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A sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. All the other finite simple groups form 18 infinite families numbered by q - power of prime number and n - natural number. Sporadic groups attach attention due to their sporadic/exceptional nature - similar to exceptional Lie groups. The first sporadic groups were found by Mathieu in 1860s. The last sporadic group J4 was discovered in 1975 by Janko.
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Why are the sporadic simple groups HUGE?
The primes dividing the order of the Monster group are precisely the primes $p$ so that the surface $\mathbb H^2/\Gamma_0(p)^*$ has genus $0$, as was observed by Ogg. See Monstrous Moonshine, which w …