# Difference between revisions of "ApCoCoA-1:Mathieu22 group"

### Mathieu group M {22}

#### Description

The Mathieu group M22 is a sporadic group with order 443520 and it is a subgroup of the monstergroup.

#### Computation

``` /*Use the ApCoCoA package ncpoly.*/

Use ZZ/(2)[a,b,c];
NC.SetOrdering("LLEX");

Define CreateRelationsMathieuM22()
Relations:=[];

// add the relation a^2 = 1
Append(Relations,[[a,a],]);
// add the relation b^4 = 1
Append(Relations,[[b,b,b,b],]);

//add the relation (ab)^11 = 1
Append(Relations,  [[a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b],]);

//add the relation (ab^2)^5 = 1
Append(Relations,[[a,b,b,a,b,b,a,b,b,a,b,b,a,b,b],]);

// the commutator of [a,bab] is a,b,a,b,b^3,a,b^3
Append(Relations,[[a,b,a,b,b,b,b,a,b,b,b,a,b,a,b,b,b,b,a,b,b,b,a,b,a,b,b,b,b,a,b,b,b],]);

//add the relation (ababab^-1)^5 = 1
//Append(Relations,[[a,b,a,b,a,b,b,b,a,b,a,b,a,b,b,b,a,b,a,b,a,b,b,b,a,b,a,b,a,b,b,b,a,b,a,b,a,b,b,b],]);
Append(Relations,[[a,b,a,b,a,b,b,b],[c]]);
Append(Relations,[[c,c,c,c,c],]);
Return Relations;
EndDefine;

Relations:=CreateRelationsMathieuM22();
Relations;

Gb:=NC.GB(Relations,31,1,39,1000);
Gb;
```

#### Example in Symbolic Data Format

``` <FREEALGEBRA createdAt="2014-03-26" createdBy="strohmeier">
<vars>a,b</vars>
<uptoDeg>12</uptoDeg>
<basis>
<ncpoly>a*a-1</ncpoly>
<ncpoly>b*b*b*b-1</ncpoly>
<ncpoly>((a*b)^11)-1</ncpoly>
<Comment>relation: (ab)^11=1 </Comment>
<ncpoly>(a*b*b)^5-1</ncpoly>
<Comment>relation: (abb)^5=1</Comment>
<ncpoly>(a*b*a*b*b*b*b*a*b*b*b)^3-1</ncpoly>
<Comment>commutator: [a,bab]=ababb^3ab^3</Comment>
<Comment>relation [a,bab]^3=1</Comment>
<ncpoly>(a*b*a*b*a*b*b*b)^5-1</ncpoly>
<Comment>relation (ababab^-1)^5=1</Comment>
</basis>
<Comment>The partial LLex Gb has 186 elements</Comment>
<Comment>Mathieu group_M_{22}</Comment>
</FREEALGEBRA>
```