  Let $H$ be a subgroup of $\text{Sym}_n$, the symmetric group of degree $n$.
For a fixed integer $l \geq 2$, the group $G$ presented with generators $x_1,
x_2, \ldots ,x_n$ and with relations $x_{i_1}x_{i_2}\cdots x_{i_l} =x_{\sigma
(i_1)} x_{\sigma (i_2)} \cdots x_{\sigma (i_l)}$, where $\sigma$ runs through
$H$, is considered. It is shown that $G$ has a free subgroup of finite index.
For a field $K$, properties of the algebra $K[G]$ are derived. In particular,
the Jacobson radical $\mathcal{J}(K[G])$ is always nilpotent, and in many cases
the algebra $K[G]$ is semiprimitive. Results on the growth and the
Gelfand-Kirillov dimension of $K[G]$ are given. Further properties of the
semigroup $S$ and the semigroup algebra $K[S]$ with the same presentation are
obtained, in case $S$ is cancellative. The Jacobson radical is nilpotent in
this case as well, and sufficient conditions for the algebra to be
semiprimitive are given.
