C₇⋊C₉ #316

order	63
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	Aut(G) = C₃×AGL(1,𝔽₇) (order 126) · Inn(G) = ΓL(1,𝔽₈) · Out(G) = C₂×C₃
holomorph	G ⋊ Aut(G) = #72224657
homology	Abelianization (H₁) = C₉ · Schur multiplier (H₂) = C₁

Aliases

C₇⋊C₉

C₇ ⋊ C₉

	subgroup	G/H
center, Frattini, hypercenter	C₃	ΓL(1,𝔽₈)
commutator, hypocenter	C₇	C₉
Fitting, socle, prime radical, cosocle	C₃×C₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₃

6 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(63, 1);