D₅₅ #1124

order	110
isoclinism	isoclinic class of D₅₅
automorphism	Aut(G) = AGL(1,𝔽₅)×AGL(1,𝔽₁₁) (order 2200) · Inn(G) = G · Out(G) = C₄×C₅
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₅₅
unambiguous semidirect product	C₅⋊D₁₁
unambiguous semidirect product	C₁₁⋊D₅

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

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 55

A minimal faithful permutation representation — 2 generators on 16 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(110, 5);