I was reading this document to answer my question. But after teaching me hell lot of jargon like subgroup, normal subgroup, cosets, factor group, direct sums, modules and all that the document says this,
You likely realize immediately that this is not a particularly easy thing to do by inspection. It turns out that there is a very straightforward and systematic way of taking a given representation and determining whether or not it is reducible, and if so, what the irreducible representations are. However, the details of how this can be done, while very interesting, are not necessary for the agenda of these notes. Therefore, for the sake of brevity, we will not pursue them.
I want to learn to do this by hand and then write a program. Please don't ask me to learn GAP or any other software instead.
How to find irreducible representation of a group from reducible one? What is that straightforward and systematic way?