Dic₅₅ #6350

order	220
isoclinism	isoclinic class of D₅₅
automorphism	Aut(G) = C₂×AGL(1,𝔽₅)×AGL(1,𝔽₁₁) (order 4400) · Inn(G) = D₅₅ · Out(G) = C₂×C₄×C₅
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₅
unambiguous semidirect product	C₁₁⋊Dic₅

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₅
commutator, hypocenter	C₅×C₁₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₁	C₂
LCS	G ▷ C₅×C₁₁
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 220

A minimal faithful permutation representation — 2 generators on 20 points:

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

Polycyclic presentation:

2^2,5,11|a^2=b,c^a=c^4,d^a=d^10

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(220, 5);