3₋¹⁺²×#501 #13145263

order	1728
nilpotency class	3
isoclinism	isoclinic class of H(3,𝔽₃)×#468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂

Aliases

direct product	3₋¹⁺²×#501
unambiguous semidirect product	(C₉×#501)⋊C₃

	subgroup	G/H
center, socle	C₂²×C₃	C₂×C₃²×D₄
commutator, Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
prime radical	C₂²×C₃²	C₂×C₃×D₄
LCS	G ▷ C₂×C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₄²×3₋¹⁺² ◁ G

360 conjugacy classes of subgroups.

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 25 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 11326);