Sylow's Third Theorem: For each prime p dividing the order of a finite group G, the number of Sylow p-subgroups of G is congruent to 1 modulo p and divides |G|.
we use the following facts here.
1. By Sylow's theorem, number of 7 sylow subgroups must be 1 or 8(Prove!)
2. number of 7 sylow subgroups must be 8, can not be 1. (Use simplicity of the group)
The problem can also be improved as follows.
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