Does every group of order $42$ have a normal subgroup of order $3$

Question

Does every group of order $42$ have a normal subgroup of order $3$?

I have both Sylow Theorems I can use. They imply that there are either $7$ or $1$ Sylow $3$-subgroups and the $7$ case is what makes me doubt whether the statement is true or false.

Can anyone please help me either prove or disprove this statement?

Answer 1

Hint. Group of order 42 $$ \begin{pmatrix} \mathbb{Z}_7^* & \mathbb{Z}_7\\ 0 & 1 \end{pmatrix} $$ has no normal subgroup of order $3$.

Check.