Generators for maximal subgroups of Conway group Co1

Abstract The Conway groups are the three sporadic simple groups Co1, Co2 and Co3. There are total of 22 maximal subgroups of Co1 and generators of 6 maximal subgroups are provided in web Atlas of finite simple groups. The aim of this paper is to give generators of remaining 16 maximal subgroups.


Introduction
The Conway group Co is one of the 26 sporadic simple groups. The largest of the Conway groups, Co , is the group of automorphisms of the Leech lattice Γ with respect to addition and inner product. It has order 8,315,553,613,086,720,000 [1] but it is not a simple group. The simple group Co of order . . . . . . is de ned as the quotient of Co by its center, which consists of the scalar matrices ± [1]. The local subgroups of Co are found in [2] and the maximal subgroups of Co in [3]. There is also a valuable discussion in [4]. The following theorem is crucial in determining the maximal subgroups of Co : Theorem 1.1. [3] If K is a non-Abelian characteristically simple subgroup of Co , then N Co (K) is contained either in a local subgroup of Co or in a conjugate of one of six particular groups: 1. NA ∼ = (A × J ). , 2. NA ∼ = (A × U ( )). , 3. NA ∼ = (A × L ( )). , 4. S( ) ∼ = Co , 5. S( ) ∼ = Co , 6. S( ) ∼ = U ( ).S . Note on notation. We use A.B to denote an arbitrary extension of A by B, while A : B and A · B denote split and non-split extensions, respectively. The symbol n denotes a cyclic group of that order, while [n] denotes an arbitrary group of order n. We follow the ATLAS [5] notation for conjugacy classes. Moreover, we denote x y by y − xy and [x, y] by x − y − xy.
To facilitate the computations in nite simple and almost simple groups, [6], [7] provides the representations and words for generators of most of the maximal subgroups. However, there are still some cases to deal with. A research problem "Words for maximal subgroups in sporadic groups" appears on the web page of R. A. Wilson. We pursue the work initiated by R.A. Wilson of nding words for maximal subgroups of Co . According to [3] there are 24 conjugacy classes of maximal subgroups, but later on R.A. Wilson pointed out a few errors (in his own paper) in the list of maximal subgroups of Co [8] in which he mentions that the two subgroups . [ . ]. A and . [ . ]. A are not the maximal subgroups of Co , so the list contains total 22 conjugacy classes of maximal subgroups. There are 22 maximal subgroups of the group Co . The maximal local subgroups have been determined in [3]. The Atlas of Group Representations [6] contains the words for maximal subgroups of Co except 16. The maximal subgroups of Co are given below. : M 21. .Suz : 22. Co Next we proceed to nd words for 16 maximal subgroups which are marked by asterisk.

Methods
Most of the maximal subgroups on our list can be generated by two elements. If the group is small enough, a random search will produce the subgroup required. This method was successfully used in [9] but in Co the subgroups are too large to use brute force. One more focused way of generating a subgroup is by choosing a pair of conjugacy classes A and B in G such that conjugates of random elements a ∈ A, b ∈ B have a reasonable high probability of generating a conjugate of the desired subgroup. In most of the cases we present here, even though the subgroup we wish to construct may be generated by two elements, it may be hard to tell which conjugacy classes they belong to. Even if we know a suitable pair of conjugacy classes it may be that the probability the random elements in these classes generate the desired subgroup is relatively small. In this case, we nd some part of the desired subgroup, and work inside another subgroup, usually an involution centralizer, to nd the rest. Once we have found a copy of the desired subgroup, we can get information regarding the generating sets. The maximal subgroups often occur as normalizers of elementary abelian groups. So normalizers, which are crux of the matter here, were mostly computed by methods given in [10] and [11]. The generators of subgroups, wherever possible, have been obtained from [6]. The subgroups can be identi ed by determining order, composition series and orbit sizes in several permutation representations. Moreover, comparing our result with the list of maximal subgroups in [5] we nd that there is only one possibility of the subgroup.

Main Results
We have extensively used the information given in [5], [3] and Atlas of nite group representations [6]. We use GAP [12] for group theoratic calculations. Throughout this paper a and b are the standard generators of Co in permutation representation on 98280 points available at [6].

