I am trying to find the number of distinct coloring in the following problem:

Consider a 1*3 grid (shown as below) using 10 sticks and 8 balls. Color sticks with $m$ colors. How many ways are there to color the sticks with $m$ colors? The balls are indistinguishable, but the edges are distinct and numbered.

I try to use Burnside's lemma to count the distinct coloring, but I cannot find all the symmetric groups. Any help would be appreciated.

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    $\begingroup$ This looks to have only four automorphisms which are easily factored into cycles. $\endgroup$ – Marko Riedel Nov 12 '18 at 14:30
  • $\begingroup$ Thank you for your comment. I think the symmetries are identity, 180 rotations, reflection with respect to the horizontal axis of symmetry and vertical axis of symmetry. Am I right?@MarkoRiedel $\endgroup$ – Morteza Soltani Nov 12 '18 at 19:20
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    $\begingroup$ That would be correct. These can be factored from the table notation or more effectively, by inspection. $\endgroup$ – Marko Riedel Nov 12 '18 at 21:25
  • $\begingroup$ @MarkoRiedel Thank you for your response. I have found all the symmetries of this figure and have solved my problem. $\endgroup$ – Morteza Soltani Nov 13 '18 at 1:04

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