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
Generators and relations for Q32
G = < a,b | a16=1, b2=a8, bab-1=a-1 >
Character table of Q32
| class | 1 | 2 | 4A | 4B | 4C | 8A | 8B | 16A | 16B | 16C | 16D | |
| size | 1 | 1 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 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 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ6 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | orthogonal lifted from D8 |
| ρ7 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | orthogonal lifted from D8 |
| ρ8 | 2 | -2 | 0 | 0 | 0 | -√2 | √2 | -ζ167+ζ16 | ζ167-ζ16 | ζ165-ζ163 | -ζ165+ζ163 | symplectic faithful, Schur index 2 |
| ρ9 | 2 | -2 | 0 | 0 | 0 | -√2 | √2 | ζ167-ζ16 | -ζ167+ζ16 | -ζ165+ζ163 | ζ165-ζ163 | symplectic faithful, Schur index 2 |
| ρ10 | 2 | -2 | 0 | 0 | 0 | √2 | -√2 | -ζ165+ζ163 | ζ165-ζ163 | -ζ167+ζ16 | ζ167-ζ16 | symplectic faithful, Schur index 2 |
| ρ11 | 2 | -2 | 0 | 0 | 0 | √2 | -√2 | ζ165-ζ163 | -ζ165+ζ163 | ζ167-ζ16 | -ζ167+ζ16 | symplectic faithful, Schur index 2 |
►Regular action on 32 points
(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 30 9 22)(2 29 10 21)(3 28 11 20)(4 27 12 19)(5 26 13 18)(6 25 14 17)(7 24 15 32)(8 23 16 31)
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), (1,30,9,22)(2,29,10,21)(3,28,11,20)(4,27,12,19)(5,26,13,18)(6,25,14,17)(7,24,15,32)(8,23,16,31)>;
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), (1,30,9,22)(2,29,10,21)(3,28,11,20)(4,27,12,19)(5,26,13,18)(6,25,14,17)(7,24,15,32)(8,23,16,31) );
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)], [(1,30,9,22),(2,29,10,21),(3,28,11,20),(4,27,12,19),(5,26,13,18),(6,25,14,17),(7,24,15,32),(8,23,16,31)]])
Q32 is a maximal subgroup of
C32⋊Q32
Dic8p: Q64 Dic24 Dic40 Dic56 Dic88 Dic104 ...
C4p.D4: SD64 C4○D16 Q32⋊C2 C3⋊Q32 C5⋊Q32 C7⋊Q32 C11⋊Q32 C13⋊Q32 ...
Q32 is a maximal quotient of
C16⋊3C4 C32⋊Q32
C8.D2p: C2.Q32 Dic24 C3⋊Q32 Dic40 C5⋊Q32 Dic56 C7⋊Q32 Dic88 ...
Matrix representation of Q32 ►in GL2(𝔽17) generated by
| 14 | 0 |
| 0 | 11 |
| 0 | 1 |
| 16 | 0 |
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
Export
Subgroup lattice of Q32 in TeX
Character table of Q32 in TeX