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There's a couple things I don't quite understand but mainly, I don't see how he knows the representations for the factor groups are 1 and -1. Infact, I don't really understand where he gets the second factor group H={a3,a4,a5} from. Is it a distinct coset from H={e,a1,a2}?

Given that this is true, he then puts a 1 in the character table under the {a1,a2} conjugacy class - is this conjugacy class related to the H={e,a1,a2} subgroup?

I find this whole subject quite a challenge so really any help you can give will be greatly appreciated. Sorry again for picture, and thanks for helping.