order	240
isoclinism	isoclinic class of Dic₆
automorphism	Aut(G) = C₂⁴×C₄×D₃ (order 384) · Inn(G) = C₂×D₃ · Out(G) = C₂³×C₄
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₅×Dic₃⋊C₄
unambiguous semidirect product	Dic₃⋊(C₄×C₅)

Splittings

Hall	C₃×C₅ ⋊ C₄⋊C₄
Sylow	C₃ ⋊ C₅×C₄⋊C₄

52 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 2 generators on 16 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 57);