C₂×C₃³×#124 #13179981

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

Aliases

C₂×C₃³×#124

	subgroup	G/H
center, socle	C₂³×C₃³	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃³
prime radical	C₂²×C₃³×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃³ ◁ G

4620 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 864

A minimal faithful permutation representation — 4 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 46044);