The following is a summary of
Proulx's classification
of groups of genus 1.
Because toroidal groups can only have two, three, or four generators, we're
able to separate the 28 cases accordingly.

- \(G = \langle x,y\:|\: y^2 = x^3yxy = 1\rangle\)
- \(G = \langle x,y\:|\: y^2 = x^{-3}yxy = 1\rangle\)
- \(y^2 = (x^2y)^2 = 1\) and \(x\) and \(xy\) have even order, and the subgroup \(S\) generated by \(\{x^2, (xy)^2\}\) is normal and \(|G|/|S| = 4\)
- \(y^2 = x^2yx^{-2}y = 1\) and \(x\) is of even order, and the subgroup \(S\) generated by \(\{x^2,yxy\}\) is such that \(|G|/|S| = 2\)
- \(x^6 = y^2 = (xy)^3 = 1\)
- \(x^4 = y^2 = (xy)^4 = 1\)
- \(x^4 = y^2 = (xy)^2(x^{-1}y)^2 = 1\)
- \(x^3 = y^2 = (xyxyxy^{-1}x)^2 = 1\)
- \(x^3 = y^2 = (xy)^3(x^{-1}y)^3\)
- \(x^4 = y^2 = (xyx^{-1}y)^2 = 1\) and the subgroup \(S\) generated by \(\{x,yxy\}\) is such that \(|G|/|S|=2\)
- \(x^3 = y^2 = (xyx^{-1}y)^3 = 1\)
- \(G = \mathbb{Z}_m \times \mathbb{Z}_n,\,\,\gcd(m,n) \geq 3\)
- \(x^{2a} = y^b = xyx^{-1}y = 1\) for integers \(a>1\), \(b>2\), and the subgroup \(S\) generated by \(\{x^2, y\}\) is such that \(|G|/|S| = 2\)
- \(x^{2a} = y^{2b}=x^2y^2=1\) for integers \(a, b > 1\) and the subgroup \(S\) generated by \(\{x^{-1}y, x^2\}\) is such that \(|G|/|S| = 2\)
- \(x^{2a} = y^{2b} = (xy)^2 = (x^{-1}y)^2 = 1\) for integers \(a,b>1\) and the subgroup \(S\) generated by \(\{x^2,y^2\}\) is such that \(|G|/|S| = 4\)
- \(x^3 = y^3 = (xy)^3 = 1\)
- \(G = \langle x,y\:|\: x^3 = y^3 = xyxy^{-1}x^{-1}y^{-1} = 1\rangle\)

- \(x^2=y^2=z^2=(xy)^3=(xz)^3=(yz)^3 = 1\) and all other reduced relations are have even length
- \(x^2 = y^2=z^2 = (xyz)^2 = 1\)
- \(x^2=y^2=z^2=xyxzyz=(xy)^{2a}=(zy)^{2a}=1\) for integer \(a>0\) and the subgroup generated by \(\{xz,zyzy\}\) is normal and is such that \(|G|/|S| = 4\)
- \(x^2=y^2=z^2=(zxzy)^2 = (xy)^2 = 1\)
- \(x^2=y^2=z^2=(xz)^2(yz)^2 = (xy)^2 = 1\)
- \(x^2=y^2=z^2= (xy)^2 = (xz)^4 = (yz)^4 = 1\) and the subgroup \(S\) generated by \(\{xy, xz, yz\}\) is such that \(|G|/|S| = 2\)
- \(x^2=y^2=z^2= (xy)^2 = (xz)^6 = (yz)^3 = 1\) and the subgroup \(S\) generated by \(\{xy, xz, yz\}\) is such that \(|G|/|S| = 2\)
- \(x^a = y^2 = z^2 = (xy)^2 = (xz)^2 = (yz)^b = 1\) for relatively prime integers \(a, b\), \(a>2\) and the subgroup \(S\) generated by \{x, yz\} is such that \(|G|/|S| = 2\)
- \(x^{2a}=y^2=z^2= (xy)^2 = xzx^{-1}z = (yz)^{2b} = 1\) for \(a,b>1\) and the subgroup \(S\) generated by \(\{x, (yz)^2\}\) is normal and such that \(|G|/|S| = 2\)
- \(x^a = y^2 = z^2 = (xy)^2 = xyx^{-1}y = xzx^{-1}z =(yz)^b= 1\) for relatively prime integers \(a, b\), \(a>2\) and the subgroup \(S\) generated by \{x, yz\} is such that \(|G|/|S| = 2\)

- \(x^2=y^2=z^2=w^2=(xy)^2=(yz)^2=(zw)^2=(wx)^2=(xz)^a=(yw)^b=1\) where \(a=2\) and \(q\) is even, or \(p,q>2\)