Order of a subgroup generated by two elements

Question

If $G$ is a group, $H\leqslant G$ and $H=\langle f,g\rangle$, and I know $$|f|=7, |g|=3, |fg|=3, |gf|=3, |G|=7!,$$ is this enough information to determine $|H|$?

If so, can someone give me a hint for how to do this?

Answer 1

No. $21$, $168$, $504$, and $2520$ are all possibilities for $|H|$. Here are examples. The first three are subgroups of $G=S_7$.

1. $f=(1,2,3,4,5,6,7)$, $g=(2,3,5)(4,7,6)$, $|H|=21$.

2. $f=(1,2,3,4,5,6,7)$, $g=(1,6,2)(3,4,7)$, $H \cong L_2(7)$, $|H|=168$.

3. $f=(1,2,3,4,5,6,7)$, $g=(1,7,4)$, $H =A_7$, $|H|=2520$.

A fourth example, with $H$ a subgroup of $A_9$, is

4. $f = (1, 4, 6, 7, 9, 3, 2)$, $g=(1, 9, 8)(2, 5, 6)(3, 4, 7)$, $H \cong L_2(8)$, $|H|=504$, and we can take $G = H \times C_{10}$.