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G = Q64order 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

C1C16 — Q64
C1C2C4C8C16Q32 — Q64
C1C2C4C8C16 — Q64
C1C2C4C8C16 — Q64
C1C2C2C2C2C2C2C2C2C4C4C4C4C8C8C16 — Q64

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

8C4
8C4
4Q8
4Q8
2Q16
2Q16

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
ρ622200-2-20000-2-2-222-222    orthogonal lifted from D8
ρ722200-2-20000222-2-22-2-2    orthogonal lifted from D8
ρ822-20000-222-21671616716ζ16716ζ165163ζ165163ζ16716165163165163    orthogonal lifted from D16
ρ922-200002-2-22ζ165163ζ165163165163ζ16716ζ167161651631671616716    orthogonal lifted from D16
ρ1022-200002-2-22165163165163ζ1651631671616716ζ165163ζ16716ζ16716    orthogonal lifted from D16
ρ1122-20000-222-2ζ16716ζ167161671616516316516316716ζ165163ζ165163    orthogonal lifted from D16
ρ122-2000-22ζ32143223210326ζ32103263214322ζ32113253211325ζ321332332253223ζ322532233213323321532ζ321532    symplectic faithful, Schur index 2
ρ132-20002-2ζ3210326ζ321432232143223210326321532ζ321532ζ322532233211325ζ3211325322532233213323ζ3213323    symplectic faithful, Schur index 2
ρ142-20002-232103263214322ζ3214322ζ321032632253223ζ32253223321532ζ32133233213323ζ3215323211325ζ3211325    symplectic faithful, Schur index 2
ρ152-2000-223214322ζ32103263210326ζ3214322ζ321332332133233211325321532ζ321532ζ3211325ζ3225322332253223    symplectic faithful, Schur index 2
ρ162-2000-22ζ32143223210326ζ321032632143223211325ζ32113253213323ζ3225322332253223ζ3213323ζ321532321532    symplectic faithful, Schur index 2
ρ172-20002-232103263214322ζ3214322ζ3210326ζ3225322332253223ζ3215323213323ζ3213323321532ζ32113253211325    symplectic faithful, Schur index 2
ρ182-2000-223214322ζ32103263210326ζ32143223213323ζ3213323ζ3211325ζ321532321532321132532253223ζ32253223    symplectic faithful, Schur index 2
ρ192-20002-2ζ3210326ζ321432232143223210326ζ32153232153232253223ζ32113253211325ζ32253223ζ32133233213323    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 56 17 40)(2 55 18 39)(3 54 19 38)(4 53 20 37)(5 52 21 36)(6 51 22 35)(7 50 23 34)(8 49 24 33)(9 48 25 64)(10 47 26 63)(11 46 27 62)(12 45 28 61)(13 44 29 60)(14 43 30 59)(15 42 31 58)(16 41 32 57)

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

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

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

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
C323C4
 C16.D2p: Q322C4  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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