H(3,𝔽₃)×#350 #13136052

order	1728
nilpotency class	3
isoclinism	isoclinic class of D₈×H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₈ · Schur multiplier (H₂) = C₂×C₃²

Aliases

direct product	H(3,𝔽₃)×#350
unambiguous semidirect product	(C₃²×#350)⋊C₃

Subgroups

	subgroup	G/H
center	C₃×C₈	C₃²×D₄
commutator	C₃×C₄	C₂×C₃²×C₈
Frattini, cosocle	C₂×C₃×C₈	C₂²×C₃²
socle	C₂×C₃	C₃²×C₂²⋊C₈
prime radical	#41×H(3,𝔽₃)	C₄
LCS	G ▷ C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₈ ◁ C₂×C₈×H(3,𝔽₃) ◁ G

308 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 119 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 — 2 generators on 41 points:

(1,2,4)(3,6,8)(5,9,7)(10,11,14,19,15,20,29,37,16,21,30,38,13,18,27,35)(12,22,24,39,23,33,40,36,25,34,28,41,26,17,31,32);(2,5,6)(4,8,9)(10,12)(11,17)(13,26)(14,28)(15,23)(16,25)(18,34)(19,36)(20,22)(21,33)(24,30)(27,40)(29,31)(32,38)(35,39)(37,41)

Conjugacy classes & character table

308 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 2115);