C₂×C₃³×#125 #13179982

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×#125

Subgroups

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

3164 conjugacy classes of subgroups; only the 2492 normal subgroups are shown.

normal subgroups only

This group has 3164 conjugacy classes of subgroups — too many to draw. Only its 2492 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) = 1728

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

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

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, 46045);