# Mathieu group M23

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vte :
In the area of modern algebra known as group theory, the Mathieu group M$_{23}$ is a sporadic simple group of order
    10,200,960 = 2$^7$ · 3$^2$ · 5 · 7 · 11 · 23
    ≈ 1 × 10$^7$.

## History and properties

M$_{23}$ is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a 4-fold transitive permutation group on 23 objects. The Schur multiplier and the outer automorphism group are both trivial.
Milgram (2000) calculated the integral cohomology, and showed in particular that M$_{23}$ has the unusual property that the first 4 integral homology groups all vanish.
The inverse Galois problem seems to be unsolved for M$_{23}$. In other words, no polynomial in Z[x] seems to be known to have M$_{23}$ as its Galois group. The inverse Galois problem is solved for all other sporadic simple groups.

### Construction using finite fields

Let F$_{211}$ be the finite field with 2$^{11}$ elements. Its group of units has order 2$^{11}$ − 1 = 2047 = 23 · 89, so it has a cyclic subgroup C of order 23.
The Mathieu group M$_{23}$ can be identified with the group of F$_2$-linear automorphisms of F$_{211}$ that stabilize C. More precisely, the action of this automorphism group on C can be identified with the 4-fold transitive action of M$_{23}$ on 23 objects.

## Representations

M$_{23}$ is the point stabilizer of the action of the Mathieu group M24 on 24 points, giving it a 4-transitive permutation representation on 23 points with point stabilizer the Mathieu group M22.
M$_{23}$ has 2 different rank 3 actions on 253 points. One is the action on unordered pairs with orbit sizes 1+42+210 and point stabilizer M$_{21}$.2, and the other is the action on heptads with orbit sizes 1+112+140 and point stabilizer 2$^4$.A$_7$.
The integral representation corresponding to the permutation action on 23 points decomposes into the trivial representation and a 22-dimensional representation. The 22-dimensional representation is irreducible over any field of characteristic not 2 or 23.
Over the field of order 2, it has two 11-dimensional representations, the restrictions of the corresponding representations of the Mathieu group M24.

## Maximal subgroups

There are 7 conjugacy classes of maximal subgroups of M$_{23}$ as follows:
Maximal subgroups of M$_{23}$ :
| No. | Structure | Order | Index | Comments |
| 1 | M$_{22}$ | 443,520 = 2$^7$·3$^2$·5·7·11 | 23 | point stabilizer |
| 2 | L$_3$(4):2 | 40,320 = 2$^7$·3$^2$·5·7 | 253 = 11·23 | has orbits of sizes 21 and 2 |
| 3 | 2$^4$:A$_7$ | 40,320 = 2$^7$·3$^2$·5·7 | 253 = 11·23 | has orbits of sizes 7 and 16; stabilizer of W$_{23}$ block |
| 4 | A$_8$ | 20,160 = 2$^6$·3$^2$·5·7 | 506 = 2·11·23 | has orbits of sizes 8 and 15 |
| 5 | M$_{11}$ | 7,920 = 2$^4$·3$^2$·5·11 | 1,288 = 2$^3$·7·23 | has orbits of sizes 11 and 12 |
| 6 | (2$^4$:A$_5$):S$_3$ ≅ M$_{20}$:S$_3$ | 5,760 = 2$^7$·3$^2$·5 | 1,771 = 7·11·23 | has orbits of sizes 3 and 20 (5 blocks of 4); one-point stabilizer of the sextet group |
| 7 | 23:11 | 253 = 11·23 | 40,320 = 2$^7$·3$^2$·5·7 | simply transitive |

## Conjugacy classes

| Order | No. elements | Cycle structure | |
| 1 = 1 | 1 | 1$^{23}$ |
| 2 = 2 | 3795 = 3 · 5 · 11 · 23 | 1$^7$2$^8$ |
| 3 = 3 | 56672 = 2$^5$ · 7 · 11 · 23 | 1$^5$3$^6$ |
| 4 = 2$^2$ | 318780 = 2$^2$ · 3$^2$ · 5 · 7 · 11 · 23 | 1$^3$2$^2$4$^4$ |
| 5 = 5 | 680064 = 2$^7$ · 3 · 7 · 11 · 23 | 1$^3$5$^4$ |
| 6 = 2 · 3 | 850080 = 2$^5$ · 3 · 5 · 7 · 11 · 23 | 1·2$^2$3$^2$6$^2$ |
| 7 = 7 | 728640 = 2$^6$ · 3$^2$ · 5 · 11 · 23 | 1$^2$7$^3$ | power equivalent |
| 7 = 7 | 728640 = 2$^6$ · 3$^2$ · 5 · 11 · 23 | 1$^2$7$^3$ | power equivalent |
| 8 = 2$^3$ | 1275120 = 2$^4$ · 3$^2$ · 5 · 7 · 11 · 23 | 1·2·4·8$^2$ |
| 11 = 11 | 927360= 2$^7$ · 3$^2$ · 5 · 7 · 23 | 1·11$^2$ | power equivalent |
| 11 = 11 | 927360= 2$^7$ · 3$^2$ · 5 · 7 · 23 | 1·11$^2$ | power equivalent |
| 14 = 2 · 7 | 728640= 2$^6$ · 3$^2$ · 5 · 11 · 23 | 2·7·14 | power equivalent |
| 14 = 2 · 7 | 728640= 2$^6$ · 3$^2$ · 5 · 11 · 23 | 2·7·14 | power equivalent |
| 15 = 3 · 5 | 680064= 2$^7$ · 3 · 7 · 11 · 23 | 3·5·15 | power equivalent |
| 15 = 3 · 5 | 680064= 2$^7$ · 3 · 7 · 11 · 23 | 3·5·15 | power equivalent |
| 23 = 23 | 443520= 2$^7$ · 3$^2$ · 5 · 7 · 11 | 23 | power equivalent |
| 23 = 23 | 443520= 2$^7$ · 3$^2$ · 5 · 7 · 11 | 23 | power equivalent |