C₃×C₉×#458 #13145132

order	1728
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₉ · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

C₃×C₉×#458

Subgroups

	subgroup	G/H
center	C₂×C₃×C₉	#120
commutator	C₂³	C₂³×C₃×C₉
Frattini, cosocle	C₂³×C₃	C₂³×C₃²
socle	C₂×C₃²	C₃×#120
prime radical	C₃²×H(5,𝔽₂)	C₂×C₃
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₉ ◁ C₂³×C₃×C₉ ◁ G

800 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 270 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) = 28
transitive degree	μ_t(G) = 432

A minimal faithful permutation representation — 3 generators on 28 points:

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

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, 11195);