Every finite group G acts as an automorphism group of several bordered Klein surfaces. The minimal genus of these surfaces is called the real genus \(\rho (G)\) of the group G. It is known that all odd positive integers are the real genus of some group. On the contrary, not all even integers are. C. L. May compiled a series of families of groups, from which he obtained arithmetic sequences of even numbers which are real genus of some group, covering a large part of the even numbers. In particular, it results that 2, 12 and 24 are not the real genus of any group. May asked on whether this is a question of small numbers, or else there are other gaps in the spectrum of the real genus, that is to say, numbers N such that there are no groups of real genus N. Recently, it has been proved that 72 is not the real genus of a group. The next two numbers on which the question remained unsolved are 84 and 108. In the present work we prove that there is no group of each real genus 84 and 108.
