Originally Posted by

**dwsmith** In my course, we didn't have time to cover classifying groups but we are expect to know it. I need some help in how this is done.

$\displaystyle |G|=70=2\times 5\times 7$

Let $\displaystyle n_i$ be the number of $\displaystyle \text{Syl}_p(G)$ for $\displaystyle i=2,5,7$

$\displaystyle n_i\equiv 1 \ (\text{mod} \ i) \ \text{and} \ n_i|\frac{70}{i}$

$\displaystyle n_2=1,5,7,35$

$\displaystyle n_5=1$

$\displaystyle n_7=1$

So G is not simple and the Sylow 5 and 7 are normal in G. Now how do I classify the groups of order 70?