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Struct/The symmetric group of order six
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Title: The group with three elements.
There is a unique nonabelian group, up to isomorphism, with six elements. One example is the group $S_3$ of permutations of the three-element set $\{ 1,2,3 \}$.
| $\circ$ | $Id$ | $(12)(3)$ | $(13)(2)$ | $(1)(23)$ | $(123)$ | $(132)$ |
|---|---|---|---|---|---|---|
| $Id$ | $Id$ | $(12)(3)$ | $(13)(2)$ | $(1)(23)$ | $(123)$ | $(132)$ |
| $(12)(3)$ | $(12)(3)$ | $Id$ | $(132)$ | $(123)$ | $(1)(23)$ | $(13)(2)$ |
| $(13)(2)$ | $(13)(2)$ | $(123)$ | $Id$ | $(132)$ | $(12)(3)$ | $(1)(23)$ |
| $(1)(23)$ | $(1)(23)$ | $(132)$ | $(123)$ | $Id$ | $(13)(2)$ | $(12)(3)$ |
| $(123)$ | $(123)$ | $(13)(2)$ | $(1)(23)$ | $(12)(3)$ | $(132)$ | $Id$ |
| $(132)$ | $(132)$ | $(1)(23)$ | $(12)(3)$ | $(13)(2)$ | $Id$ | $(123)$ |
