# C-groups of $\mathrm{PSL}(2,q)$ and $\mathrm{PGL}(2,q)$

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Connor Thomas

Universit\'e libre de Bruxelles

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Jambor Sebastian

University of Auckland

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Leemans Dimitri

University of Auckland

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**Minisymposium:**
POLYTOPES AND GRAPHS

**Content:**
C-groups are smooth quotients of Coxeter groups. It is well known that there is a one-to-one correspondence between string C-groups and abstract regular polytopes.
Recently, some results were obtained in the more general case of C-groups without any condition on the diagram.
I will present the classification of C-group representations of the groups $\mathrm{PSL}(2,q)$ and $\mathrm{PGL}(2,q)$. We obtain that the C-rank of $\mathrm{PSL}(2,q)$ and $\mathrm{PGL}(2,q)$ is $3$ except when $q \in \{7,9,11,19,31\}$ for $\mathrm{PSL}(2,q)$ and when $q=5$ for $\mathrm{PGL}(2,q)$, in which case it is $4$. We provide all representations of rank four.