# Sylow $q-$radical subgroup of a solvable group

Let ‎‎$G‎‎$ ‎be a‎ ‎finite ‎solvable ‎group ‎of ‎order ‎‎$p^2q^2‎‎$‎, where ‎$p>q‎$ ‎and ‎$‎q\nmid‎‎ ‎p-1‎$‎‎. ‎Let‎ ‎‎$G‎‎$ ‎has ‎the ‎following ‎presentation‎:‎

$‎‎‎\langle a‎ , ‎b‎ ,‎c \vert a^p=b^p=c^q=1‎, ‎ab=ba‎, ‎cac^{-1}=a^{i}b^{j}‎, ‎cbc^{-1}=a^{k}b^{l}\rangle$‎‎

‎and‎ $‎ ‎{\left( {\begin{array}{*{20}c}‎ ‎i & j \\‎ ‎k & l ‎\end{array}}\right)}$ has order $q^2$ in $GL(2,p)$‎. ‎ Is it possible to classify such groups with ‎$‎‎O_q(G)=1$‎‎? (Recall that ‎$‎‎‎O_q(G)‎=\cap_{g\in G} Q^g$, ‎where ‎$‎Q\in {\rm ‎Syl}_q(G)‎$‎ ‎).‎ ‎ Many thanks for your thoughts on this!

There is an old paper "Classification of groups of order $p^2q^2$" by G.Cheissin where all such groups were described (unfortunately, in Russian). See http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=3913&option_lang=eng