H(3,𝔽₃)×#468 #13145163

order	1728
nilpotency class	3
isoclinism	isoclinic class of H(3,𝔽₃)×#468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃²

Aliases

direct product	H(3,𝔽₃)×#468
unambiguous semidirect product	(C₃²×#468)⋊C₃

	subgroup	G/H
center, socle	C₂²×C₃	C₂×C₃²×D₄
commutator, Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
prime radical	H(3,𝔽₃)×#121	C₂
LCS	G ▷ C₂×C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₄×H(3,𝔽₃) ◁ G

660 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)(10,11)(12,13)(15,18)(16,20)(19,21);(2,5,6)(4,8,9)(14,15,16,18,17,19,20,21);(11,13)(15,19)(18,21)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 11226);