C₂×C₅×Dic₆ #6804

order	240
isoclinism	isoclinic class of Dic₆
automorphism	Aut(G) = C₄×D₃×B(4,𝔽₂) (order 1536) · Inn(G) = C₂×D₃ · Out(G) = C₄×#120
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₂×C₅×Dic₆
unambiguous semidirect product	(C₂×C₃)⋊(C₅×Q₈)

Hall	C₃×C₅ ⋊ C₂×Q₈
Sylow	C₃ ⋊ C₂×C₅×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)(6,7,8,9,10)(11,13,14,18)(12,15,16,17);(1,2)(4,5)(11,12,14,16)(13,17,18,15);(1,2)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 165);