H(3,𝔽₃)×#472 #13145175

order	1728
nilpotency class	3
isoclinism	isoclinic class of H(3,𝔽₃)×#471
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃²

Aliases

direct product	H(3,𝔽₃)×#472
unambiguous semidirect product	(C₃²×#472)⋊C₃

Subgroups

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

594 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 25 points:

(1,2,4)(3,6,8)(5,9,7)(11,15)(12,19)(13,16)(18,23)(20,21)(22,24);(2,5,6)(4,8,9)(10,11,16,23,14,18,13,15)(12,17,19,24,21,25,20,22);(10,12)(11,17)(13,20)(14,21)(15,22)(16,19)(18,25)(23,24)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 11238);