Geodesic
An algorithm for decomposition of finite group representations by means of invariant projectors
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIX, Tome 468 (2018), pp. 228-248

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We describe an algorithm for decomposition of permutation representations of finite groups over fields of characteristic zero into irreducible components. The algorithm is based on the fact that the components of the invariant inner product in invariant subspaces are operators of projection into these subspaces. This allows us to reduce the problem to solving systems of quadratic equations. The current implementation of the proposed algorithm allows us to split representationы of dimensions up to hundreds of thousands. Computational examples are given. 
@article{ZNSL_2018_468_a15,
     author = {V. V. Kornyak},
     title = {An algorithm for decomposition of finite group representations by means of invariant projectors},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {228--248},
     publisher = {mathdoc},
     volume = {468},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2018_468_a15/}
}
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V. V. Kornyak. An algorithm for decomposition of finite group representations by means of invariant projectors. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIX, Tome 468 (2018), pp. 228-248. http://geodesic.mathdoc.fr/item/ZNSL_2018_468_a15/