# Mathieu group:M10

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## Definition

This group, denoted $M_{10}$, is the Mathieu group of degree $10$. Explicitly, it can be defined as the isotropy subgroup (stabilizer) of two points under the natural action of Mathieu group:M12 on the projective line over field:F11. Since the action is a 2-transitive group action, it does not matter which two points we choose, but for concreteness, we can choose $0$ and $\infty$.

Although it is one of the Mathieu groups, it is not among the five sporadic simple Mathieu groups. Rather, it is among the two non-simple Mathieu groups. The other is Mathieu group:M9. The Mathieu group $M_{21}$ is isomorphic to projective special linear group:PSL(3,4) and is hence simple but not a sporadic simple group.

## Arithmetic functions

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 720 groups with same order As Mathieu group $M_n, n = 10, n \in \{ 9,10,11,12 \}$: $n!/7! = n(n-1) \dots 8 = (10)(9)(8) = 720$
exponent of a group 120 groups with same order and exponent of a group | groups with same exponent of a group
Frattini length 1 groups with same order and Frattini length | groups with same Frattini length

## Group properties

Property Satisfied? Explanation
abelian group No
nilpotent group No
solvable group No derived subgroup is alternating group:A6.
simple group No

## Subgroups

Further information: subgroup structure of Mathieu group:M10

### Quick summary

Item Value
number of subgroups 793
number of conjugacy classes of subgroups 25
number of automorphism classes of subgroups 24
isomorphism classes of Sylow subgroups and the corresponding Sylow numbers 2-Sylow: semidihedral group:SD16, Sylow number is 45
3-Sylow: elementary abelian group:E9, Sylow number is 10
5-Sylow: cyclic group:Z5, Sylow number is 36
Hall subgroups No Hall subgroups other than the trivial subgroup, whole group, and Sylow subgroups
maximal subgroups maximal subgroups have order 16, 20, 72, 360
normal subgroups the only proper nontrivial normal subgroup is of index two and is isomorphic to alternating group:A6 (order 360)

## Linear representation theory

Further information: linear representation theory of Mathieu group:M10

Item Value
degrees of irreducible representations over a splitting field (such as $\overline{\mathbb{Q}}$ or $\mathbb{C}$) 1, 1, 9, 9, 10, 10, 10, 16
grouped form: 1 (2 times), 9 (2 times), 10 (3 times), 16 (1 time)
maximum: 16, number: 8, lcm: 720, sum of squares: 720

## GAP implementation

### Group ID

This finite group has order 720 and has ID 765 among the groups of order 720 in GAP's SmallGroup library. For context, there are 840 groups of order 720. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(720,765)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(720,765);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [720,765]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

### Other descriptions

Description Functions used
MathieuGroup(10) MathieuGroup