Do these two permutations generate $A_n$?

Question

Let $n$ be odd and not a multiple of $3$. Do the cycle $\sigma:=(1, 2, \dots, n)$ and any cycle of length $3$ generate $A_n$?

Answer 1

$\newcommand{\Set}[1]{\left\{ #1 \right\}}$$\newcommand{\Span}[1]{\left\langle #1 \right\rangle}$No.

$$ \Span{(1,2,3,4,5,6,\dots,35), (1,6,11)} $$ is not the alternating group on $35$ letters.

The reason is that it is imprimitive. A block is $$ \Set{1,6,11,16,21,26,31} $$