C₃³×#535 #13179990

order	1728
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃³ · Schur multiplier (H₂) = C₂⁴×C₃³×C₄

Aliases

	subgroup	G/H
center, socle	C₂²×C₃³	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃³ ◁ G

4676 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 432

A minimal faithful permutation representation — 4 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 46053);