D₃×D₇ #749

order	84
isoclinism	isoclinic class of D₃×D₇
automorphism	Aut(G) = D₃×AGL(1,𝔽₇) (order 252) · Inn(G) = G · Out(G) = C₃
holomorph	G ⋊ Aut(G) = D₃²×#64423⋊C₂
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

direct product	D₃×D₇
unambiguous semidirect product	(D₃×C₇)⋊C₂
unambiguous semidirect product	(C₃×D₇)⋊C₂
unambiguous semidirect product	D₂₁⋊C₂

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇	C₂²
LCS	G ▷ C₃×C₇
UCS	C₁

20 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);(1,2,4,6,7,5,3)(9,10)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(84, 8);