C₁₃×D₁₁ #63348

order	286
isoclinism	isoclinic class of D₁₁
automorphism	Aut(G) = C₃×C₄×AGL(1,𝔽₁₁) (order 1320) · Inn(G) = D₁₁ · Out(G) = C₃×C₄×C₅
homology	Abelianization (H₁) = C₂×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₃×D₁₁
unambiguous semidirect product	C₁₁⋊(C₂×C₁₃)

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

	subgroup	G/H
center, hypercenter	C₁₃	D₁₁
commutator, cosocle, hypocenter	C₁₁	C₂×C₁₃
Frattini	C₁	G
Fitting, socle	C₁₁×C₁₃	C₂
LCS	G ▷ C₁₁
UCS	C₁ ◁ C₁₃

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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(286, 1);