H(3,𝔽₃)×#470 #13145169

order	1728
nilpotency class	3
isoclinism	isoclinic class of H(3,𝔽₃)×#468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃²

Aliases

direct product	H(3,𝔽₃)×#470
unambiguous semidirect product	(C₃²×#470)⋊C₃

Subgroups

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

605 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 605 conjugacy classes of subgroups — too many to draw. Only its 189 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) = 288

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

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

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, 11232);