Regular maps of order $2$-powers

In this paper, we consider the possible types of regular maps of order $2^n$, where the order of a regular map is the order of automorphism group of the map. For $n le 11$, M. Conder classified all regular maps of order $2^n$. It is easy to classify regular maps of order $2^n$ whose valency or covalency is $2$ or $2^{n-1}$. So we assume that $n geq 12$ and $2leq s,tleq n-2$ with $sleq t$ to consider regular maps of order $2^n$ with type ${2^s, 2^t}$. We show that for $s+tleq n$ or for $s+tu003en$ with $s=t$, there exists a regular map of order $2^n$ with type ${2^s, 2^t}$, and furthermore, we classify regular maps of order $2^n$ with types ${2^{n-2},2^{n-2}}$ and ${2^{n-3},2^{n-3}}$. We conjecture that, if $s+tu003en$ with $su003ct$, then there is no regular map of order $2^n$ with type ${2^s, 2^t}$, and we confirm the conjecture for $t=n-2$ and $n-3$.