H(3,𝔽₃)×#508 #13145283

order	1728
nilpotency class	4
isoclinism	isoclinic class of H(3,𝔽₃)×D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃²

Aliases

direct product	H(3,𝔽₃)×#508
unambiguous semidirect product	(C₃²×#508)⋊C₃

Subgroups

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

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

normal subgroups only

This group has 462 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) = 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)(12,21)(13,18)(14,26)(15,30)(16,20)(17,23)(19,28)(22,37)(24,38)(25,36)(27,32)(29,40)(31,35)(33,41)(34,39);(2,5,6)(4,8,9)(10,12)(11,17)(13,22)(14,27)(15,31)(16,24)(18,32)(19,35)(20,34)(21,26)(23,28)(25,39)(29,41)(30,33)(36,37)(38,40);(10,13,16,15)(11,18,20,30)(12,22,24,31)(14,25,29,28)(17,32,34,33)(19,26,36,40)(21,37,38,35)(23,27,39,41)

Conjugacy classes & character table

242 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 11346);