3₋¹⁺²×#391 #13144933

order	1728
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)×#120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product	3₋¹⁺²×#391
unambiguous semidirect product	(C₉×#391)⋊C₃

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³×C₃	C₂³×C₃²
commutator	C₂²×C₃	C₂²×C₃²×C₄
prime radical	C₂³×C₃²	C₂³×C₃
LCS	G ▷ C₂²×C₃ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

744 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)(11,13)(14,15,17,16)(20,24)(22,25);(2,5,6)(4,8,9)(10,11,12,13)(14,16,17,15)(18,19,21,23)(20,22,24,25);(18,20,21,24)(19,22,23,25)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 10996);