H(3,𝔽₃)×#495 #13145244

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)×#495
unambiguous semidirect product	(C₃²×#495)⋊C₃

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×C₃²×D₄
commutator, Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
prime radical	C₂×D₈×H(3,𝔽₃)	C₂
LCS	G ▷ C₂×C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₄²×H(3,𝔽₃) ◁ G

715 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 715 conjugacy classes of subgroups — too many to draw. Only its 203 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)(11,13)(14,15)(16,22)(17,25)(18,20)(19,23)(21,26)(24,28)(27,29);(2,5,6)(4,8,9)(14,16,17,23,18,24,26,29)(15,19,21,22,20,27,25,28);(10,11,12,13)(15,20)(19,27)(21,25)(22,28)

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, 11307);