Symmetries of the square

As symmetries of the square we find rotations over 0° 90°, 180°, 270°, reflections in lines through the middle of two opposite edges, as well reflections in the diagonals. Together they form the 8 elements of the group D₄. Below you find three of these symmetries.

The symmetries of the square are visualized in the logo of idA. You can use the above symmetries to obtain the 8 different instances of the logo. Just click on a symmetry and it will act on the logo. (To restart, just reload the page.) Use this to answer the questions below.

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• Can you obtain all 8 instances with just the rotation? What is the order of the subgroup generated by the rotation? What is the order of the rotation?
• What is the order of a reflection?
• Can you obtain all different instances with just the two reflections. Do the two reflection generate the whole group?
• Are the two reflections conjugate?