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Below is a character table created using the MAGMA computer algebra system.

What is the meaning of the rows starting with $p=2$, $p=3$, etc.? I've tried looking in the documentation, but couldn't find any reference to this.

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I think these are the power maps described in the GAP manual here:

http://www.math.niu.edu/help/math/gap4/ref/CHAP069.htm#SECT001

If I'm right line $p$, column $i$ of that section describes which conjugacy class $p$th powers of elements of class $i$ belong to. This seems to fit with the example you give.