order	246
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₅×D₃×C₈ (order 240) · Inn(G) = D₃ · Out(G) = C₅×C₈
holomorph	G ⋊ Aut(G) = D₃²×AGL(1,𝔽₄₁)
homology	Abelianization (H₁) = C₂×C₄₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×C₄₁
unambiguous semidirect product	C₃⋊(C₂×C₄₁)

Splittings

Fitting, socle, Hall	C₃×C₄₁ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂×C₄₁

	subgroup	G/H
center, hypercenter	C₄₁	D₃
commutator, cosocle, hypocenter	C₃	C₂×C₄₁
Frattini	C₁	G
Fitting, socle	C₃×C₄₁	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₄₁

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 123

A minimal faithful permutation representation — 2 generators on 44 points:

(2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44);(1,2,3)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(246, 1);