Question: If you have a finite group, how do you name it?

If, for whatever reason, you have to list all subgroups of $GL_2({\mathbb F}_5)$ up to isomorphism in a paper, you are likely to write something along the lines of

$$ C_1, C_2, C_2, C_3, C_{2,2}, C_4, C_5, C_6, S_3, Q_8, C_8, C_{2,4}, D_4, $$ $$ C_{10}, D_5, D_6, C_{12}, C_3\rtimes C_4, C_{2,4}\rtimes C_2, OMC_{16}, C_{4,4}, $$ $$ C_{20}, D_{10}, G_{20}, C_5\rtimes C_4, SL_2(F_3), C_4\times S_3, C_3\rtimes C_8, C_{24}, $$ $$ Q_8\rtimes C_4, C_2\times G_{20}, C_2\times G_{20}, C_4\times D_5, (C_{2,4}\rtimes C_2)\rtimes C_3, C_3\rtimes OMC_{16}, $$ $$ C_4\times G_{20}, C_2.A_5, SL_2(F_3)\rtimes C_4, (C_2.A_5)\rtimes C_2, GL_2(F_5). $$

Computer algebra packages tend to produce a human-unfriendly output of
generators and relations or generating permutations in $S_n$. How do
**you** convert from one to the other and decide how to name complicated groups?
I am looking for standard names, standard constructions, conventions and
notations. For me a good notation is informative,
human friendly, short and is generally as close as possible to what you would
use in a paper. I am also looking for any kind of canonical conventions: e.g. $(C_5\times C_5)\rtimes C_4$ or $(C_5\rtimes C_4)\times C_5$?

(The reason I am asking is that I seem to have to work with funny groups all the time recently. I have a Magma function for personal use that analyzes and names finite groups; e.g. it produces the list above for $GL_2({\mathbb F}_5)$, and I personally find this really useful.

Currently it knows various standard groups: cyclic, abelian, dihedral, alternating, symmetric, special $p$-groups (semi-dihedral, generalized quaternion, "other maximal cyclic", Heisenberg), simple groups, linear groups (SL, GL, O, SP) and eventually their projective versions; it tries to recognize direct, semidirect (and eventually wreath) products if the group is not too large, and reverts to chief series if everything else fails.

Recently sufficiently many people asked me to share the code that I'll make it public domain. But before that I'd very much like to get suggestions from the MO community how to make it as useful as possible for most people.)

**Edit (6 years later)**: The names are finally in public domain (groupnames.org), and comments and suggestions are still very much welcome.

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