. Construction of (A × S ) inside Co
The required maximal subgroup is the normalizer of an element of class D. Here we use power maps to nd the representative of class D to say that a is the element of said class given by a = ((ba) b) . Now the normalizer of a inside Co gives us the required maximal subgroup. The normalizer can be computed by the technique given in [10] and the programs given in [11]. Before computing the normalizer we will give some random words of Co which will be used later. These words are given below: Consider the group generated by a and b say H =< a , b >, then compute the normalizer of a inside H . From here we get the partial normalizer of a inside Co . Before computing the partial normalizer we will give some random elements of H which will facilitates our computations. These elements are given by: The words for partial normalizer are given below: Next we will nd an involution inside the above partial normalizer. This involution is given by d = k , then nd the centralizer of d inside Co by using the method given by J.Bray [13]. The generators of the centralizer of d Inside Co are given by: The words for the normalizer of a inside the above centralizer (H ) are given below: By looking at the [5] we see that these words generate only partial normalizer so we repeat the above process untill we nd the complete normalizer. Now we give some other elements of Co which will be used in further computations.
Consider the group generated by a and m say H =< a , m >, then compute the normalizer of a inside H . From here we get the partial normalizer of a inside Co . Before computing the partial normalizer we will give some random elements of H . These elements are given by: The word for the normalizer of a inside H is given by: k = a n a n a n a n a n . Now by combining the words for the normalizer of a inside H , H and H we get the required complete normalizer given by k , k and k . The generators for (A × S ) are k and k k .

. Construction of (D × (A × A ). ). inside Co
The required maximal subgroup is the normalizer of an element of class 5B. Here we rst nd an element of class 5B and then nd the normalizer of that element inside Co , which gives us the required maximal subgroup. The element of class 5B can be calculated by using the power maps. The normalizer can be computed by the technique given in [10] and the programs given in [11] will facilitate us in computing the normalizer. Before computing the normalizer we will give some elements of Co which will be used later. These elements are given below: The element of class C is given by c = (b b ) . Next we will give the strategy of nding the normalizer of c inside Co . Consider the group generated by c and a say H =< c, a >, then compute the normalizer of c inside H . From here we get the partial normalizer of c inside Co . Before computing the partial normalizer we will give some random elements of H which will facilitates us in computations.
The words for normalizer of c inside H are given by: k = cc cc cc cc cc , k = cc cc cc cc cc , k = cc cc cc cc cc , k = ac ac ac ac ac , k = ac ac ac ac ac .
Here we will give some more elements of Co .
Consider the group generated by xing c and a is replaced by d say H =< c, d >, then compute the normalizer of c inside H . Some elements of H , which are used in further computations, are given below: The words for normalizer of c inside H are given by:

. Construction of + GL ( ) inside Co
From the information given in [5], the required maximal subgroup is the normalizer of an element of class 5c. The construction of this group consist of two steps given below.
Step 1 Here we rst nd an element of class 5c. For that we give some words of Co . These words are given by: Then we use the power maps to nd an element of order 5 and next check its centralizer order which con rms that the element belongs to class 5c. The element of class 5c is given by c = (b b ) .
Step 2 In this step we will nd the normalizer of 5c inside Co . The normalizer can be found by using the technique given in [10] i.e. we construct the partial normalizer of 5c inside di erent subgroups of of Co . Then we combine these partial normalizer to get the required normalizer. The computations of these partial normalizers are given below. Consider the group generated by c and c say H =< c, c >, then compute the normalizer of c inside H . Before computing the partial normalizer we will give some words of H which will facilitates us in computations. These words are given by: Next we use the "TKnormalizertest" given in [11] to compute the words for the partial normalizer of c inside H . These words are given by: Again consider the group generated by c inside k say H =< c, k >. Here we compute the partial normalizer of c inside H by giving the similar arguments as mentioned above. We will give some words for H which are used in computations. These words are given below: The words for the partial normalizer are given below: Consider the group generated by k , k and k say H =< k , k , k >. The words for H are given below: The words for the normalizer of c inside H are given below: Consider the involution c = k . Since c is an involution so its normalizer can easily be calculated by using the method given by J.Bray [13]. The generators of the normalizer of c inside Co are given below: Consider the group generated by f and f say H =< f , f >. Now compute the normalizer of 5c inside H . Before computing the normalizer we give some words of H . These words are given below: The word for the normalizer of 5c inside H is given below: Now combining the above partial normalizers will give us the words for the required maximal subgroup. These words are given by k , k and k .

