  Given a map with underlying graph $\mathcal{G}$, if the set of prime divisors
of $|V(\mathcal{G}|$ is denoted by $\pi$, then we call the map a {\it
$\pi$-map}.
  An orientably-regular (resp. A regular ) $\pi$-map is called {\it solvable}
if the group $G^+$ of all orientation-preserving automorphisms (resp. the group
$G$ of automorphisms) is solvable; and called {\it normal} if $G^+$ (resp. $G$)
contains a normal $\pi$-Hall subgroup.
  In this paper, it will be proved that orientably-regular $\pi$-maps are
solvable and normal if $2\notin \pi$ and regular $\pi$-maps are solvable if
$2\notin \pi$ and $G$ has no sections isomorphic to ${\rm PSL}(2,q)$ for some
prime power $q$. In particular, it's shown that a regular $\pi$-map with
$2\notin \pi$ is normal if and only if $G/O_{2^{'}}(G)$ is isomorphic to a
Sylow $2$-group of $G$.
  Moreover, nonnormal $\pi$-maps will be characterized and some properties and
constructions of normal $\pi$-maps will be given in respective sections.

Combinatorics