C₃×C₉×#383 #13144907

order	1728
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄×C₉ · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

C₃×C₉×#383

Subgroups

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

690 conjugacy classes of subgroups; only the 410 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 410 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) = 32
transitive degree	μ_t(G) = 1728

A minimal faithful permutation representation — 3 generators on 32 points:

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

Conjugacy classes & character table

756 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 10970);