The centralizer C(g) of g = (12) in S_6 is C(g) = Z/2Z x S_4. The centralizer C(h) of h = (12)(34)(56) in S_6 also has order 48. Are the groups C(g) and C(h) isomorphic as abstract groups? (Equivalently, is there an isomorphism C(h) = Z/2Z x S_4?)
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