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G = Q8order 8 = 23

Quaternion group

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

C1C2 — Q8
C1C2C4 — Q8
C1C2 — Q8
C1C2 — Q8
C1C2 — Q8

Generators and relations for Q8
 G = < a,b | a4=1, b2=a2, bab-1=a-1 >


Character table of Q8

 class 124A4B4C
 size 11222
ρ111111    trivial
ρ211-11-1    linear of order 2
ρ3111-1-1    linear of order 2
ρ411-1-11    linear of order 2
ρ52-2000    symplectic faithful, Schur index 2

Permutation representations of Q8
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);

Polynomial with Galois group Q8 over ℚ
actionf(x)Disc(f)
8T5x8-12x6+36x4-36x2+9224·314

Matrix representation of Q8 in GL2(𝔽3) generated by

22
21
,
02
10
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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