3₋¹⁺²×#383 #13144909

order	1728
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)×#120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

3₋¹⁺²×#383

Subgroups

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

552 conjugacy classes of subgroups; only the 287 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 287 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) = 29
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 29 points:

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

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, 10972);