C₅⋊#272 #63201

order	270
isoclinism	isoclinic class of C₅⋊#272
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₃	C₃⋊D₁₅
commutator, Fitting, cosocle, hypocenter	C₅×H(3,𝔽₃)	C₂
socle	C₃×C₅	C₃⋊D₃
LCS	G ▷ C₅×H(3,𝔽₃)
UCS	C₁ ◁ C₃

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:

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

Polycyclic presentation:

2,3^2,5,3|b^a=b^2,c^a=c^2,d^a=d^4,c^b=ce

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(270, 19);