H(3,𝔽₃)×#496 #13145247

order	1728
nilpotency class	3
isoclinism	isoclinic class of H(3,𝔽₃)×#468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂×C₃²×C₄

Aliases

direct product	H(3,𝔽₃)×#496
unambiguous semidirect product	(C₃²×#496)⋊C₃

	subgroup	G/H
center, socle	C₂²×C₃	C₂×C₃²×D₄
commutator, Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
LCS	G ▷ C₂×C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₄²×H(3,𝔽₃) ◁ G

792 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 203 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 11310);