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Now I'd like to show that the order of C' divides C. I know from the class equation that |C| divides |G|, that |C'| divides |G'|, and that due to surjectivity, |G'| divides |G|, so |C| divides |G|. I can't see how to piece these facts together to show that |C'| divides |C| though, or how to use what I proved in the first part, so there's definitely something I'm missing.

Any help is appreciated!