I am struggling with following problem. Consider $2A$ class in $Co\_1$ having $819\*759\*75$ elements. Each element $a$ from $2A$ have two representatives in $Co\_0$. Element $a$ corresponds to $E\_8$ sublattice in Leech lattice defined as $\{v: av=-v\}$ where I call by $a$ also proper preimage in $Co\_0$. Now the opposite: having $E\_8$ sublattice $L$ in Leech lattice I can find element $a\_L$ in $2A$ class. 


My goal is to find relation between $Order(ab)$ for $a,b$ in $2A$ and corresponding geometry of two $E\_8$ sublattices. The $Order(ab)$ can be $2,3,4,5,6$. (BTW in 2017 I have found this relation already).


Take any other sporadic group $g$ and certain conjugacy class $C$ of involutions. The possible values of $Order(ab)$ for $a,b$ in $C$ can be obtained from character table. Can this help to connect some "lattice" to the group $g$ ?



Geometric interpretation of $2A$ conjugacy class in Conway group $Co_1$
Here is the answer for $Co\_1$ which I obtained in GAP.


For given $2A$ involution $a$ in $Co\_1$ the number $n$ of such $2A$ involutions $b$ that product $ab$ is in conjugacy class $C$ is presented in format $[C, n]$. I call *hook* the order of product. 


$[ [ "1a", 1 ], [ "2a", 12870 ], [ "2c", 60480 ], [ "3b", 573440 ], [ "3d", 491520 ], [ "4a", 34560 ], [ "4c", 4838400 ], [ "4d", 2419200 ], [ "5b", 12386304 ], [ "6e", 25804800 ] ]$


Next the relation between two $E\_8$ sublattices $L\_a$ and $L\_b$ of Leech lattice is presented in following format.
[hook of a and b, number of vectors belonging to intersection of $L\_a$ and $L\_b$, dimension of vector space generated by both $L\_a$ and $L\_b$, $[k,l]$ where $k$ is number of vectors in $L\_a$ having exactly $l$ perpendicular vectors in $L\_b$. 


First two rows correspond to $2A$ hook in $Co\_1$. These are two cases in $Co\_0$. For given $E\_8$ sublattice there are $270$ sublattices perpendicular to it (first row) and $12600$ intersecting in $D\_4$ sublattice (second row). 


$[ 2, 0, 16, [ [ 240, 240 ] ] ]$


$[ 2, 24, 12, [ [ 192, 84 ], [ 24, 126 ], [ 24, 240 ] ] ]$


$[ 2, 4, 14, [ [ 48, 84 ], [ 132, 126 ], [ 60, 240 ] ] ]$


$[ 3, 0, 16, [ [ 240, 126 ] ] ]$


$[ 3, 6, 14, [ [ 162, 84 ], [ 78, 126 ] ] ]$


$[ 4, 0, 16, [ [ 240, 84 ] ] ]$


$[ 4, 0, 16, [ [ 24, 84 ], [ 192, 126 ], [ 24, 240 ] ] ]$


$[ 4, 2, 15, [ [ 120, 84 ], [ 114, 126 ], [ 6, 240 ] ] ]$


$[ 5, 0, 16, [ [ 120, 84 ], [ 120, 126 ] ] ]$


$[ 6, 0, 16, [ [ 66, 84 ], [ 168, 126 ], [ 6, 240 ] ] ]$


The numbers $84$ and $126$ which appear on second $l$ position correspond to $D\_7$ and $E\_7$ sublattices of $E\_8$. The first case happens when projection of the vector on $E\_8$ is collinear with "type 4" vector which is sum of two perpendicular vectors from the lattice. The second case happens when projection of the vector on $E\_8$ is collinear with lattice vector.


As we can see one of the 4-hook is obtained by intersection in one line of the two 8-planes. This feature can be used to define the Leech lattice from $Co\_0$ in 24-dim space. I don't know whether this is useful for something yet.


