  Supergroups are defined in the framework of $\dZ_2$-graded Clifford algebras
over the fields of real and complex numbers, respectively. It is shown that
cyclic structures of complex and real supergroups are defined by Brauer-Wall
groups related with the modulo 2 and modulo 8 periodicities of the complex and
real Clifford algebras. Particle (fermionic and bosonic) representations of a
universal covering (spinor group $\spin_+(1,3)$) of the proper orthochronous
Lorentz group are constructed via the Clifford algebra formalism. Complex and
real supergroups are defined on the representation system of $\spin_+(1,3)$. It
is shown that a cyclic (modulo 2) structure of the complex supergroup is
equivalent to a supersymmetric action, that is, it converts fermionic
representations into bosonic representations and vice versa. The cyclic action
of the real supergroup leads to a much more high-graded symmetry related with
the modulo 8 periodicity of the real Clifford algebras. This symmetry acts on
the system of real representations of $\spin_+(1,3)$.
