C₂×C₃³×#127 #13179985

order	1728
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃³ · Schur multiplier (H₂) = C₂⁵×C₃³×C₄

Aliases

C₂×C₃³×#127

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

6972 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 864

A minimal faithful permutation representation — 4 generators on 19 points:

(1,2)(11,12,13);(4,6)(8,9)(14,15,16);(7,8,10,9)(17,18,19);(3,4,5,6)(7,9,10,8)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 46048);