C₃×C₅×#41 #6827

order	240
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂²×C₄×S₄ (order 384) · Inn(G) = C₂² · Out(G) = C₂²×C₄×D₃
homology	Abelianization (H₁) = C₂³×C₃×C₅ · Schur multiplier (H₂) = C₂²

Aliases

C₃×C₅×#41

	subgroup	G/H
center	C₃×C₄×C₅	C₂²
commutator, Frattini, cosocle	C₂	C₂³×C₃×C₅
socle	C₂×C₃×C₅	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄×C₅ ◁ G

80 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 3 generators on 16 points:

(1,2)(3,6)(4,5)(7,8)(9,10,11,12,13)(14,15,16);(2,5)(6,8);(1,3,4,7)(2,6,5,8)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 188);