order	182
isoclinism	isoclinic class of D₁₃
automorphism	Aut(G) = C₂×C₃×AGL(1,𝔽₁₃) (order 936) · Inn(G) = D₁₃ · Out(G) = C₂²×C₃²
homology	Abelianization (H₁) = C₂×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×D₁₃
unambiguous semidirect product	C₁₃⋊(C₂×C₇)

Splittings

Fitting, socle, Hall	C₇×C₁₃ ⋊ C₂
derived, Sylow	C₁₃ ⋊ C₂×C₇

	subgroup	G/H
center, hypercenter	C₇	D₁₃
commutator, cosocle, hypocenter	C₁₃	C₂×C₇
Frattini	C₁	G
Fitting, socle	C₇×C₁₃	C₂
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₇

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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(182, 2);