** Authors:**
RECEP ŞAHİN

** Abstract: **
Let $q\geq 3$ be a prime number and let $\overline{H}(\lambda _{q})$ be the
extended Hecke group associated with $q.$ In this paper, we determine the
presentation of the commutator subgroup ($H$($\lambda _{q})\alpha )^{\prime
} $ of the normal subgroup $H$($\lambda _{q})\alpha $, where $H$($\lambda
_{q})\alpha $ is a subgroup of index $2$ in $\overline{H}$($\lambda _{q}).$
Next we discuss the commutator subgroup ($\overline{H}_{2})^{\prime }$($%
\lambda _{q})$ of the principal congruence subgroup $\overline{H}_{2}$($%
\lambda _{q})$ of $\overline{H}$($\lambda _{q})$. Then we show that some
quotient groups of $\overline{H}$($\lambda _{q})$ are generalized $M^{\ast
}- $groups. Finally, we prove some results related to some normal subgroups
of $\overline{H}$($\lambda _{q})$, especially in the case $q=5.$

** Keywords: **
Extended Hecke groups, commutator subgroups,
principal congruence subgroups, generalized $M^{\ast }-$groups

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