Geodesic
The Mathieu group $M_{12}$
Matematičeskie zametki, Tome 15 (1974) no. 4, pp. 651-660

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Let $G$ be a finite simple non-Abelian group. $t$ is an involution of $G$, and $L=O^2(C_G(t)/O(C_G(t)))$. If the center $Z(L)$ is cyclic and $L/Z(L)\simeq PGL(2,q)$, $q$ odd, then either a Sylow 2-subgroup of $G$ is semidihedral or $C_G(t)\simeq Z_2\times PGL(2,5)$ and $G$ is isomorphic to the Mathieu group $M_{12}$ of degree 12. 
@article{MZM_1974_15_4_a16,
     author = {V. M. Sitnikov},
     title = {The {Mathieu} group $M_{12}$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {651--660},
     publisher = {mathdoc},
     volume = {15},
     number = {4},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1974_15_4_a16/}
}
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V. M. Sitnikov. The Mathieu group $M_{12}$. Matematičeskie zametki, Tome 15 (1974) no. 4, pp. 651-660. http://geodesic.mathdoc.fr/item/MZM_1974_15_4_a16/