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.
Best Answer
It is difficult to come up with a consistent notation for all groups of a certain order since their construction is somewhat chaotic. We might be able to describe all the groups of order $p^3$ or $p^4$ but what about all groups of order $p^6$? Or order $p^4q^2$?
The software package GAP (http://www.gap-system.org/) has a catalogue of all groups of order up to 2000 or so and so I've sometimes referred to groups by their catalogue number, for example, SmallGroup(96, 33) refers to a particular group in that library. (As does SmallGroup(512, 1000000)!)