# Mathieu group:M12

## Definition

This group is defined as the Mathieu group of degree $12$. It is one of the five simple Mathieu groups. It is given by the following permutation representation, as a subgroup of symmetric group:S12 as follows. Consider the following set of size 12: the projective line over field:F11. Explicitly, this set can be written as: $\{ 0,1,2,3,4,5,6,7,8,9,10,\infty \}$

We have projective special linear group:PSL(2,11), acting naturally on this set as fractional linear transformations by: $\begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mapsto \left(x \mapsto \frac{ax + b}{cx + d}\right)$

This defines an embedding of $PSL(2,11)$ inside $S_{12}$.

The group $M_{12}$ is defined as the subgroup of $S_{12}$ generated by the image of $PSL(2,11)$ and the permutation given by $x \mapsto 4x^2 - 3x^7$, which as a permutation is: $\! (2,6,10,7)(3,9,4,5)$

Note that this permutation fixes the points $0,1,\infty$.

Note further that since all generating permutations are even permutations, this group is in fact a subgroup of alternating group:A12.

## Arithmetic functions

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 95040 groups with same order As Mathieu group $M_n, n \in \{ 9,10,11,12 \}$: $n!/7! = n(n-1)\dots 8 = (12)(11)(10)(9)(8) = 95040$
exponent of a group 1320 groups with same order and exponent of a group | groups with same exponent of a group

## Group properties

Property Satisfied? Explanation
abelian group No
nilpotent group No
solvable group No
simple group Yes
minimal simple group No

## Subgroups

Further information: subgroup structure of Mathieu group:M12

## Linear representation theory

Further information: linear representation theory of Mathieu group:M12

Item Value
degrees of irreducible representations over a splitting field (such as $\overline{\mathbb{Q}}$ or $\mathbb{C}$) 1,11,11,16,16,45,54,55,55,55,66,99,120,144,176
grouped form (occurs once by default): 1, 11 (2 times), 16 (2 times), 45, 54, 55 (3 times), 66, 99, 120, 144, 176
maximum: 176, quasirandom degree: 11, number: 15, sum of squares: 95040

## GAP implementation

Unfortunately, GAP's SmallGroup library is not available for this order of group (95040) so the group cannot be constructed that way. It can be constructed in other ways:

Description Functions used
MathieuGroup(12) MathieuGroup
PerfectGroup(95040) or equivalently PerfectGroup(95040,1) PerfectGroup

The group is somewhat cumbersome to manipulate directly because of its large size. Information about the group, including its character table, can be accessed using the symbol "M12" -- see linear representation theory of Mathieu group:M12#GAP implementation for more.