The fundamental group of a band

The fundamental group of a band

I need some help with this please, my problem is to find the fundamental
group of a "band".I have the circular region that we obtain from taking
away an open disk from a closed disk in R^2 and then identify in the
"external border" the antipodes, and then identify in the "inner border"
the antipodes too, but separatly(i. e. the identify process on the the
external and internal border), Well i have to compute the fundamental
group of this quotient.