B. Fischer, in his work on finite groups which contain a conjugacy class of
3
3
-transpositions, discovered three new sporadic finite simple groups, usually denoted
M
(
22
)
M(22)
,
M
(
23
)
M(23)
and
M
(
24
)
′
M(24)’
. In Part I two of these groups,
M
(
22
)
M(22)
and
M
(
23
)
M(23)
, are characterized by the structure of the centralizer of a central involution. In addition, the simple groups
U
6
(
2
)
{U_6}(2)
(often denoted by
M
(
21
)
)
M(21))
and
P
Ω
(
7
,
3
)
P\Omega (7,3)
, both of which are closely connected with Fischer’s groups, are characterized by the same method. The largest of the three Fischer groups
M
(
24
)
M(24)
is not simple but contains a simple subgroup
M
(
24
)
′
M(24)’
of index two. In Part II we give a similar characterization by the centralizer of a central involution of
M
(
24
)
M(24)
and also a partial characterization of the simple group
M
(
24
)
′
M(24)’
. The purpose of Part III is to complete the characterization of
M
(
24
)
′
M(24)’
by showing that our abstract group
G
G
is isomorphic to
M
(
24
)
′
M(24)’
. We first prove that
G
G
contains a subgroup
X
≅
M
(
23
)
X \cong M(23)
and then we construct a graph (on the cosets of
X
X
) which is shown to be isomorphic to the graph for
M
(
24
)
M(24)
.