Wythoff polytopes and low-dimensional homology of Mathieu groups

We describe two methods for computing the low-dimensional integral homology of the Mathieu simple groups and use them to make computations such as $H_5(M_{23},\text{ZZ})={\text{ZZ}}_7$ and $H_3(M_{24},\text{ZZ})={\text{ZZ}}_{12}$. One method works via Sylow subgroups. The other method uses a Wythoff polytope and perturbation techniques to produce an explicit free $\text{ZZ}{M}_n$-resolution. Both methods apply in principle to arbitrary finite groups.