C₃²×C₅×#123 #13088232

order	1440
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃²×C₅ · Schur multiplier (H₂) = C₂²×C₃

Aliases

C₃²×C₅×#123

	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

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

normal subgroups only

This group has 468 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)(13,15)(17,20)(18,22);(4,5,6)(16,17)(18,21)(19,20)(22,23);(12,13)(14,15)(16,18,19,22)(17,21,20,23)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 4549);