H(3,𝔽₃)×#491 #13145232

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)×#491
unambiguous semidirect product	(C₃²×#491)⋊C₃

Subgroups

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

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

normal subgroups only

This group has 495 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) = 33
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 33 points:

(1,2,4)(3,6,8)(5,9,7)(10,11)(12,15)(13,14)(16,17)(18,19,24,29)(20,26,31,32)(21,25,30,33)(22,28,23,27);(2,5,6)(4,8,9)(11,14)(15,17)(18,20,22,21,24,31,23,30)(19,25,27,26,29,33,28,32);(10,12,13,16)(11,15,14,17)(18,21,22,31,24,30,23,20)(19,26,27,33,29,32,28,25)

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, 11295);