D₃×C₇⋊C₉ #67363

order	378
isoclinism	isoclinic class of D₃×ΓL(1,𝔽₈)
automorphism	Aut(G) = C₃×D₃×AGL(1,𝔽₇) (order 756) · Inn(G) = D₃×ΓL(1,𝔽₈) · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×C₇⋊C₉
unambiguous semidirect product	C₃⋊(C₂×C₇⋊C₉)
unambiguous semidirect product	(D₃×C₇)⋊C₉

	subgroup	G/H
center, Frattini, hypercenter	C₃	D₃×ΓL(1,𝔽₈)
commutator, hypocenter	C₃×C₇	C₂×C₉
Fitting, socle, cosocle	C₃²×C₇	C₂×C₃
prime radical	C₃×D₃×C₇	C₃
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₃

28 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 126

A minimal faithful permutation representation — 2 generators on 19 points:

(2,3)(5,6,7)(8,10,9)(11,15,19,13,18,14,16,12,17);(1,2,3)(4,5,6,8,7,9,10)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 16);