GL(2,𝔽₃) #227

order	48
isoclinism	isoclinic class of #226
automorphism	Aut(G) = C₂×S₄ (order 48) · Inn(G) = S₄ · Out(G) = C₂
holomorph	G ⋊ Aut(G) = #72215280
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

GL(2,𝔽₃)

Fitting	Q₈ ⋊ D₃
derived	SL(2,𝔽₃) ⋊ C₂

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂	S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂
Fitting	Q₈	D₃
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 8

A minimal faithful permutation representation — 2 generators on 8 points:

(1,2,8,3,4,7,6,5);(2,5,8)(3,6,7)

Polycyclic presentation:

2,3,2^3|c^2=e,d^2=e,b^a=b^2,c^a=d,d^a=c,c^b=de,d^b=cd,d^c=de

Here i is the imaginary unit (i² = −1).

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(48, 29);