C₃×C₉×#491 #13145231

order	1728
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₉ · Schur multiplier (H₂) = C₃×C₄

Aliases

direct product

C₃×C₉×#491

Subgroups

	subgroup	G/H
center	C₂²×C₃×C₉	C₂×D₄
commutator	C₂×C₄	C₂³×C₃×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
socle, prime radical	C₂²×C₃²	C₂×C₃×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₉ ◁ C₃×C₄²×C₉ ◁ G

450 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 250 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) = 36
transitive degree	μ_t(G) = 1728

A minimal faithful permutation representation — 3 generators on 36 points:

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

Conjugacy classes & character table

432 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 11294);