order	273
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	Aut(G) = C₃×C₄×AGL(1,𝔽₇) (order 504) · Inn(G) = ΓL(1,𝔽₈) · Out(G) = C₂×C₃×C₄
homology	Abelianization (H₁) = C₃×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₃×ΓL(1,𝔽₈)
unambiguous semidirect product	C₇⋊(C₃×C₁₃)

Splittings

Fitting, socle, Hall	C₇×C₁₃ ⋊ C₃
derived, Sylow	C₇ ⋊ C₃×C₁₃

	subgroup	G/H
center, hypercenter	C₁₃	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₃×C₁₃
Frattini	C₁	G
Fitting, socle	C₇×C₁₃	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₁₃

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 91

A minimal faithful permutation representation — 2 generators on 20 points:

(2,3,4)(5,7,6)(8,9,10,11,12,13,14,15,16,17,18,19,20);(1,2,3,5,4,6,7)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(273, 1);