C₇×D₉ #1254

order	126
isoclinism	isoclinic class of D₉
automorphism	Aut(G) = C₂×C₃×AGL(1,ℤ/9) (order 324) · Inn(G) = D₉ · Out(G) = C₂×C₃²
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₇)×#92265⋊C₂
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×D₉
unambiguous semidirect product	C₉⋊(C₂×C₇)

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

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

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 63

A minimal faithful permutation representation — 2 generators on 16 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(126, 3);