  Those fundamental physical properties, such as phase transitions, Weyl
fermions, and spin excitation, in all magnetic ordered materials, were
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 underlying properties of
magnetic ordered materials. However, the basic theory of SSG has seldom been
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 translation groups with a maximum order of 8, we establish an
extensive collection of over 100000 SSGs under a four-index nomenclature as
well as the International notation. We then identify inequivalent SSGs
specifically applicable to collinear, coplanar, and noncoplanar magnetic
configurations. To facilitate the identification of SSG, we develop an online
program (findspingroup.com) that can determine the SSG symmetries of any
magnetic ordered crystals. Moreover, we derive the irreducible
co-representations of the little group in momentum space within the SSG
framework. Finally, we illustrate the SSG symmetries and physical effects
beyond the framework of magnetic space groups through several representative
material examples, including a well-known altermagnet RuO2, spiral spin
polarization in the coplanar antiferromagnet CeAuAl3, and geometric Hall effect
in the noncoplanar antiferromagnet CoNb3S6. 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.
