H(3,𝔽₃)×#385 #13144914

order	1728
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)×#120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂⁴×C₃²

Aliases

direct product	H(3,𝔽₃)×#385
unambiguous semidirect product	(C₃²×#385)⋊C₃

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³×C₃	C₂³×C₃²
commutator	C₂²×C₃	C₂²×C₃²×C₄
prime radical	C₂⁴×H(3,𝔽₃)	C₂²
LCS	G ▷ C₂²×C₃ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

1287 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 3 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 10977);