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I have read some material on Coset Enumeration. Unfortunately I could not follow the steps in Todd-Coexeter Algorithm, and also in Handbook of Computational Group Theory by Derek Holt. The problem is how we scan an element of subgroup and deduce the results by coset tables. How we make cost tables.
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$\begingroup$The last table tells us that if we set $H=\langle x\rangle$ of order $2$, then $[G:H]=3$. So $|G|=6$. But $$1(xy)=(1x)y=1y=2\neq3=2x=(1y)x=1(yx)$$ so the group is not abelian an therefore $G$ could be $S_3$.
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