C₂³×C₅×3₋¹⁺² #11762016

order	1080
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃²×C₅ · Schur multiplier (H₂) = C₂³

Aliases

C₂³×C₅×3₋¹⁺²

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₃²
commutator, Frattini, cosocle	C₃	C₂³×C₃²×C₅
prime radical	C₂³×C₃²×C₅	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ G

256 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 360

A minimal faithful permutation representation — 3 generators on 20 points:

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

440 conjugacy classes. The character table is too large to display.

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1080, 362);