order	507
isoclinism	isoclinic class of C₁₃⋊C₃
automorphism	Aut(G) = C₃×C₄×AGL(1,𝔽₁₃) (order 1872) · Inn(G) = C₁₃⋊C₃ · Out(G) = C₃×C₄²
homology	Abelianization (H₁) = C₃×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₃×C₁₃⋊C₃
unambiguous semidirect product	C₁₃⋊(C₃×C₁₃)

Splittings

Fitting, socle, Sylow	C₁₃² ⋊ C₃
derived	C₁₃ ⋊ C₃×C₁₃

	subgroup	G/H
center, hypercenter	C₁₃	C₁₃⋊C₃
commutator, cosocle, hypocenter	C₁₃	C₃×C₁₃
Frattini	C₁	G
Fitting, socle	C₁₃²	C₃
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₁₃

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 39

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(507, 3);