order	399
isoclinism	isoclinic class of C₁₉⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×C₁₉⋊C₃
unambiguous semidirect product	C₁₉⋊(C₃×C₇)

Splittings

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

	subgroup	G/H
center, hypercenter	C₇	C₁₉⋊C₃
commutator, cosocle, hypocenter	C₁₉	C₃×C₇
Frattini	C₁	G
Fitting, socle	C₇×C₁₉	C₃
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₇

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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(399, 2);