D₆₃ #1256

order	126
isoclinism	isoclinic class of D₆₃
automorphism	Aut(G) = AGL(1,𝔽₇)×AGL(1,ℤ/9) (order 2268) · Inn(G) = G · Out(G) = C₂×C₃²
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₆₃
unambiguous semidirect product	C₇⋊D₉
unambiguous semidirect product	C₉⋊D₇

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×C₉	C₂
Frattini	C₃	D₂₁
socle	C₃×C₇	D₃
LCS	G ▷ C₇×C₉
UCS	C₁

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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(126, 5);