C₃²×D₅ #778

order	90
isoclinism	isoclinic class of D₅
automorphism	Aut(G) = AGL(1,𝔽₅)×GL(2,𝔽₃) (order 960) · Inn(G) = D₅ · Out(G) = C₂×GL(2,𝔽₃)
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃²×D₅
unambiguous semidirect product	C₅⋊(C₂×C₃²)

Fitting, socle, Hall	C₃²×C₅ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₃²

	subgroup	G/H
center, hypercenter	C₃²	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃²×C₅	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₃²

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 45

A minimal faithful permutation representation — 2 generators on 11 points:

(4,5,6)(8,9)(10,11);(1,2,3)(7,8,10,11,9)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(90, 5);