  In the first section of this senior thesis the author provides some new
efficient algorithms for calculating with finite permutation groups. They
cannot be found in the computer algebra system MAGMA, but they can be
implemented there. For any finite group G with a given set of generators, the
algorithms calculate generators of a fixed subgroup of G as short words in
terms of original generators. Another new algorithm provides such a short word
for a given element of G. In the later sections, the author gives a
self-contained existence proof for Fischer's sporadic simple group Fi_{23}
using G. Michler's Algorithm [11] constructing finite simple groups from
irreducible subgroups of GL_n(2). This sporadic group was originally discovered
by B. Fischer in [6] by investigating 3-transposition groups, see also [5].
This thesis gives a representation theoretic and algorithmic existence proof
for his group. The author constructs the three non-isomorphic extenstions E_i
by the two 11-dimensional non-isomorphic simple modules of the Mathieu group
M_{23} over F=GF(2). In two cases Michler's Algorithm fails. In the third case
the author constructs the centralizer H=C_G(z) of a 2-central involution z of
E_i in any target simple group G. This allows the author to construct G inside
GL_{782}(17). Its four generating matrices, character table and representatives
for conjugacy classes are computed. It follows that G and Fi_{23} have the same
character table.
