Basic Examples (4) 

Construct Klein's 4-group:

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Its multiplication table:

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CyclicGroup[4] also has order 4 but differs from Klein's 4-group:

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Construct direct products of explicit permutation groups:

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Find its order:

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Easily deal with large groups:

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This is as expected:

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Compute direct products of large named groups:

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Scope (1) 

Construct direct product of more than 2 groups:

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Applications (4) 

Show that CyclicGroup[2] × CyclicGroup[2] × CyclicGroup[2] shares the same structure as that of AbelianGroup[{2,2,2}]:

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Index of the subgroup {e}×CyclicGroup[2] inside CyclicGroup[4]×CyclicGroup[2] yields the expected order of the quotient group (CyclicGroup[4]):

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Construct the symmetry group of a regular n-gonal prism:

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For n=6:

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Show that the symmetry group of a disjoint union of two separate complete graphs is the direct product of their respective symmetric groups:

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Verify that these are isomorphic to each other:

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Properties and Relations (3) 

Order of the direct product equals product of orders of the constituent groups:

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Direct product of a single group is nothing but that group itself:

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The direct product with the trivial group returns a structure isomorphic to that of the group itself:

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Possible Issues (2) 

For some groups, data may not be available in the Wolfram language:

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As such, it is not possible to compute direct products of these:

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Neat Examples (2) 

When gcd(m,n)=1 we have that CyclicGroup[mn] is isomorphic to CyclicGroup[m]×CyclicGroup[n]:

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However, otherwise this equivalence is not guaranteed:

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