H(3,𝔽₃)×#509 #13145286

order	1728
nilpotency class	4
isoclinism	isoclinic class of H(3,𝔽₃)×#509
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃²

Aliases

direct product	H(3,𝔽₃)×#509
unambiguous semidirect product	(C₃²×#509)⋊C₃

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	C₂×C₃²×D₈
commutator, Frattini, cosocle	C₃×C₈	C₂³×C₃²
LCS	G ▷ C₃×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₄×H(3,𝔽₃) ◁ C₂×C₈×H(3,𝔽₃) ◁ G

495 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 161 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)(10,11)(12,15)(13,22)(14,16)(17,19)(18,20)(21,23)(24,25);(2,5,6)(4,8,9)(11,15)(12,19)(13,20)(16,23)(17,18)(21,25)(22,24);(11,16)(15,23)(17,24)(18,22)

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, 11349);