C₁₁×D₁₃ #63349

order	286
isoclinism	isoclinic class of D₁₃
automorphism	Aut(G) = C₂×C₅×AGL(1,𝔽₁₃) (order 1560) · Inn(G) = D₁₃ · Out(G) = C₂²×C₃×C₅
homology	Abelianization (H₁) = C₂×C₁₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₁×D₁₃
unambiguous semidirect product	C₁₃⋊(C₂×C₁₁)

Fitting, socle, Hall	C₁₁×C₁₃ ⋊ C₂
derived, Sylow	C₁₃ ⋊ C₂×C₁₁

	subgroup	G/H
center, hypercenter	C₁₁	D₁₃
commutator, cosocle, hypocenter	C₁₃	C₂×C₁₁
Frattini	C₁	G
Fitting, socle	C₁₁×C₁₃	C₂
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₁₁

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 143

A minimal faithful permutation representation — 2 generators on 24 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(286, 2);