<div class="thumb tright"><div class="thumbinner" style="width:222px;"><a href="//commons.wikimedia.org/wiki/File:Symmetric_group_S4;_Cayley_table_submatrix_with_all_elements_(right).svg" class="image"><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/5/55/Symmetric_group_S4%3B_Cayley_table_submatrix_with_all_elements_%28right%29.svg/220px-Symmetric_group_S4%3B_Cayley_table_submatrix_with_all_elements_%28right%29.svg.png" decoding="async" width="220" height="300" class="thumbimage" srcset="https://upload.wikimedia.org/wikipedia/commons/thumb/5/55/Symmetric_group_S4%3B_Cayley_table_submatrix_with_all_elements_%28right%29.svg/330px-Symmetric_group_S4%3B_Cayley_table_submatrix_with_all_elements_%28right%29.svg.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/5/55/Symmetric_group_S4%3B_Cayley_table_submatrix_with_all_elements_%28right%29.svg/440px-Symmetric_group_S4%3B_Cayley_table_submatrix_with_all_elements_%28right%29.svg.png 2x" data-file-width="195" data-file-height="266"></a>  <div class="thumbcaption"><div class="magnify"><a href="//commons.wikimedia.org/wiki/File:Symmetric_group_S4;_Cayley_table_submatrix_with_all_elements_(right).svg" class="internal" title="Enlarge"></a></div>A submatrix containing all 24 elements</div></div></div>
<p><a href="https://en.wikipedia.org/wiki/Cayley_table" class="extiw" title="w:Cayley table">Cayley table</a> of the <a href="https://en.wikipedia.org/wiki/Symmetric_group" class="extiw" title="w:Symmetric group">symmetric group</a> <a href="https://en.wikiversity.org/wiki/Symmetric_group_S4" class="extiw" title="v:Symmetric group S4">S<sub>4</sub></a> showing all 24 permutations of 4 elements
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<table class="wikitable" style="text-align: center;"><tbody><tr><td><div style="opacity: 50%;">
<table><tbody><tr><td style="width:80px;"><a href="//commons.wikimedia.org/wiki/File:Perm_2_3_1_left.svg" class="image"><img alt="Perm 2 3 1 left.svg" src="https://upload.wikimedia.org/wikipedia/commons/thumb/7/72/Perm_2_3_1_left.svg/50px-Perm_2_3_1_left.svg.png" decoding="async" width="50" height="67" srcset="https://upload.wikimedia.org/wikipedia/commons/thumb/7/72/Perm_2_3_1_left.svg/75px-Perm_2_3_1_left.svg.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/7/72/Perm_2_3_1_left.svg/100px-Perm_2_3_1_left.svg.png 2x" data-file-width="223" data-file-height="297"></a>
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<td>There is also:<br><a href="https://en.wikipedia.org/wiki/Group_action#Left_group_action" class="extiw" title="w:Group action">left action</a><br><abbr title="prefix notation (the usual way functions are composed)">last ⋅ first</abbr>
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<table><tbody><tr><td>This is:<br><b><a href="https://en.wikipedia.org/wiki/Group_action#Right_group_action" class="extiw" title="w:Group action">right action</a></b><br><abbr title="postfix notation (sometimes used in group theory)">first ⋅ last</abbr>
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<td style="width:80px;"><a href="//commons.wikimedia.org/wiki/File:Perm_2_3_1_right.svg" class="image"><img alt="Perm 2 3 1 right.svg" src="https://upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Perm_2_3_1_right.svg/67px-Perm_2_3_1_right.svg.png" decoding="async" width="67" height="50" srcset="https://upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Perm_2_3_1_right.svg/101px-Perm_2_3_1_right.svg.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Perm_2_3_1_right.svg/134px-Perm_2_3_1_right.svg.png 2x" data-file-width="297" data-file-height="223"></a>
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</td></tr></tbody></table><p><small>For <b>left action</b> (the usual <a href="https://en.wikipedia.org/wiki/function_composition" class="extiw" title="w:function composition">way to compose functions</a>) see: <a href="//commons.wikimedia.org/wiki/File:Symmetric_group_4;_Cayley_table_(left);_numbers.svg" title="File:Symmetric group 4; Cayley table (left); numbers.svg">File:Symmetric group 4; Cayley table (left); numbers.svg</a></small>
</p><p><a href="https://en.wikipedia.org/wiki/Parity_of_a_permutation" class="extiw" title="w:Parity of a permutation">Even permutations</a> are white:
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<ul><li>the identity</li>
<li>eight <a href="https://en.wikipedia.org/wiki/Cycles_and_fixed_points" class="extiw" title="w:Cycles and fixed points">3-cycles</a></li>
<li>three double-<a href="https://en.wikipedia.org/wiki/Transposition_(mathematics)" class="extiw" title="w:Transposition (mathematics)">transpositions</a></li></ul><p>Odd permutations are colored:
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<ul><li>six transpositions (green)</li>
<li>six 4-cycles (orange)</li></ul><p>Calculation of the matrix with Python: <a rel="nofollow" class="external free" href="http://pastebin.com/BGkvfdgH">http://pastebin.com/BGkvfdgH</a>
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