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I'm implementing a class of groups into Sage (, a computer algebra system, and I'm wondering if it has any official names. I found it in Gorenstein's "Finite Groups." It is there called $M_m(p)$, and has the presentation $\langle x, y \shortmid x^{p^{m-1}} = y^p = 1, x^y = x^{1+p^{m-2}} \rangle$. The family is notable because they are the only nonabelian p-groups (for odd p) that have maximal cyclic subgroups. I'm looking for any common names for this family of groups, perhaps even what Gorenstein meant by his $M_m(p)$ notation.

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Well, for one thing they are split metacyclic groups... – Arturo Magidin Jul 19 '12 at 3:00
I believe Tim Dokchitser decided to call them OMC (other maximal cyclic), though this might be because of something I wrote which was only a convenience to me. I assumed M stands for maximal. I suggest that it does not stand for modular. See – Jack Schmidt Jul 19 '12 at 16:17

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