= Monster group
{c}
{title2=$\mathbb M$}
The <Monster group> $\mathbb M$ is the largest of the twenty-six <sporadic simple groups>. Its order is
$$
|\mathbb M|=2^{46}3^{20}5^97^6 11^2 13^3 17\cdot19\cdot23\cdot29\cdot31\cdot41\cdot47\cdot59\cdot71.
$$
It is a <finite nonabelian simple group>: it has no nontrivial proper <normal subgroups>. Size alone does not specify its multiplication; its structure is studied through representations and subgroups.<ATLAS of Finite Group Representations>[Monster group page, order and standard generators: https://brauer.maths.qmul.ac.uk/Atlas/spor/M/]{ref}
The ATLAS specifies generators $a,b$ in conjugacy classes $2A,3B$, with $ab$ of order $29$. The letters distinguish different <conjugacy classes> having the same element order; the relations $a^2=b^3=(ab)^{29}=1$ alone are not a presentation of the <Monster group>. This illustrates how computational data about enormous <groups> can still start from two carefully chosen elements.
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