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Given the group $G= \langle (1,3,6,7,4,2,5),(1,2,3)(4,5,7) \rangle$.
How do i calculate the order of the group, the subgroups and normal subgroups?
The order is the number of elements. We have the elements 1-7. But is it that easy that $|G|=7$?
I hope you can help me.

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The conjugate $(6742513)^{(123)(457)} = (6742513)^2$. Hence the group is $\langle (6742513)\rangle:\langle (123)(457)\rangle\cong\mathbb{Z}_7:\mathbb{Z}_3$, a semi-direct product.

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