p-group, metacyclic, nilpotent (class 3), monomial
Aliases: Q16, Dic4, C8.C2, Q8.C2, C2.5D4, C4.3C22, 2-Sylow(SL(2,7)), SmallGroup(16,9)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q16
G = < a,b | a8=1, b2=a4, bab-1=a-1 >
Character table of Q16
| class | 1 | 2 | 4A | 4B | 4C | 8A | 8B | |
| size | 1 | 1 | 2 | 4 | 4 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ5 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ6 | 2 | -2 | 0 | 0 | 0 | √2 | -√2 | symplectic faithful, Schur index 2 |
| ρ7 | 2 | -2 | 0 | 0 | 0 | -√2 | √2 | symplectic faithful, Schur index 2 |
►Regular action on 16 points - transitive group 16T14
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 12 5 16)(2 11 6 15)(3 10 7 14)(4 9 8 13)
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,12,5,16)(2,11,6,15)(3,10,7,14)(4,9,8,13)>;
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,12,5,16)(2,11,6,15)(3,10,7,14)(4,9,8,13) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,12,5,16),(2,11,6,15),(3,10,7,14),(4,9,8,13)]])
G:=TransitiveGroup(16,14);
Q16 is a maximal subgroup of
CSU2(𝔽3) C32⋊Q16 C52⋊Q16
Dic4p: Q32 Dic12 Dic20 Dic28 Dic44 Dic52 Dic68 Dic76 ...
C4.D2p: SD32 C4○D8 C8.C22 C3⋊Q16 C5⋊Q16 C7⋊Q16 C11⋊Q16 C13⋊Q16 ...
Q16 is a maximal quotient of
C2.D8 C32⋊Q16 C52⋊Q16
C4.D2p: Q8⋊C4 Dic12 C3⋊Q16 Dic20 C5⋊Q16 Dic28 C7⋊Q16 Dic44 ...
Matrix representation of Q16 ►in GL2(𝔽7) generated by
| 0 | 6 |
| 1 | 3 |
| 6 | 5 |
| 1 | 1 |
G:=sub<GL(2,GF(7))| [0,1,6,3],[6,1,5,1] >;
Q16 in GAP, Magma, Sage, TeX
Q_{16} % in TeX
G:=Group("Q16"); // GroupNames label
G:=SmallGroup(16,9);
// by ID
G=gap.SmallGroup(16,9);
# by ID
G:=PCGroup([4,-2,2,-2,-2,32,49,37,146,78,34]);
// Polycyclic
G:=Group<a,b|a^8=1,b^2=a^4,b*a*b^-1=a^-1>;
// generators/relations
Export
Subgroup lattice of Q16 in TeX
Character table of Q16 in TeX