D₁₁ #57

order	22
isoclinism	isoclinic class of D₁₁
automorphism	Aut(G) = AGL(1,𝔽₁₁) (order 110) · Inn(G) = G · Out(G) = C₅
holomorph	G ⋊ Aut(G) = #11925892⋊C₂
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁
unambiguous semidirect product	C₁₁⋊C₂

C₁₁ ⋊ C₂

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

4 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 11

A minimal faithful permutation representation — 2 generators on 11 points:

(1,2,3,4,5,6,7,8,9,10,11);(2,11)(3,10)(4,9)(5,8)(6,7)

Polycyclic presentation:

2,11|b^a=b^10

Here ζ is a primitive root of unity of order 11 (ζ¹¹ = 1).

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22, 1);