order	399
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	Aut(G) = C₂×C₉×AGL(1,𝔽₇) (order 756) · Inn(G) = ΓL(1,𝔽₈) · Out(G) = C₂²×C₉
homology	Abelianization (H₁) = C₃×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₉×ΓL(1,𝔽₈)
unambiguous semidirect product	C₇⋊(C₃×C₁₉)

Splittings

Fitting, socle, Hall	C₇×C₁₉ ⋊ C₃
derived, Sylow	C₇ ⋊ C₃×C₁₉

	subgroup	G/H
center, hypercenter	C₁₉	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₃×C₁₉
Frattini	C₁	G
Fitting, socle	C₇×C₁₉	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₁₉

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 133

A minimal faithful permutation representation — 2 generators on 26 points:

(2,3,4)(5,7,6)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26);(1,2,3,5,4,6,7)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(399, 1);