H(3,𝔽₃)×#504 #13145271

order	1728
nilpotency class	2
isoclinism	isoclinic class of D₄×H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₈ · Schur multiplier (H₂) = C₂²×C₃²

Aliases

direct product

H(3,𝔽₃)×#504

Subgroups

	subgroup	G/H
center	C₃×C₁₆	C₂²×C₃²
commutator, socle	C₂×C₃	C₂²×C₃²×C₈
Frattini, cosocle	C₃×C₈	C₂³×C₃²
prime radical	#41×H(3,𝔽₃)	C₄
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₁₆ ◁ G

462 conjugacy classes of subgroups; only the 273 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 273 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)(10,11,13,17,14,18,23,29,15,19,24,30,25,31,35,39)(12,16,21,27,22,28,34,38,20,26,32,36,33,37,40,41);(2,5,6)(4,8,9)(10,12)(11,16)(13,21)(14,22)(15,20)(17,27)(18,28)(19,26)(23,34)(24,32)(25,33)(29,38)(30,36)(31,37)(35,40)(39,41);(12,20)(16,26)(21,32)(22,33)(27,36)(28,37)(34,40)(38,41)

Conjugacy classes & character table

440 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 11334);