C₅×#6860 #11925933

order	1215
nilpotency class	3
isoclinism	isoclinic class of #6856
automorphism	?
homology	Abelianization (H₁) = C₃²×C₅ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, socle	C₃²×C₅	H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃²×C₅
prime radical	C₃³×C₅	C₃²
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₅ ◁ C₃³×C₅ ◁ G

70 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 2 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1215, 7);