  The $k\cdot p$ effective Hamiltonians have been widely applied to predict a
large variety of phenomena in condensed matter systems. Currently, the popular
way to construct a $k\cdot p$ Hamiltonian is in a case-by-case manner, which
significantly limits its applications especially for magnetic systems. In this
work, we first explicitly tabulate all the representation matrices for all
single-valued and double-valued irreducible representations (irreps) and
co-irreps for the little groups of all special $k$ points in 1651 magnetic
space groups (including nonmagnetic 230 space groups). Then through group
theory analysis, we obtain 4 857 832 elementary $k\cdot p$ matrix blocks, and
directly using these matrix blocks given in this work one can obtain any
$k\cdot p$ Hamiltonian for any periodic system, including bulk or boundary. We
believe our work will accelerate the studies in various fields in condensed
matter physics, such as semiconductors, topological physics, spintronics, etc.
We also expect our exhaustive results on $k\cdot p$ models will play vital
roles in connecting other fields with condensed matter physics and promote
realizations of diverse theoretical models which possess exotic properties but
lack practical materials.
