Mathieu group:M12

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

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.