3₋¹⁺²×#474 #13145182

order	1728
nilpotency class	3
isoclinism	isoclinic class of H(3,𝔽₃)×#447
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²

Aliases

direct product	3₋¹⁺²×#474
unambiguous semidirect product	(C₉×#474)⋊C₃

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

432 conjugacy classes of subgroups.

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 288

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 11245);