D₄₄ #765

order	88
isoclinism	isoclinic class of Dic₂₂
automorphism	Aut(G) = D₄×AGL(1,𝔽₁₁) (order 880) · Inn(G) = C₂×D₁₁ · Out(G) = C₂²×C₅
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₄₄
unambiguous semidirect product	C₄⋊D₁₁

Fitting	C₄×C₁₁ ⋊ C₂
Sylow	C₁₁ ⋊ D₄

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

Polycyclic presentation:

2^3,11|b^2=c,b^a=bc,d^a=d^10

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(88, 5);