D₃×C₇ #185

order	42
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂×C₃×D₃ (order 36) · Inn(G) = D₃ · Out(G) = C₂×C₃
holomorph	G ⋊ Aut(G) = D₃²×AGL(1,𝔽₇)
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×C₇
unambiguous semidirect product	C₃⋊(C₂×C₇)

Fitting, socle, Hall	C₃×C₇ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂×C₇

	subgroup	G/H
center, hypercenter	C₇	D₃
commutator, cosocle, hypocenter	C₃	C₂×C₇
Frattini	C₁	G
Fitting, socle	C₃×C₇	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₇

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 21

A minimal faithful permutation representation — 2 generators on 10 points:

(2,3)(4,5,6,7,8,9,10);(1,2,3)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(42, 3);