Searching for groups

Type a group expression — a name, a cyclic group, or a combination built with operators. A query that names one group jumps straight to it; one that matches several lists them.

By name

Standard names resolve directly — case does not matter (d4 = D4):

Symmetric / alternating: S3, S4, A5
Dihedral / quaternion / dicyclic: D4, Q8, Dic3
Lie type: SL(2,3), GL(2,3), PSL(2,7)
ATLAS names, e.g. for extraspecial groups: 2^(1+2)_+, 3^(1+2)_-

Cyclic groups

Cn is the cyclic group of order n, for any n: C6, C12, C100.

By id

#556 is the group with that id — the number shown beside a group on its own page. This is how to write a group that has no name: #556.

Operators

Combine expressions — A and B below are themselves any expression, so an id may stand in for a group anywhere a name may (C2*#556, #556:C2):

Meaning	Write	Example → result
Direct product `A × B`	`A*B`	`C2*C3` → `C6`
`k` copies, `A × ⋯ × A`	`A^k`	`C2^3` → `C2×C2×C2`
Semidirect product `A ⋊ B`	`A:B` or `A⋊B`	`C3:C2` → `S3`
Non-split extension `A . B`	`A.B`	`C2.C2` → `C4`

A*B and A^k name a single group (the direct product).
A:B and A.B list every group that is an extension of A by B of that kind — often more than one, shown as a list to pick from.

Try these live: C2*C3, C2^3, C3:C2, C2.C4.

Combining expressions

Operators chain, and parentheses group. Binding order, tightest first:

^ — repeat
: ⋊ . — extension
* — direct product

So C3*C2^2:C4 reads as C3 × (C2² ⋊ C4). Parentheses override the order: (C3*C2):C4.

Notes

Case-insensitive, and spaces are ignored: c3 : c2 is the same as C3:C2.
An id is fixed only for as long as the catalog stands as it is; a name, where a group has one, is the durable way to refer to it.
An exact name always wins, so ATLAS names that contain the operators' own ^ / * symbols (like 2^(1+2)_+) resolve as written.
In any search box, matching groups appear in a dropdown as you type.

To search by what a group is rather than what it is called — its polycyclic presentation or permutation generators, or a set of properties such as order, number of conjugacy classes, and solvability — use Advanced Group Search.

See also: named series, and the worked examples on the group, abelian, and p-group pages.