Mathieu group:M9

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Definition

This group, denoted $M_9$, and termed the Mathieu group of degree nine, is defined in the following equivalent ways:

• It is the external semidirect product of elementary abelian group:E9 (a two-dimensional vector space over field:F3) by quaternion group where the latter acts on the former via the faithful irreducible representation of quaternion group (a two-dimensional irreducible representation) over field:F3 (this representation can be thought of as the embedding Q8 in GL(2,3)).
• It is the subgroup of the symmetric group of degree nine given by the following generating set:

$\! M_9 := \langle (1,4,9,8)(2,5,3,6), (1,6,5,2)(3,7,9,8) \rangle$.

• It is the projective special unitary group $PSU(3,2)$ of degree three for the quadratic extension field:F4 over field:F2.
• It is the special unitary group $SU(3,2)$ of degree three for the quadratic extension field:F4 over field:F2.

This is one of the Mathieu groups, but is not one of the five sporadic simple Mathieu groups. Rather, it is among the two Mathieu groups (the other being Mathieu group:M10) that are not simple. The Mathieu group $M_{21}$ is simple, but not a sporadic simple group -- it is isomorphic to projective special linear group:PSL(3,4).

Arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	72	groups with same order	As semidirect product of two-dimensional vector space over field:F3 by quaternion group: $3^2 \cdot 8 = 72$. See order of semidirect product is product of orders As Mathieu group $M_n, n = 9, n \in \{ 9,10,11,12 \}$: $n!/7! = n(n-1) \dots 8 = (9)(8) = 72$ As $\! PSU(n,q)$, $\! n = 3, q = 2$: PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
exponent of a group	12	groups with same order and exponent of a group | groups with same exponent of a group	Elements of order $1,2,3,4$.
nilpotency class	not a nilpotent group.
derived length	3	groups with same order and derived length | groups with same derived length
Frattini length	1	groups with same order and Frattini length | groups with same Frattini length	Frattini-free group
Fitting length	2	groups with same order and Fitting length | groups with same Fitting length
minimum size of generating set	2	groups with same order and minimum size of generating set | groups with same minimum size of generating set	Two elements of order four from different $2$-Sylow subgroups.
subgroup rank of a group	2	groups with same order and subgroup rank of a group | groups with same subgroup rank of a group
max-length of a group	5	groups with same order and max-length of a group | groups with same max-length of a group

Arithmetic functions of a counting nature

Function	Value	Explanation
number of subgroups	68	See subgroup structure of Mathieu group:M9
number of conjugacy classes	6
number of conjugacy classes of subgroups	14	See subgroup structure of Mathieu group:M9

Group properties

Property	Satisfied	Explanation	Comment
Abelian group	No
Nilpotent group	No
Supersolvable group	No
Solvable group	Yes
T-group	No	The $3$-Sylow subgroup is abelian normal, has subgroups that are not normal.
Monolithic group	Yes	Unique minimal normal subgroup is $3$-Sylow subgroup.
One-headed group	No	Three maximal normal subgroups of order $36$.
Rational group	Yes
Rational-representation group	No	Its quotient, quaternion group, is not a rational-representation group.
Ambivalent group	Yes
Any two elements generating the same cyclic subgroup are automorphic	Yes	Follows from being a rational group
Every element is automorphic to its inverse	Yes	Follows from being an ambivalent group
Frobenius group	Yes	Frobenius kernel is $3$-Sylow subgroup, complement is $2$-Sylow subgroup
Camina group	Yes	Derived subgroup is of order $18$, three non-identity cosets are conjugacy classes.

Linear representation theory

Further information: linear representation theory of Mathieu group:M9

Summary

Item	Value
Degrees of irreducible representations over a splitting field (such as $\overline{\mathbb{Q}}$ or $\mathbb{C}$)	1,1,1,1,2,8 maximum: 8, lcm: 8, number: 6, sum of squares: 72
Ring generated by character values	$\mathbb{Z}$ -- ring of integers
Field generated by character values	$\mathbb{Q}$ -- field of rational numbers. So the group is a rational group.

GAP implementation

Group ID

This finite group has order 72 and has ID 41 among the groups of order 72 in GAP's SmallGroup library. For context, there are 50 groups of order 72. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(72,41)

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

gap> G := SmallGroup(72,41);

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

IdGroup(G) = [72,41]

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(9)	MathieuGroup
PSU(3,2)	PSU
SU(3,2)	SU