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Let  G  be a (2,  m, n )‐group and let  x  be the number of distinct primes dividing χ, the Euler characteristic of  G . We prove first that, apart from a finite number of known exceptions, a non‐abelian simple composition factor  T  of  G  is a finite group of Lie type with rank  n  ⩽  x . This result is proved using new results connecting the prime graph of  T  to the integer  x .  We then study the particular cases  x  = 1 and  x  = 2. We give a general structure statement for (2,  m, n )‐groups which have Euler characteristic a prime power, and we construct an infinite family of these objects. We also give a complete classification of those (2,  m, n )‐groups that are almost simple and for which the Euler characteristic is a prime power (there are four such).  Finally, we announce a result pertaining to those (2,  m, n )‐groups that are almost simple and for which |χ| is a product of two prime powers. All such groups which are not isomorphic to PSL 2 ( q ) or PGL 2 ( q ) are completely classified.

Orientably regular maps with Euler characteristic divisible by few primes