C₁₃×D₁₃ #66763

order	338
isoclinism	isoclinic class of D₁₃
automorphism	Aut(G) = C₃×C₄×AGL(1,𝔽₁₃) (order 1872) · Inn(G) = D₁₃ · Out(G) = C₂×C₃²×C₄
homology	Abelianization (H₁) = C₂×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

affine special semilinear	AΣL(1,𝔽₁₆₉)
direct product	C₁₃×D₁₃
unambiguous semidirect product	C₁₃⋊(C₂×C₁₃)

Fitting, socle, Sylow	C₁₃² ⋊ C₂
derived	C₁₃ ⋊ C₂×C₁₃

	subgroup	G/H
center, hypercenter	C₁₃	D₁₃
commutator, cosocle, hypocenter	C₁₃	C₂×C₁₃
Frattini	C₁	G
Fitting, socle	C₁₃²	C₂
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₁₃

14 conjugacy classes of subgroups.

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

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(338, 3);