The rotation $g$ has order six. The possible <cycle types> in the <alternating group> $A_4$ are the identity, a three-cycle, and a product of two disjoint transpositions; their orders are one, three and two. Since a <group isomorphism> preserves the <order of a group element>, \b[$D_{12}$ is not isomorphic to $A_4$].

Solution

Past exam of the mathematics course of the University of Cambridge / 2004 / ia / Paper 3 / 5D / iv / Solution

 Solution (source code)