If $G $ is a group of order $12$ not isomorphic to $A_4$ then does $G$ have an element of order $6$ ?

Question

If $G $ is a group of order $12$ not isomorphic to $A_4$ then does $G$ have an element of order $6$ ? ( By Cauchy's theorem I can show that there are elements of order $2$ and $3$ but cant proceed further , please help )

Answer 1

There are five isotopy classes of groups of order $12$. One is the alternating group $A_4$, whose elements are the identity (order $1$), three products of two disjoint transpositions (order $2$) and eight $3$-cycles (order $3$).

All of the remaining four have elements of order $6$:

• abelian group with $2$ elements of order $6$,
• abelian group with $6$ elements of order $6$,
• non-abelian group with $2$ elements of order $6$ (1),
• non-abelian group with $2$ elements of order $6$ (2).