C₅×#6900 #11925973

order	1215
nilpotency class	2
isoclinism	isoclinic class of C₃²⋊H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃³×C₅ · Schur multiplier (H₂) = C₃

Aliases

	subgroup	G/H
center, socle	C₃²×C₅	C₃³
commutator, Frattini, cosocle	C₃²	C₃³×C₅
prime radical	C₃³×C₅	C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₅ ◁ G

140 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 405

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2,3,4,5)(7,9,12)(10,14,13)(16,18,21)(19,23,22);(6,7,10,8,9,13,11,12,14);(15,16,19,17,18,22,20,21,23)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1215, 47);