The group has six conjugacy classes. They almost correspond to the cycle type,
but there are two different conjugacy classes with 7-cycles.
(For the distinction between them, see here.)
In a symmetric representation like the Fano plane, there are 33 different cycle shapes (or 24 if the direction is ignored).
A complete list can be found here.
They are denoted by city names, followed by a qualifier of the direction, where needed.
$A$ and $B$ have the same cycle shape, if there is a $P$ from the symmetric subgroup, so that $B=P^{-1}AP$.
(The 3×3 matrix of $P$ is a permutation matrix.)
This is a refinement of matrix similarity, where $P$ is allowed to be any element of the group.
(Cycle shapes are a refinement of conjugacy classes.)
The cycle graph of this group has 28 triangles, 21 squares and 8 heptagons.
Each of the following rows is an example of a cycle.
(Each one is closed by the neutral element, which is not shown.)
It shows consecutive powers of the first element from left to right.
(Also of the last element from right to left.)
Elements in symmetric positions are inverse to each other.