PSU(3,𝔽₂) #644

order	72
isoclinism	isoclinic class of PSU(3,𝔽₂)
automorphism	Aut(G) = AGL(2,𝔽₃) (order 432) · Inn(G) = G · Out(G) = D₃
holomorph	G ⋊ Aut(G) = #72220496
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₃

Aliases

PSU(3,𝔽₂)

C₃² ⋊ Q₈

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, prime radical, cosocle	C₃⋊D₃	C₂²
Fitting, socle, hypocenter	C₃²	Q₈
LCS	G ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁

14 conjugacy classes of subgroups.

minimal degree	μ(G) = 9
transitive degree	μ_t(G) = 9

A minimal faithful permutation representation — 2 generators on 9 points:

(1,4,5,6)(3,8,7,9);(2,4,5,9)(3,7,8,6)

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(72, 41);