Find a nontrivial presentation of this group of permutations: the identity with $(1234), (13)(24), (4321), (13), (24), (12)(34),$ and $(14)(23)$.

Question

The Question:

Find a nontrivial presentation of the permutation group $G$ consisting of the identity with $(1234), (13)(24), (4321), (13), (24), (12)(34),$ and $(14)(23)$.

My Attempt:

Under the mapping $a\mapsto (13), b\mapsto (24), c\mapsto (1234)$, I have $$G=\langle a, b, c\mid a^2, b^2, (ab)^2, c^4, (ac)^2\rangle,$$ but I don't think that quite captures it all; is it right?

Please help :)