H(3,𝔽₃)×#473 #13145178

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)×#473
unambiguous semidirect product	(C₃²×#473)⋊C₃

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	C₃²×#121
commutator, Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
prime radical	#41×H(3,𝔽₃)	C₂²
LCS	G ▷ C₂×C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₄×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) = 41
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 3 generators on 41 points:

(1,2,4)(3,6,8)(5,9,7)(10,11,16,21)(12,22,26,32)(13,27,29,35)(14,20,15,19)(17,25,33,24)(18,30,34,28)(23,37,39,31)(36,40,41,38);(2,5,6)(4,8,9)(10,12,15,25,16,26,14,24)(11,17,20,32,21,33,19,22)(13,23,30,40,29,39,28,38)(18,31,35,41,34,37,27,36);(10,13)(11,18)(12,23)(14,28)(15,30)(16,29)(17,31)(19,27)(20,35)(21,34)(22,36)(24,38)(25,40)(26,39)(32,41)(33,37)

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, 11241);