Proving Solvability of Group Order 12p

[fireisland27]

Homework Statement

Show that a group of order 12p is solvable for any prime p greater than 11

Homework Equations

I'm not very good about solvability questions so if anybody has any good ideas I'd be interested to hear them.

The Attempt at a Solution

I know that that every group of order 12 is either isomorphic to A₄ or has an element of order 6, but I'm not really sure how to use this.

 

[rasmhop]

Call the group in question G. You know that G has a subgroup H of order p.

If you can show that H is a normal subgroup of G, then you know that G is solvable if and only if H and G/H are solvable. It's considerably simpler to show that H and G/H are solvable (hint: H has prime order and G/H has order 12).