Dic₁₁ #190

order	44
isoclinism	isoclinic class of D₁₁
automorphism	Aut(G) = C₂×AGL(1,𝔽₁₁) (order 220) · Inn(G) = D₁₁ · Out(G) = C₂×C₅
holomorph	G ⋊ Aut(G) = #72213693
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₁₁
unambiguous semidirect product	C₁₁⋊C₄

C₁₁ ⋊ C₄

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₁
commutator, hypocenter	C₁₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁	C₂
LCS	G ▷ C₁₁
UCS	C₁ ◁ C₂

6 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 44

A minimal faithful permutation representation — 2 generators on 15 points:

(1,2)(3,4)(5,6,7,8,9,10,11,12,13,14,15);(1,3,2,4)(6,15)(7,14)(8,13)(9,12)(10,11)

Polycyclic presentation:

2^2,11|a^2=b,c^a=c^10

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(44, 1);