. Construction of + : .S ( ). inside Co
From the information given in Atlas [5] the required maximal subgroup is the normalizer of an element of class 3B. So here we rst nd an element of class 3B and then nd the normalizer of it. We will give some words of Co . These words are given by: The construction of this group consist of two steps given below.
Step 1 In this step we will nd an element of class 3B. This can be done by using the power maps of the above generated elements, then checking whether the centralizer order con rms that the element under consideration belongs to class 3B or not. The element of class 3B is given by "b".
Step 2 In this step we will nd the normalizer of b inside Co . The normalizer can be found by using the technique given in [10] i.e. we construct the partial normalizer of b inside di erent subgroups of of Co . Then we combine these partial normalizer to get the required normalizer. The computations of these partial normalizers are given below. Consider the group generated by b and c say H =< b, c >, then compute the normalizer of b inside H . Before computing the partial normalizer we will give some words of H which will facilitates our computations. These words are given below: e = bc , e = bc b, e = bc bc .
Next we use the "TKnormalizertest" [11] to compute the words for the partial normalizer of b inside H . k = be be be be be , k = be be be be be , k = be be be be be , k = be be be be be , k = be be be be be , k = be be be be be .
Again consider the group generated by b and c say H =< b, c >. Here we compute the partial normalizer of b inside H by giving similar arguments as those mentioned above. We will give some words for H which are used in further computations. These words are given by: The words for the partial normalizer are given below: k = be be be be be , k = be be be be be , k = be be be be be , k = be be be be be . Now combining the above partial normalizers will give the words for the required maximal subgroup. These words are given by k and k .

. Construction of (A × G ( )) : inside Co
From the information given in Atlas [5] the required maximal subgroup is the normalizer of B (a four group whose involutions are in class 2B). The construction of this subgroup consist of two steps.

Step1
In this step we nd B . First we nd an involution of class 2B. This involution is given by a, then we nd the centralizer inside Co . This can be done by using the technique given by J. Bray given in [13]. The generators of the centralizer inside Co are given by: Then we search inside this centralizer for an involution of class 2B to nd an elementary abelian group of order 4. This involution is given by c = a . Now we have the required B generated by a and c where c = a .
Step2 In this step we will calculate the normalizer of B inside Co . The normalizer can be found by using the technique given in [10] i.e. we construct the partial normalizer of B inside di erent subgroups of Co . Then we combine these partial normalizer to get the required normalizer. We also give some elements of Co which are used in computations: Consider the group generated by a, c and d say H =< a, c, d >, then compute the normalizer of B inside H . Before computing the partial normalizer we will some words of H which will facilitates our computations. These words are given below: e = ad , e = cd , e = cd a, e = cd ac.
Next we use the "TKnormalizertest" [11] to compute the words for the partial normalizer of B inside Co . These words are given below: k = ae ae ae ae ae , k = ae ae ae ae ae , k = ae ae ae ae ae k = ae ae ae ae ae , k = ae ae ae ae ae Again consider the group generated by a, c and d say H =< a, c, d >. Here we compute the partial normalizer of B inside H by giving similar arguments to those mentioned above. We give some words for H which are used in further computations: The words for the partial normalizer are given by: k = ae ae ae ae ae , k = ae ae ae ae ae . Now combining the above partial normalizers gives us the words for the required maximal subgroup. These words are given by k , k and k .

. Construction of ( + )A × S inside Co
From the information given in Atlas [5] the required maximal subgroup is the normalizer of A (a four group whose involutions are in class 2A). The construction of this subgroup consist of two steps.

Step1
In this step we nd A . First we nd an involution of class 2A. This involution is given by c = (ab) , then we nd the centralizer of c inside Co . This can be done by using the technique in [13]. The generators of the centralizer of c inside Co are given below: Then searching inside this centralizer for an involution of class 2A, which combines with c, gives an elementary abelian group of order 4. This involution is given by d = (a a ) . Now we have the required A generated by c and d. Step2 In this step we will calculate the normalizer of A inside Co . The normalizer can be found by using the technique given in [10] i.e. we construct the partial normalizer of A inside di erent subgroups of of Co . Then we combine these partial normalizer to get the required normalizer. We also give some elements of Co which are used in computations: Consider the group generated by c, d and d say H =< c, d, d >, then compute the normalizer of A inside H . Before computing the partial normalizer we give some words of H which will facilitate our computations: e = cd , e = cd d, e = cd dc, e = cd dcdd .
Next we use the "TKnormalizertest" [11] to compute the words for the partial normalizer of B inside Co . These words are given by: The words for the partial normalizer are given below: k = ce ce ce ce ce , k = ce ce ce ce ce , k = de de de de de . Now combining the above partial normalizers gives the words for the required maximal subgroup. These words are given by k , k and k .

