Those fundamental properties, such as phase transitions, Weyl fermions and spin excitation, in all magnetic ordered materials was ultimately believed to rely on the symmetry theory of magnetic space groups. Recently, it has come to light that a more comprehensive group, known as the spin space group (SSG), which combines separate spin and spatial operations, is necessary to fully characterize the geometry and physical properties of magnetic ordered materials such as altermagnets. However, the basic theory of SSG has been seldomly developed. In this work, we present a systematic study of the enumeration and the representation theory of SSG. Starting from the 230 crystallographic space groups and finite translational groups with a maximum order of 8, we establish an extensive collection of over 80,000 SSGs under a four-segment nomenclature. We then identify inequivalent SSGs specifically applicable to collinear, coplanar, and noncoplanar magnetic configurations. Moreover, we derive the irreducible co-representations of the little group in momentum space within the SSG framework. Finally, we illustrate the SSGs and band degeneracies resulting from SSG symmetries through several representative material examples, including a well-known altermagnet RuO2, and a spiral magnet CeAuAl3. Our work advances the field of group theory in describing magnetic ordered materials, opening up avenues for deeper comprehension and further exploration of emergent phenomena in magnetic materials.
29 pages, 1 table, 5 figures and a Supplementary table with 1508 pages