# What's the automorphism group of this covering?

What's the automorphism group of this covering?

I know why this is a covering, but I don't know how to find the automorphism group of this covering.

I need help, thanks

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Two automorphisms of a path-connected covering coincide iff they coincide at one point. 6 points lie above the node, thus the automorphism group can be identified as a subgroup of the permutation group on a 6 point set. Call the six points $\lbrace i_1,i_2,i_3,o_1,o_2,o_3\rbrace$ : the nodes labeled $i$ are those on the inner circle and one labeled $o$ lie on the outer circle. The lift of $b$ produces the permutation $(i_1i_2i_3)(o_1o_2o_3)$, while the lift of $a$ gives the permutation $(i_1o_1)(i_2o_2)(i_3o_3)$, so it seems that the group $G$ of the covering is isomorphic to the subgroup of $S_6$ generated by these two permutations. They commute, and so we should have $G\simeq\Bbb Z/6\Bbb Z$.