  In Ref. [arXiv:1802.05554v3] one of the authors (N.S.M.B.) studies the second
quantization of fermions with integer spin while describing the internal
degrees of freedom of fermions in Grassmann space. In this contribution we
study the representations in Grassmann space of the groups $SO(5,1)$,
$SO(3,1)$, $SU(3) \times U(1)$, and $SO(4)$, which are of particular interest
as the subgroups of the group $SO(13,1)$. The second quantized integer spin
fermions, appearing in Grassmann space, not observed so far, could be an
alternative choice to the half integer spin fermions, appearing in Clifford
space. The spin-charge-family theory, using two kinds of Clifford operators ---
$\gamma^a$ and $\tilde{\gamma}^a$ --- for the description of spins and charges
(first) and family quantum numbers (second), offers the explanation for not
only the appearance of families but also for all the properties of quarks and
leptons, the gauge fields, scalar fields and others. In both cases the gauge
fields in $d \ge(13+1)$ --- the spin connections $\omega_{ab \alpha}$ (of the
two kinds in Clifford case and of one kind in Grassmann case) and the vielbeins
$f^{\alpha}{}_{\alpha}$ --- determine in $d=(3+1)$ scalars, those with the
space index $\alpha=(5,6,\cdots,d)$, and gauge fields, those with the space
index $\alpha=(0,1,2,3)$. While states of the Lorentz group and all its
subgroups (in any dimension) are in Clifford space in the fundamental
representations of the groups, with the family degrees of freedom included,
states in Grassmann space manifest with respect to the Lorentz group adjoint
representations, allowing no families.
