# Smallest finite group larger than the Quaternion group that isn't an outer Semi Direct Product

Define the outer-semi direct product in the standard way

https://en.wikipedia.org/wiki/Semidirect_product#Outer_semidirect_products

I found out that $Q_8$ is the smallest finite group that is neither finite simple, nor can be expressed as an outer-semi-direct product of other groups.

What is the next smallest finite group (/set of smallest finite groups if they share the same order) that cannot be expressed an outer-semi-direct product?

(i'm not looking for a proof, just the names of these group(s))

• What about a cyclic group of order $4$? – Eric Wofsey Feb 19 '17 at 6:15
• ah i'm silly, and thanks! that works – frogeyedpeas Feb 19 '17 at 6:28

It's not true that $Q_8$ is the smallest such group: the smallest such group is a cyclic group of order $4$. The next smallest such group after $Q_8$ is a cyclic group of order $9$.