p-group, metacyclic, nilpotent (class 2), monomial, rational
Aliases: Q8, Dic2, 2- 1+2, C4.C2, C2.2C22, 2-Sylow(SL(2,3)), SmallGroup(8,4)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8
G = < a,b | a4=1, b2=a2, bab-1=a-1 >
Character table of Q8
| class | 1 | 2 | 4A | 4B | 4C | |
| size | 1 | 1 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ5 | 2 | -2 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►Regular action on 8 points - transitive group 8T5
Generators in S8
(1 2 3 4)(5 6 7 8)
(1 7 3 5)(2 6 4 8)
G:=sub<Sym(8)| (1,2,3,4)(5,6,7,8), (1,7,3,5)(2,6,4,8)>;
G:=Group( (1,2,3,4)(5,6,7,8), (1,7,3,5)(2,6,4,8) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8)], [(1,7,3,5),(2,6,4,8)]])
G:=TransitiveGroup(8,5);
Q8 is a maximal subgroup of
SD16 C4○D4 SL2(𝔽3) PSU3(𝔽2) C52⋊Q8 C72⋊Q8
Dic2p: Q16 Dic6 Dic10 Dic14 Dic22 Dic26 Dic34 Dic38 ...
Q8 is a maximal quotient of
PSU3(𝔽2) C52⋊Q8 C72⋊Q8 A5⋊Q8
C2.D2p: C4⋊C4 Dic6 Dic10 Dic14 Dic22 Dic26 Dic34 Dic38 ...
| action | f(x) | Disc(f) |
|---|---|---|
| 8T5 | x8-12x6+36x4-36x2+9 | 224·314 |
Matrix representation of Q8 ►in GL2(𝔽3) generated by
| 2 | 2 |
| 2 | 1 |
| 0 | 2 |
| 1 | 0 |
G:=sub<GL(2,GF(3))| [2,2,2,1],[0,1,2,0] >;
Q8 in GAP, Magma, Sage, TeX
Q_8
% in TeX
G:=Group("Q8"); // GroupNames label
G:=SmallGroup(8,4);
// by ID
G=gap.SmallGroup(8,4);
# by ID
G:=PCGroup([3,-2,2,-2,12,37,16]);
// Polycyclic
G:=Group<a,b|a^4=1,b^2=a^2,b*a*b^-1=a^-1>;
// generators/relations
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