C₅×C₁₃×3₋¹⁺² #13182178

order	1755
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₅×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×C₁₃×3₋¹⁺²
unambiguous semidirect product	C₉⋊(C₃×C₅×C₁₃)
unambiguous semidirect product	(C₅×C₉)⋊(C₃×C₁₃)

	subgroup	G/H
center, socle	C₃×C₅×C₁₃	C₃²
commutator, Frattini, cosocle	C₃	C₃²×C₅×C₁₃
prime radical	C₃²×C₅×C₁₃	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₅×C₁₃ ◁ G

32 conjugacy classes of subgroups.

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 585

A minimal faithful permutation representation — 2 generators on 27 points:

(1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,22,21,24,26,25,23,27);(20,23,24)(22,26,27)

715 conjugacy classes. The character table is too large to display.

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1755, 11);