A PERMUTATION REPRESENTATION OF THE GROUP OF THE BITANGENTS .sx W. L. Edge .sx .sx The group 5G of the bitangents has been studied in two recent papers ( [3] and [4]) .sx It was represented in [4] as a subgroup of index 2 of the group of symmetries of a regular polytope in Euclidean space of dimension 6 , in [3] as the group of automorphisms of a non-singular quadric Q in the finite projective space [6] over F- the Galois Field GF(2) .sx The culmination of [4] is the compilation , for the first time , of the complete table of characters of 5G , and Frame uses this table to suggest possible degrees for permutation representations .sx Such representations , of degrees 28 , 36 , 63 , 135 , 288 are patent once the geometry of Q is known ; but Frame , having observed that there is a combination of the characters that satisfies the several conditions known to be necessary , had proposed also 120 as a possible degree .sx As there is no guarantee that the set of necessary conditions is sufficient , and as no representation of 5G of degree 120 seems yet to have appeared in the literature , a description is here submitted of one that is incorporated with the geometry of Q. Q consists , as explained in [3] , of 63 points m ; 315 lines g ( all three points on a g being m ) lie on Q , while through each g pass three planes d lying wholly on Q ( in that all seven points in d are m , and all seven lines in d are g) .sx These three d form the complete intersection of Q with E , the polar [4] of g. There are , and it is intended to construct them , 120 figures F ; each F includes all 63 m together with 63 d , one d being associated with each m- having m for its focus as one may say .sx Those g in d that pass through its focus may be called rays ; all three d containing a ray belong to F , their foci being those three m that constitute the ray , so that , there being three rays in each of 63 d , there are 63 rays in F. The plane of any two intersecting rays is on Q , and the third line therein through the intersection is a ray too .sx None of the 72 d extraneous to F includes a ray ; of those d that pass through a g which is not a ray only one belongs to F , the other two being extraneous to F. Although such a figure as F may not have been previously described it has been encountered , so to say , by implication , being obtainable when Q is regarded as a section of a ruled quadric S in [7] ; one has then only to take , on S , those points that are autoconjugate ( i.e. incident with their corresponding solids ) in a certain triality .sx That such points make up a prime section of S is known ( see 5.2.2 in [5] ) , and that there are 63 of them accords with putting 5k = 5l = 2 in 8.2.4 of [5] ; 8.2.6 then says that , of 63 m , 32 lie outside the tangent prime T;0 ; to Q at a given point m;0 ; while 8.2.5 says that there are 63 rays , or " fixed lines " in Tits' phraseology .sx .sx Let 5d , 5d@7 be any two of the 135 planes on Q that are skew to one another ; they span a [5] C and , being skew , belong to opposite systems on K , the Klein section of Q by C. Through any line g of 5d passes another plane of K which , belonging to the opposite system to 5d , is in the same system as 5d@7 and so meets 5d@7 at a point m@7 ; moreover , the points m@7 so arising from g in 5d concurrent at m lie on g@7 , the line of intersection of 5d@7 with the tangent space [15dg@7] of K at m. The plane , other than 5d@7 , on K that contains g@7 is [mg@7] .sx So there is set up a correlation between 5d and 5d@7 ; each point of either is correlative to a line of the other .sx If m in 5d and m@7 in 5d@7 each lie on the line correlative to the other their join is on K. There are 21 such joins ; through each point m of 5d there pass three , lying in the plane joining m to its correlative g@7 , and likewise there pass three coplanar joins through each point m@7 of 5d@7 .sx Since K consists of 35 m there are 21 , which may be labelled temporarily as points 5m , that lie neither in 5d nor in 5d@7 ; through each 5m passes one transversal to 5d and 5d@7 ; these 21 lines , one through each 5m , are the joins mm@7 of points each on the line correlative to the other .sx Through each point on K pass nine lines lying on K ; if m is in 5d three of them lie in 5d while another three join m to the points on its correlative g@7 ; there remain three others , so that 21 g on K meet 5d in points and are skew to 5d@7 .sx