H(3,𝔽₃)×#494 #13145241

order	1728
nilpotency class	3
isoclinism	isoclinic class of H(3,𝔽₃)×#465
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂×C₃²×C₄

Aliases

direct product	H(3,𝔽₃)×#494
unambiguous semidirect product	(C₃²×#494)⋊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

671 conjugacy classes of subgroups; only the 231 normal subgroups are shown.

normal subgroups only

This group has 671 conjugacy classes of subgroups — too many to draw. Only its 231 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) = 29
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 29 points:

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

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, 11304);