C₂×C₃×Dic₁₀ #6794

order	240
isoclinism	isoclinic class of Dic₁₀
automorphism	Aut(G) = C₂×AGL(1,𝔽₅)×B(4,𝔽₂) (order 2560) · Inn(G) = C₂×D₅ · Out(G) = C₂×#525
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₂×C₃×Dic₁₀
unambiguous semidirect product	(C₂×C₅)⋊(C₃×Q₈)

Hall	C₃×C₅ ⋊ C₂×Q₈
Sylow	C₅ ⋊ C₂×C₃×Q₈

76 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 3 generators on 18 points:

(3,4,5)(6,7,9,10,8)(11,13,14,18)(12,15,16,17);(1,2)(7,8)(9,10)(11,12,14,16)(13,17,18,15);(1,2)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 155);