The symmetric crosscap number of the families of groups \(DC_{3}\times C_{n}\) and \(A_{4}\times C_{n}\) (Q2901913)

Every finite group \(G\) acts faithfully as a group of automorphisms of some non-orientable Klein surface without boundary. The minimum genus of all such surfaces is called the \textit{symmetric cross-cap number} of \(G\), and denoted by \(\tilde{\sigma}(G)\). This parameter is known for several classes of finite groups and for all groups of order up to 127 (following work by Coy May, the authors of this paper, and the reviewer).NEWLINENEWLINESome recent attention has been paid to the inverse problem of determining which positive integers can occur as values of \(\tilde{\sigma}\). It was proved by \textit{C. L. May} [Glasg. Math. J. 43, No.3, 399-410 (2001; Zbl 0997.57034)] that there are groups with symmetric cross-cap number 1, 2 and any integer from 4 to 16 inclusive, but no finite group \(G\) with \(\tilde{\sigma}(G) = 3\). Subsequently the authors of this paper proved in [Proc. R. Soc. Edinb., Sect. A, Math. 138, No. 6, 1197-1213 (2008; Zbl 1165.30022)] that every integer \(g > 10\) such that \(g \equiv 0, 1\) or \(2\) mod \(4\) is the symmetric cross-cap number of \(C_m \times D_n\) for some \(m\) and \(n\), and that for every positive integer \(k\), the group \(C_{6k} \times C_3\) has symmetric cross-cap number \(12k-1\) (\(\equiv 11\) mod \(12\)). Also the reviewer has proved (in unpublished work) that for every positive integer \(k\), there is a semi-direct product \(C_n \rtimes S_4\) with symmetric cross-cap number \(3k-2\) (\(\equiv 1\) mod \(3\)). It follows that every positive integer not congruent to \(3\) mod \(12\) is now known to be \(\tilde{\sigma}(G)\) for some \(G\).NEWLINENEWLINEIn this paper, the authors determine the symmetric cross-cap numbers of the groups \(DC_3 \times C_n\) (where \(DC_3\) is the dicyclic group \(C_3 \rtimes_{-1} C_4\) of order \(12\)) and \(A_4 \times C_n\), for all \(n\). In doing so, they prove that these numbers cover a quarter of all positive integers congruent to \(3\) mod \(12\) (and three quarters of of all positive integers congruent to \(7\) mod \(12\)).NEWLINENEWLINENote: the reviewer has found (in computer-assisted work) that every integer \(g\) in the range \(3 < g \leq 100\) is \(\tilde{\sigma}(G)\) for some \(G\), and conjectures that \(3\) is the only positive integer that does not occur in the range of values of \(\tilde{\sigma}\).