D₁₁×C₂₃ #92781

order	506
isoclinism	isoclinic class of D₁₁
automorphism	Aut(G) = C₂×C₁₁×AGL(1,𝔽₁₁) (order 2420) · Inn(G) = D₁₁ · Out(G) = C₂×C₅×C₁₁
homology	Abelianization (H₁) = C₂×C₂₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₁₁×C₂₃
unambiguous semidirect product	C₁₁⋊(C₂×C₂₃)

Fitting, socle, Hall	C₁₁×C₂₃ ⋊ C₂
derived, Sylow	C₁₁ ⋊ C₂×C₂₃

	subgroup	G/H
center, hypercenter	C₂₃	D₁₁
commutator, cosocle, hypocenter	C₁₁	C₂×C₂₃
Frattini	C₁	G
Fitting, socle	C₁₁×C₂₃	C₂
LCS	G ▷ C₁₁
UCS	C₁ ◁ C₂₃

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 253

A minimal faithful permutation representation — 2 generators on 34 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34);(1,2,4,6,8,10,11,9,7,5,3)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(506, 3);