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A [[Finite group|finite group]] isomorphic to one of the five groups discovered by E. Mathieu . The series of Mathieu groups consists of the groups denoted by

$$ 
M _ {11} , M _ {12} , M _ {22} , M _ {23} , M _ {24} .
$$

They are representable as permutation groups (cf. [[Permutation group|Permutation group]]) on sets with 11, 12, 22, 23, and 24 elements, respectively. The groups  $  M _ {12} $
and  $  M _ {24} $
are five-fold transitive.  $  M _ {11} $
is realized naturally as the [[Stabilizer|stabilizer]] in  $  M _ {12} $
of an element of the set on which  $  M _ {12} $
acts; similarly,  $  M _ {23} $
and  $  M _ {22} $
are stabilizers of elements of  $  M _ {24} $
and  $  M _ {23} $, 
respectively. The Mathieu groups have the respective orders

$$ 
7  920 , 95  040 , 443  520 , 10  200  960 , 244  823  040.
$$

When considering a Mathieu group, one often uses (see ) its representation as the group of automorphisms of the corresponding [[Steiner system|Steiner system]]  $  S( l, m, n) $, 
i.e. of the set of  $  n $
elements in which there is distinguished a system of

$$ 
{\left ( \begin{array}{c}
m \\
 l 
\end{array}
 \right )  ^ {-1} } {\left ( \begin{array}{c}
n \\
 l 
\end{array}

\right ) }
$$

subsets, called blocks, consisting of  $  m $
elements of the set, and such that every set of  $  l $
elements is contained in one and only one block. An automorphism of a Steiner system is defined as a permutation of the set of its elements which takes blocks into blocks. The list of Mathieu groups and corresponding Steiner systems for which they are automorphism groups is as follows:  $  M _ {11} $— 
$  S( 4, 5, 11) $; 
$  M _ {12} $— 
$  S( 5, 6, 12) $; 
$  M _ {22} $— 
$  S( 3, 6, 22) $; 
$  M _ {23} $— 
$  S( 4, 7, 23) $; 
$  M _ {24} $— 
$  S( 5, 8, 24) $.

The Mathieu groups were the first (and for over 80 years the only) known sporadic finite simple groups (cf. also [[Sporadic simple group|Sporadic simple group]]).

====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  E. Mathieu,   "Mémoire sur l'étude des fonctions de plusieures quantités, sur la manière de les formes et sur les substitutions qui les laissant invariables"  ''J. Math. Pures Appl.'' , '''6'''  (1861)  pp. 241–323</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  E. Mathieu,   "Sur la fonction cinq fois transitive des 24 quantités"  ''J. Math. Pures Appl.'' , '''18'''  (1873)  pp. 25–46</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  E. Witt,   "Die $5$-fach transitiven Gruppen von Matthieu"  ''Abh. Math. Sem. Univ. Hamburg'' , '''12'''  (1938)  pp. 256–264</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  E. Witt,   "Ueber Steinersche Systeme"  ''Abh. Math. Sem. Univ. Hamburg'' , '''12'''  (1938)  pp. 265–275</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.D. Mazurov,   "Finite groups"  ''Itogi Nauk. i Tekhn. Algebra. Topol. Geom.'' , '''14'''  (1976)  pp. 5–56  (In Russian)</TD></TR></table>

====Comments====
For more information (e.g. character tables and maximal subgroups) see [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.H. Conway,   R.T. Curtis,   S.P. Norton,   R.A. Parker,   R.A. Wilson,   "Atlas of finite groups" , Clarendon Press  (1985)</TD></TR></table>