A wallpaper group is a crystallographic group (space group) in dimension 2, hence a subgroup $G \subset Iso(\mathbb{R}^2)$ of the isometry group of the Euclidean plane such that
the part of $G$ inside the translation group is generated by two linearly independent vectors;
the point group is finite.
e. g.
Richard Tilley, Sec. 3.5 in: Crystals and Crystal Structure, Wiley (2020) [ISBN:978-1-119-54838-6]
Patrick Morandi, The Classification of Wallpaper Patterns: From Group Cohomology to Escherís Tessellations (pdf)
See also
Discussion of meromorphic functions on the complex plane invariant under wallpaper groups:
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