζ167-ζ16ζ167-ζ16-ζ165+ζ163-ζ167+ζ16-ζ167+ζ16    orthogonal lifted from D16
ρ1022-20000√2-√2-√2√2-ζ165+ζ163-ζ165+ζ163ζ165-ζ163-ζ167+ζ16-ζ167+ζ16ζ165-ζ163ζ167-ζ16ζ167-ζ16    orthogonal lifted from D16
ρ1122-20000-√2√2√2-√2ζ167-ζ16ζ167-ζ16-ζ167+ζ16-ζ165+ζ163-ζ165+ζ163-ζ167+ζ16ζ165-ζ163ζ165-ζ163    orthogonal lifted from D16
ρ122-2000-√2√2ζ3214-ζ322-ζ3210+ζ326ζ3210-ζ326-ζ3214+ζ322ζ3211-ζ325-ζ3211+ζ325ζ3213-ζ323-ζ3225+ζ3223ζ3225-ζ3223-ζ3213+ζ323-ζ3215+ζ32ζ3215-ζ32    symplectic faithful, Schur index 2
ρ132-2000√2-√2ζ3210-ζ326ζ3214-ζ322-ζ3214+ζ322-ζ3210+ζ326-ζ3215+ζ32ζ3215-ζ32ζ3225-ζ3223-ζ3211+ζ325ζ3211-ζ325-ζ3225+ζ3223-ζ3213+ζ323ζ3213-ζ323    symplectic faithful, Schur index 2
ρ142-2000√2-√2-ζ3210+ζ326-ζ3214+ζ322ζ3214-ζ322ζ3210-ζ326-ζ3225+ζ3223ζ3225-ζ3223-ζ3215+ζ32ζ3213-ζ323-ζ3213+ζ323ζ3215-ζ32-ζ3211+ζ325ζ3211-ζ325    symplectic faithful, Schur index 2
ρ152-2000-√2√2-ζ3214+ζ322ζ3210-ζ326-ζ3210+ζ326ζ3214-ζ322ζ3213-ζ323-ζ3213+ζ323-ζ3211+ζ325-ζ3215+ζ32ζ3215-ζ32ζ3211-ζ325ζ3225-ζ3223-ζ3225+ζ3223    symplectic faithful, Schur index 2
ρ162-2000-√2√2ζ3214-ζ322-ζ3210+ζ326ζ3210-ζ326-ζ3214+ζ322-ζ3211+ζ325ζ3211-ζ325-ζ3213+ζ323ζ3225-ζ3223-ζ3225+ζ3223ζ3213-ζ323ζ3215-ζ32-ζ3215+ζ32    symplectic faithful, Schur index 2
ρ172-2000√2-√2-ζ3210+ζ326-ζ3214+ζ322ζ3214-ζ322ζ3210-ζ326ζ3225-ζ3223-ζ3225+ζ3223ζ3215-ζ32-ζ3213+ζ323ζ3213-ζ323-ζ3215+ζ32ζ3211-ζ325-ζ3211+ζ325    symplectic faithful, Schur index 2
ρ182-2000-√2√2-ζ3214+ζ322ζ3210-ζ326-ζ3210+ζ326ζ3214-ζ322-ζ3213+ζ323ζ3213-ζ323ζ3211-ζ325ζ3215-ζ32-ζ3215+ζ32-ζ3211+ζ325-ζ3225+ζ3223ζ3225-ζ3223    symplectic faithful, Schur index 2
ρ192-2000√2-√2ζ3210-ζ326ζ3214-ζ322-ζ3214+ζ322-ζ3210+ζ326ζ3215-ζ32-ζ3215+ζ32-ζ3225+ζ3223ζ3211-ζ325-ζ3211+ζ325ζ3225-ζ3223ζ3213-ζ323-ζ3213+ζ323    symplectic faithful, Schur index 2

Smallest permutation representation of Q64
►Regular action on 64 points
Generators in S64
(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 53 17 37)(2 52 18 36)(3 51 19 35)(4 50 20 34)(5 49 21 33)(6 48 22 64)(7 47 23 63)(8 46 24 62)(9 45 25 61)(10 44 26 60)(11 43 27 59)(12 42 28 58)(13 41 29 57)(14 40 30 56)(15 39 31 55)(16 38 32 54)
 
G:=sub<Sym(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), (1,53,17,37)(2,52,18,36)(3,51,19,35)(4,50,20,34)(5,49,21,33)(6,48,22,64)(7,47,23,63)(8,46,24,62)(9,45,25,61)(10,44,26,60)(11,43,27,59)(12,42,28,58)(13,41,29,57)(14,40,30,56)(15,39,31,55)(16,38,32,54)>;
 
G:=Group( (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), (1,53,17,37)(2,52,18,36)(3,51,19,35)(4,50,20,34)(5,49,21,33)(6,48,22,64)(7,47,23,63)(8,46,24,62)(9,45,25,61)(10,44,26,60)(11,43,27,59)(12,42,28,58)(13,41,29,57)(14,40,30,56)(15,39,31,55)(16,38,32,54) );
 
G=PermutationGroup([[(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)], [(1,53,17,37),(2,52,18,36),(3,51,19,35),(4,50,20,34),(5,49,21,33),(6,48,22,64),(7,47,23,63),(8,46,24,62),(9,45,25,61),(10,44,26,60),(11,43,27,59),(12,42,28,58),(13,41,29,57),(14,40,30,56),(15,39,31,55),(16,38,32,54)]])
 

Q64 is a maximal subgroup of
 Dic16p: Q128  Dic48  Dic80  Dic112 ...
 C4p.D8: SD128  C4○D32  Q64⋊C2  C3⋊Q64  C5⋊Q64  C7⋊Q64 ...
Q64 is a maximal quotient of
 C32⋊3C4
 C16.D2p: Q32⋊2C4  Dic48  C3⋊Q64  Dic80  C5⋊Q64  Dic112  C7⋊Q64 ...

Matrix representation of Q64 ►in GL2(𝔽31) generated by

1612
1211
,
030
10
G:=sub<GL(2,GF(31))| [16,12,12,11],[0,1,30,0] >;
 

Q64 in GAP, Magma, Sage, TeX 

Q_{64}
 
% in TeX
 
G:=Group("Q64");
 
// GroupNames label
 
G:=SmallGroup(64,54);
 
// by ID
 
G=gap.SmallGroup(64,54);
 
# by ID
 
G:=PCGroup([6,-2,2,-2,-2,-2,-2,192,73,199,218,116,122,579,297,165,1444,730,88]);
 
// Polycyclic
 
G:=Group<a,b|a^32=1,b^2=a^16,b*a*b^-1=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of Q64 in TeX
Character table of Q64 in TeX

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