D₇⋊C₉ #1252

order	126
isoclinism	isoclinic class of AGL(1,𝔽₇)
automorphism	Aut(G) = C₃×AGL(1,𝔽₇) (order 126) · Inn(G) = AGL(1,𝔽₇) · Out(G) = C₃
holomorph	G ⋊ Aut(G) = #72229823
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	D₇⋊C₉
unambiguous semidirect product	(C₇⋊C₉)⋊C₂

	subgroup	G/H
center, Frattini, hypercenter	C₃	AGL(1,𝔽₇)
commutator, hypocenter	C₇	C₂×C₉
Fitting, socle, cosocle	C₃×C₇	C₂×C₃
prime radical	C₃×D₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₃

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:

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

Polycyclic presentation:

2,3^2,7|b^3=c,d^a=d^6,d^b=d^4

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(126, 1);