. Construction of : .M inside Co
From the information given in Atlas [5] the required maximal subgroup is the normalizer of (elementary abelian group of order 729). The construction of this subgroup consist of two steps given below. Step1 Here we will construct . To construct we adopt the following strategy. i) Find an arbitrary element of order 3. This element is given by b. ii) Find centralizer of b inside Co . Before calculating the centralizer we give some elements of Co as follows: The centralizer of b can be found by using the technique given in [10] i.e. we start by constructing the partial centralizer of inside di erent subgroups of of Co . Next we combine these partial centralizer to get the required centralizer of b inside Co . We also give some elements of Co which are used in computations. The generators of centralizer of b inside Co are given below: k = be be be be be , k = be be be be be , k = be be be be be , k = be be be be be , k = be be be be be , k = be be be be be .
Again consider the group generated by b and c say H =< b, c >. Here we compute the partial normalizer of b inside H . We give some random elements of H which are used in further computations: e = bc , e = bc b, e = bc bc .
The generators of centralizer of b inside H are given below: k = be be be be be , k = be be be be be , k = be be be be be , k = be be be be be . Now combining the above partial normalizers gives us the generators for . These generators are given by k ,k ,k ,k ,k ,k ,k ,k ,k and k . Then we can easily nd inside the above centralizer. The generators for are given by: Step2 In this step we will nd the normalizer of inside Co which is the required maximal subgroup. The words for the normalizer of are given below: The words for : .M are w = z z and w = z z .

. Construction of .U ( ).D inside Co
From the information given in Atlas [5] the required maximal subgroup is the normalizer of (elementary abelian group of order 9). Similarly as in the previous cases we will construct this group into two steps given below.

Step1
In this step we will nd . This can be done by taking which we constructed in section 3.7, then searching inside these we can easily nd the required given by f and f . Step2 In this step we will nd the normalizer of H =< f , f > inside Co . The normalizer can be found by using the technique given in [10] i.e., we construct the partial normalizer of H inside di erent subgroups of of Co . Then we combine these partial normalizer to get the required normalizer. The computations of these partial normalizers are given below. Before computing the partial normalizer we give some words of Co which will facilitate our computations. These words are given below: the words for the words for the partial normalizer of H are: We nd an involution inside the above calculated normalizer and then calculate its centralizer. This involution is given by g = k , generators of the centralizer of g inside Co are: . Now we will nd the partial normalizer of H inside Co . The words for the centralizer are given below: The words for the partial normalizer of H inside Co are given by: The words for the required maximal subgroup .U ( ).D are k and k k .

. Construction of + : .(S × S ) inside Co
Following [5], we see that the required subgroup is the normalizer of . We can easily nd from calculated above in 3.7, then the normalizer of it gives us the required subgroup. The generators of are f , f and f . Before computing the normalizer we give some elements: The generators for the normalizer of are given below: The words for + : .(S × S ) are given by w = z z and w = z . .

Construction of + .(S × S ) inside Co
From [5], the required subgroup is the normalizer of A (inside Co ) and can be constructed by taking an involution of class A, then searching inside its centralizer. We have already constructed N Co ( A ) in section 3.6, and now we will search A inside N Co ( A ). Now A = k , k , g , g , where k , k are same as in section 3.6 and g = (k k ) , g = (k k ) . The words for k and k are given in section 3.6.
The generator for the normalizer of A are given below.

k = h h h h h h h h h h , k = h h h h h h h h h h ,
k = l l l l l l l l l l .
The words for + .(S × S ) are found to be k and k = k k . .

Construction of : ( × A ). inside Co
From [5], : ( × A ). is the normalizer of inside Co . We give some random elements of Co below.  ,

Construction of : ( × .S ) inside Co
Following [5], the required subgroup is the normalizer of . It is constructed by taking an element of class B and by searching inside its centralizer we nd B . Before computations we give some random elements of Co : .

Construction of : A inside Co
The required subgroup is the normalizer of C [5] and it can be constructed by taking an element of order and then search inside its centralizer. We can easily nd C .
The generators of C are given by c and z .
l = j j j , k = j l j l j l j l j l .
The words for : A are k and k . .

Construction of (A × L ( )) : inside Co
The whole strategy for locating (A × L ( )) : is given in [3]. First we take an A of the type ( B, A, A) which is in the unique class of Co with normalizer N(A ) = (A × J ). [5], then nd A inside A . Then we nd an element of class A which commutes with A but not with A . This A element with A extends A to our required A . Finally we found that N(A ) = (A × L ( )) : . We give some random elements of Co to facilitate computations. , g = ab[g , ab] , g = g g , k = g g g g g g g g g g .
The words for (A × L ( )) : are k and k = k k . .

Construction of (A × U ( )) : inside Co
The required group is the normalizer of A which lies in the suzuki chain [3]. We easily nd A inside 3.14. The .

Construction of (A × J ) : inside Co
The required group is the normalizer of A [5] which we already constructed in 3.14. It can also be constructed by taking an involution of class B, an element of class A and product of these two elements belongs to class A [5], then N( B, A, A) = (A × J ). . b = (ab) a, b = bab (ab) , b = bab (ab) ba, The generators of the required group whose normalizer is to be computed are given below: Before computing the normalizer we give some elements: