C₅×#272 #63199

order	270
isoclinism	isoclinic class of #272
automorphism	Aut(G) = C₄×AGL(2,𝔽₃) (order 1728) · Inn(G) = C₃⋊D₃ · Out(G) = C₄×S₄
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁

Aliases

Fitting, Hall	C₅×H(3,𝔽₃) ⋊ C₂
derived, Sylow	H(3,𝔽₃) ⋊ C₂×C₅

	subgroup	G/H
center, socle, hypercenter	C₃×C₅	C₃⋊D₃
commutator, cosocle, hypocenter	H(3,𝔽₃)	C₂×C₅
Frattini	C₃	C₅×C₃⋊D₃
Fitting	C₅×H(3,𝔽₃)	C₂
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃×C₅

44 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 45

A minimal faithful permutation representation — 3 generators on 14 points:

(1,2,3,4,5)(7,9)(10,14)(11,13);(7,10,11)(9,13,14);(6,7,9)(8,11,13)(10,14,12)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(270, 17);