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The Fano plane duality group includes both incidence-preserving permutations of points and projective lines and the dualities that exchange those two types. Its type-preserving subgroup is $G{L}_3({\mathbb{F}}_2)$, of order $(8-1)(8-2)(8-4)=168$. A polarity supplies the inverse-transpose automorphism $A\mapsto (A^{-1}{)}^{\mathsf{T}}$, giving the semidirect productThere are $7\cdot 3=21$ unordered incident point-line pairs and $7\cdot 4=28$ unordered nonincident point-line pairs. The two transitive actions therefore have stabilizers of orders $336/21=16$ and $336/28=12$, respectively. These are dihedral groups. The normalizer of a Sylow subgroup of order seven has order42 and is $C_7⋊C_6$.

Every maximal proper subgroup is a conjugate subgroup of one of these three types or of the index-two collineation subgroup. The extension is also isomorphic to $PG{L}_2(7)$, identifying the geometric stabilizers with the standard finite-group classification.^[ref]

The Monster group $\mathbb{M}$ is the largest of the twenty-six sporadic simple groups. Its order isIt 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.^[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.