C₂×C₃³×#126 #13179983

order	1728
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃³ · Schur multiplier (H₂) = C₂⁴×C₃³

Aliases

direct product

C₂×C₃³×#126

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃³	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃³
prime radical	C₂⁴×C₃³	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃³ ◁ G

3444 conjugacy classes of subgroups; only the 2268 normal subgroups are shown.

normal subgroups only

This group has 3444 conjugacy classes of subgroups — too many to draw. Only its 2268 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) = 27
transitive degree	μ_t(G) = 864

A minimal faithful permutation representation — 4 generators on 27 points:

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

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, 46046);