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The examples $C_n, D_n, A_4, S_4, A_5$ are the first nice examples of groups to relate with 2-D and 3-D Euclidean geometry. These groups can be investigated by studying action on sphere as described in M. Artin's book on Algebra. Next, Fano plane is another nice object to relate the simple group of order 168 with (finite) geometry. Then, by considering the double cover $SU(2)$of $SO(3)$, we get few more finite groups (called double/binary groups). The good references to study these groups and geometry (as far as I saw) are

Geometries and Groups (Viacheslav, Shafarevich)

Groups: A path to Geometry (R. P. Burn)

Groups and Geometry (P.M. Neumann, etc.)

On Quaternions and Octonions (Conway, Smith).

Can one suggest more references (books, notes, articles) to see more finite groups with geometry?

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Iterated wreath products often appear in symmetry groups of rooted trees. For example, $D_4 \simeq \mathbb{Z}_2 \wr \mathbb{Z}_2$ is such a symmetry group. – Seirios Aug 22 '12 at 8:20
The Geometry of the Classical Groups by D.E. Taylor. – j.p. Aug 23 '12 at 6:38

You might enjoy "Finite Reflection Groups" by Benson and Grove.

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Non-euclidean tessellations and their groups by William Magnus. For example, the author shows that $T^*(\ell,m,n)=\langle a,b,c | (ab)^{\ell}=(bc)^m=(ac)^n=1 \rangle$ is the group of symmetries of a triangular tessellation on the plan if $\displaystyle \frac{1}{\ell}+ \frac{1}{m} + \frac{1}{n} =1$, on the hyperbolic plane if $\displaystyle \frac{1}{\ell}+ \frac{1}{m} + \frac{1}{n} <1$ or on the sphere if $\displaystyle \frac{1}{\ell}+ \frac{1}{m}+ \frac{1}{n}>1$. Thus, $T^*(\ell,m,n)$ is finite iff $\displaystyle \frac{1}{\ell}+ \frac{1}{m}+ \frac{1}{n}>1$!

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