  We investigate the groups generated by the sets of $CP$, $CNOT$ and
$SWAP^\alpha$ (power-of-SWAP) quantum gate operations acting on $n$ qubits.
Isomorphisms to standard groups are found, and using techniques from
representation theory, we are able to determine the invariant subspaces of the
$n-$qubit Hilbert space under the action of each group. For the $CP$ operation,
we find isomorphism to the direct product of $n(n-1)/2$ cyclic groups of order
$2$, and determine $2^n$ $1$-dimensional invariant subspaces corresponding to
the computational state-vectors. For the $CNOT$ operation, we find isomorphism
to the general linear group of an $n$-dimensional space over a field of $2$
elements, $GL(n,2)$, and determine two $1$-dimensional invariant subspaces and
one $(2^n-2)$-dimensional invariant subspace. For the $SWAP^\alpha$ operation
we determine a complex structure of invariant subspaces with varying dimensions
and occurrences and present a recursive procedure to construct them. As an
example of an application for our work, we suggest that these invariant
subspaces can be used to construct simple formal verification procedures to
assess the operation of quantum computers of arbitrary size.
