Hint:
You should be able to convince yourself that none of the rotations apart from the identity could possibly when applied to an arrangement result in an identical arrangement.
Mirror symmetries where there are seven beads to either side of the axis of symmetry, i.e. the axis of symmetry passes through spaces between beads, also cannot result in an identical arrangement as there would necessarily be an odd number of blue beads on one side and an even on the other.
The mirror symmetry where beads lie along the axis of symmetry however can result in an identical arrangement.
Consider the following arrangement:
$$\begin{array}{cccccccc}~&\color{blue}{*}&*&*&*&\color{red}{*}&\color{red}{*}\\\color{blue}{*}&-&-&-&-&-&-&\color{red}{*}\\~&\color{blue}{*}&*&*&*&\color{red}{*}&\color{red}{*}\end{array}$$
(black is substituted in place of white)
The mirror symmetry when flipping along the horizontal axis will result in an identical arrangement. Similarly, there will be other arrangements which are symmetrical with regards to flipping. You should be able to convince yourself that each of these must have one of the beads along the axis of symmetry be blue and the other be red since the number of blue and red beads are both odd numbers.
I get an answer of $6036$ possible arrangements.