How is Conway numbering the hyperbolic groups in his Table 18.1?

Question

Conway lists many hyperbolic groups in his Table 18.1 on pages 239-240 of The Symmetries of Things. Here are scans of those pages: and The groups are sorted in decreasing order of their characteristic, from $0$ down to $-1/8$. I understand the orbifold-based names of the groups and I know how to compute their characteristics. But I don't understand how Conway is numbering them. His numbers, in the leftmost column, jump up in steps that don't seem to correspond to anything --- including a step of 39, in going from $\omega+63$ to $\omega+102$. How is Conway computing those numbers?

Answer 1

I finally realized: Conway is numbering the characteristics in numeric order; but Table 18.1, while it does list "all possibilities" in a sense, omits some lines without including ellipses to warn of those omissions. For example, there are three omitted lines between $\omega+4$ and $\omega+8$, where those lines, numbered $\omega+5$ through $\omega+7$, are for the groups $\ast 2\; 3\; 9$, $\ast 2\; 3\; 10$, and $\ast 2\; 3\; 11$, with second-column entries $36$, $30$, and $26\;\,2/5$.

A few lines down, the line numbered just "$\omega+\hbox{}$" should be numbered $\omega+13$, where the omitted lines $\omega+12$ and $\omega+14$ are for the groups $\ast 2\;3\;16$ and $\ast 2\;3\;17$, with second-column entries $19\;\,1/5$ and $18\;\,6/11$.