In general, groups are referred to by standard name:

That is equivalent to this simpler syntax:

Other acceptable forms of input are called alternate standard names:

This is the English name for a group, always a string:

Some groups have alternate names:

There is also a short string form of the name:

Every group has a short notation form:

The default notation for the 32 three-dimensional crystallographic point groups is Schoenflies notation, but other notations are possible:

Those are only applicable to the crystallographic groups:

Two important properties of a group are its order and multiplication table:

The multiplication table defines the group as an abstract group (that is, up to isomorphism). For example, the group is Abelian if and only if its multiplication table is symmetric:

The elements of the group are abstractly identified by their position in the multiplication table:

These are their inverses:

This allows identification of which elements form the various subgroups. For example, the fourth element commutes with all other group elements:

More information on the commutativity of a group is given by the group of all commutators in the group. This group is not Abelian because the commutator (or derived) subgroup is not trivial:

However, upon further commutation we get the trivial group:

Hence, the original group is solvable:

These are the cycles obtained by repeated multiplication of each element with itself:

They have a natural representation as a graph. The identity element is highlighted:

Presentations in terms of generators and relations:

The group product is represented as a small circle, also used to denote its associated power:

With those two generators (of respective orders 4 and 2) we have this Cayley graph:

Naming the generators, it is possible to give all elements of the groups as words:

Permutation group representations as the complete set of permutation lists:

Or as a permutation group that can be used for further computation:

It is a transitive representation, because it has a single orbit:

However, any stabilizer is trivial:

Hence, this is not a multiply-transitive representation:

The cyclic structure of a permutation representation gives the so-called cycle index polynomial:

This is a matrix representation of a cyclic group:

Obtain groups isomorphic to a given one:

They all share the same properties as abstract groups:

Conjugacy classes of a group, given as respective lists of elements:

The class number is defined as the number of conjugacy classes:

Those elements that are self-conjugate, hence commuting with all other elements in the group, form the center of that group:

The Klein 4-group only has one type of proper subgroup:

But that type is realized three times:

For n≥5 the alternating groups are equal to their commutator subgroups:

Groups with such property are called perfect:

All simple groups are perfect:

Normal subgroups of the "Tetrahedral" group:

It has nontrivial normal subgroups, and therefore it is not simple:

Sylow subgroups are those whose order is a maximal power of a prime:

Possible quotient groups, obtained by finding the quotient with respect to normal subgroups:

Schur multiplier group:

Automorphism groups:

These are various properties only applicable to the crystallographic groups: