D₉×C₁₉ #66785

order	342
isoclinism	isoclinic class of D₉
automorphism	Aut(G) = C₂×C₉×AGL(1,ℤ/9) (order 972) · Inn(G) = D₉ · Out(G) = C₂×C₃×C₉
homology	Abelianization (H₁) = C₂×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₉×C₁₉
unambiguous semidirect product	C₉⋊(C₂×C₁₉)

Fitting, Hall	C₉×C₁₉ ⋊ C₂
derived, Sylow	C₉ ⋊ C₂×C₁₉

	subgroup	G/H
center, hypercenter	C₁₉	D₉
commutator, cosocle, hypocenter	C₉	C₂×C₁₉
Frattini	C₃	D₃×C₁₉
Fitting	C₉×C₁₉	C₂
socle	C₃×C₁₉	D₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₁₉

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 171

A minimal faithful permutation representation — 2 generators on 28 points:

(2,4)(3,7)(5,8)(6,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28);(1,6,4,3,8,5,7,2,9)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(342, 3);