C₃²×C₅×#124 #13088233

order	1440
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃²×C₅ · Schur multiplier (H₂) = C₂×C₃×C₄

Aliases

C₃²×C₅×#124

	subgroup	G/H
center, socle	C₂²×C₃²×C₅	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₃²×C₅
prime radical	C₂×C₃²×C₅×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃²×C₅ ◁ G

456 conjugacy classes of subgroups; only the 264 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 264 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 720

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 4550);