C₇×C₃⋊D₉ #67387

order	378
isoclinism	isoclinic class of C₃⋊D₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₇×C₃⋊D₉
unambiguous semidirect product	C₃⋊(C₇×D₉)
unambiguous semidirect product	C₉⋊(D₃×C₇)

Fitting, Hall	C₃×C₇×C₉ ⋊ C₂
derived, Sylow	C₃×C₉ ⋊ C₂×C₇

	subgroup	G/H
center, hypercenter	C₇	C₃⋊D₉
commutator, cosocle, hypocenter	C₃×C₉	C₂×C₇
Frattini	C₃	C₇×C₃⋊D₃
Fitting	C₃×C₇×C₉	C₂
socle	C₃²×C₇	D₃
LCS	G ▷ C₃×C₉
UCS	C₁ ◁ C₇

40 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 189

A minimal faithful permutation representation — 3 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 40);