C₃×#689 #6686

order	240
isoclinism	isoclinic class of #686
automorphism	Aut(G) = C₂²×D₄×AGL(1,𝔽₅) (order 640) · Inn(G) = C₂²⋊D₅ · Out(G) = C₂²×C₄
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₁

Aliases

Hall	C₃×C₅ ⋊ Q₁₆
Sylow	C₅ ⋊ C₃×Q₁₆

36 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 2 generators on 24 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 47);