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The exceptional group $G_4$ or $\langle 3,3,2 \rangle$, abstractly presented as:
$$
\langle R,S \ |\ R^3=S^3=(RS)^2 \rangle \ .
$$
It is finite of order 24.  It has the [[alternating group]] $A_4$ as quotient by the centre  and the [[quaternion group]] of order 8 as a quotient.

This group may be realised as the group of invertible [[Hurwitz number]]s:
$$
\pm 1\,,\ \pm i\,,\ \pm j\,,\ \pm k\,,\ \frac{\pm1\pm i\pm j\pm k}{2} \ .
$$

The group has an action on the three-sphere with [[tetrahedral space]] as quotient.

==References==
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  H.S.M. Coxeter,   "Regular complex polytopes" , Cambridge Univ. Press  (1991)  pp. 76 ISBN 0-521-20125-X  {{ZBL|0732.51002}}</TD></TR>
</table>