H(3,𝔽₃)×#489 #13145226

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

Aliases

direct product	H(3,𝔽₃)×#489
unambiguous semidirect product	(C₃²×#489)⋊C₃

Subgroups

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

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

normal subgroups only

This group has 550 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) = 288

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

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

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, 11289);