H(3,𝔽₃)×#548 #13154627

order	1728
nilpotency class	2
isoclinism	isoclinic class of D₄²×H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁵×C₃²

Aliases

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²×C₃	C₂⁴×C₃²
LCS	G ▷ C₂²×C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

1595 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 288

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 20690);