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Let $N$ be a positive integer and $\Gamma$ be a subgroup of$\mathrm{SL}_2(\mathbb{Z})$ containing $\Gamma_1(N)$. Let $K$ be an imaginaryquadratic field and $\mathcal{O}$ be an order of discriminant $D_\mathcal{O}$in $K$. Under some assumptions, we show that $\Gamma$ induces a form classgroup of discriminant $D_\mathcal{O}$ (or, of order $\mathcal{O}$) and level$N$ if and only if there is a certain canonical model of the modular curve for$\Gamma$ defined over a suitably small number field. In this way we can find aninteresting link between two different subjects.

Gauss' form class groups and Shimura's canonical models