\titleofpresentation{Quasi-Permutation Representations of
$2$-Groups of order $64$}

\presenter{Ghodrat Ghaffarzadeh}

By a quasi-permutation matrix we mean a square matrix over the
complex field $\Bbb C$ with non-negative integral trace. Thus
every permutation matrix over $\Bbb C$ is a quasi-permutation
matrix. For a given finite group $G$, let $p(G)$ denote the
minimal degree of a faithful permutation representation of $G$ (or
of a faithful representation of $G$ by permutation matrices), let
$q(G)$ denote the minimal degree of a faithful representation of
$G$ by quasi-permutation matrices over the rational field $\Bbb
Q$, and let $c(G)$ denote the minimal degree of a faithful
representation of $G$ by complex quasi-permutation matrices. It is
easy to see that
\[c(G) \leq q(G) \leq p(G) \]
where $G$ is a finite group. In \cite{1}, we gave algorithms to calculate 
$c(G)$, $q(G)$ and $p(G)$ for a finite group $G$. In this paper, we calculate 
$c(G)$, $q(G)$ and $p(G)$ for non-abelian groups of oeder $64$. We will show that 
for the groups of order $64$ not both of $c(G)<q(G)$ and $q(G)<p(G)$ can hold. So
still one can ask that when both of the mentioned inequalities can hold for $2$-groups.
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\begin{ourbiblio}{99}
\bibitem{1}
{\bf H.BEHRAVESH}. {\em Quasi-permutation  representations of
$p$-groups of class $2$}. {\rm  J.London Math. Soc. (2) 55 (1997)
251-260}.
\bibitem{2}
{\bf H.BEHRAVESH}. {\em Quasi-permutation representations of metacyclic 
$2$-groups with cyclic center}. {\rm Bull. Iranian Math. Soc. 24(1)(1998), 1-14}.
\bibitem{3}
{\bf M. Schonert et al.}. {\em Gap- Groups, Algorithms and
 Programming}. {\rm Lehrstuhl D fur Mathematik, RWTH Aachen, Germany, 1994}. 
\end{ourbiblio}
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\presenteraddress{University of Urmia, Urmia, Iran}

\email{q_gafarzadeh@yahoo.com}

\web{}
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