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Recent methods (e.g. MaterialGAN) have used unconditional GANs to generate per-pixel material maps, or as a prior to reconstruct materials from input photographs. These models can generate varied random material appearance, but do not have any mechanism to constrain the generated material to a specific category or to control the coarse structure of the generated material, such as the exact brick layout on a brick wall. Furthermore, materials reconstructed from a single input photo commonly have artifacts and are generally not tileable, which limits their use in practical content creation pipelines. We propose TileGen, a generative model for SVBRDFs that is specific to a material category, always tileable, and optionally conditional on a provided input structure pattern. TileGen is a variant of StyleGAN whose architecture is modified to always produce tileable (periodic) material maps. In addition to the standard "style" latent code, TileGen can optionally take a condition image, giving a user direct control over the dominant spatial (and optionally color) features of the material. For example, in brick materials, the user can specify a brick layout and the brick color, or in leather materials, the locations of wrinkles and folds. Our inverse rendering approach can find a material perceptually matching a single target photograph by optimization. This reconstruction can also be conditional on a user-provided pattern. The resulting materials are tileable, can be larger than the target image, and are editable by varying the condition.