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G = Q32order 32 = 25

Generalised quaternion group

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

Aliases: Q32, Dic8, C16.C2, C4.3D4, C2.5D8, Q16.C2, C8.4C22, 2-Sylow(SL(2,17)), SmallGroup(32,20)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — Q32
C1C2C4C8Q16 — Q32
C1C2C4C8 — Q32
C1C2C4C8 — Q32
C1C2C2C2C2C4C4C8 — Q32

Generators and relations for Q32
 G = < a,b | a16=1, b2=a8, bab-1=a-1 >

4C4
4C4
2Q8
2Q8

Character table of Q32

 class 124A4B4C8A8B16A16B16C16D
 size 11288222222
ρ111111111111    trivial
ρ21111-111-1-1-1-1    linear of order 2
ρ3111-1111-1-1-1-1    linear of order 2
ρ4111-1-1111111    linear of order 2
ρ522200-2-20000    orthogonal lifted from D4
ρ622-20000-2-222    orthogonal lifted from D8
ρ722-2000022-2-2    orthogonal lifted from D8
ρ82-2000-2216716ζ16716ζ165163165163    symplectic faithful, Schur index 2
ρ92-2000-22ζ1671616716165163ζ165163    symplectic faithful, Schur index 2
ρ102-20002-2165163ζ16516316716ζ16716    symplectic faithful, Schur index 2
ρ112-20002-2ζ165163165163ζ1671616716    symplectic faithful, Schur index 2

Smallest permutation representation of Q32
Regular action on 32 points
Generators in S32
(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)
(1 23 9 31)(2 22 10 30)(3 21 11 29)(4 20 12 28)(5 19 13 27)(6 18 14 26)(7 17 15 25)(8 32 16 24)

G:=sub<Sym(32)| (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), (1,23,9,31)(2,22,10,30)(3,21,11,29)(4,20,12,28)(5,19,13,27)(6,18,14,26)(7,17,15,25)(8,32,16,24)>;

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), (1,23,9,31)(2,22,10,30)(3,21,11,29)(4,20,12,28)(5,19,13,27)(6,18,14,26)(7,17,15,25)(8,32,16,24) );

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)], [(1,23,9,31),(2,22,10,30),(3,21,11,29),(4,20,12,28),(5,19,13,27),(6,18,14,26),(7,17,15,25),(8,32,16,24)])

Matrix representation of Q32 in GL2(𝔽17) generated by

140
011
,
01
160
G:=sub<GL(2,GF(17))| [14,0,0,11],[0,16,1,0] >;

Q32 in GAP, Magma, Sage, TeX

Q_{32}
% in TeX

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

G:=SmallGroup(32,20);
// by ID

G=gap.SmallGroup(32,20);
# by ID

G:=PCGroup([5,-2,2,-2,-2,-2,80,61,86,182,97,102,483,248,58]);
// Polycyclic

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

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