Another 21 meet 5d@7 in points and are skew to 5d .sx There are also among the 105g on K seven in 5d , seven in 5d@7 , 21 transversal to 5d and 5d@7 ; there remain 28 , which may be labelled g/ , skew to both 5d and 5d@7 .sx These 28 g/ may be identified as follows .sx Take any g in 5d ; the solid that joins it to any g@7 through its correlative m@7 in 5d meets K in two planes through mm@7 , m being that point on g to which g@7 is correlative .sx But there are four lines g@7 in 5d@7 that do not contain m@7 ; then the solid [gg@7] meets K in a hyperboloid whereon the regulus that includes g and g@7 is completed by g/ .sx As there are seven g in 5d , and four g@7 in 5d@7 not containing the correlative m , the 28 g/ are accounted for .sx There being three 5m on each g/ , but only 21 5m in all , one expects there to be four g/ through each 5m ; this is so .sx For let the transversal from 5m to 5d , 5d@7 meet 5d in m , 5d@7 in m@7 ; through m , and in 5d , are lines g;1 ; , g;2 ; other than the correlative g to m@7 ; through m@7 , and in 5d@7 , are lines g;1;@7 , g;2;@7 other than the correlative g@7 to m ; each solid meets K in a hyperboloid whereon a regulus is completed by a g/ through 5m .sx .sx Take , now , one of these g/ :sx the transversals from its three 5m to 5d , 5d@7 form a regulus whose complement includes g in 5d and g@7 in 5d@7 , neither g nor g@7 being correlative to any point on the other .sx The correlative m in 5d of g@7 is conjugate to every point of g and , by the defining property of the correlation , to every point of g@7 ; so , likewise , is the correlative m@7 in 5d@7 of g. Hence the polar plane j;0 ; ( [3] , @136 ) of [gg@7] with respect to Q contains both m and m@7 ; there is one remaining point m/ of Q in j;0 ; , and it lies outside C- for to suppose that it belonged to C would put the whole of j;0 ; in C , whereas the kernel of Q , which is in j;0 ; , is outside C. Now there are 63-25 = 28 points m/ on Q that are not on K ; thus each m/ is linked to a g/ , and m/g/ is a plane d on Q. There are three planes on Q through any line thereon ; if this line is a transversal m15mm@7 from one of the 21 5m to 5d and 5d@7 two of these planes are on K , while the third contains a quadrangle m;1;/m;2;/m;3;/m;4;/ with its diagonal points at m , 5m , m@7 .sx The tangent prime to Q at any vertex of this quadrangle contains m15mm@7 and meets 5d , 5d@7 in lines belonging to a regulus completed by g/ through 5m .sx Thus four concurrent g/ are linked with coplanar m/ whose plane , containing the transversal to 5d and 5d@7 from the point of concurrence , lies on Q but not on K. 4 .sx Choose now , from among the 315 g on Q , the 21 transversals of 5d , 5d@7 and those , three through each m/ , that join m/ to those 5m on the g/ that is linked with it .sx Each such join contains two m/ , the g/ that are linked therewith both passing through 5m ; hence , under this second heading , the number of g selected is [FORMULA] .sx So 63 g are chosen :sx call them rays .sx Through each m on Q pass three rays , and they are coplanar .sx If m is m/ this is manifest from the prescription of choice , as it is too if m is in 5d or 5d@7 .sx If m is 5m the rays are , say , [FORMULA] and lie in that d through m15mm@7 that is not on K. So 63 d are chosen from among the 135 on Q ; each contains three concurrent rays .sx Call the m wherein the rays concur the focus of d. Through any g there pass three d ; if g is a ray these d are those having the m on the ray for foci .sx The points of d other than its focus m are foci of those other d which belong to F and contain m ; if d , d@7 in F are such that the focus of d@7 is in d then the focus of d is in d@7 .sx Whenever two rays meet the third line through their intersection and lying in their plane is a ray too .sx It is these 63 d , with the 63 rays and foci , that constitute the figure F. Each d in F contains , as well as three concurrent rays , a quadrilateral of g that are not rays ; thus , by four in each of 63 d , the 315-63 = 252 g that are not rays are accounted for .sx Through each such g pass two planes on Q in addition to d , but they are extraneous to F. The 135-63 = 72 extraneous planes may be labelled 5d ; the planes above denominated by 5d and 5d@7 are in this category .sx No g in 5d is a ray and only one of the planes on Q that pass through it belongs to F whereas , were g a ray , all three would do so .sx .sx Label the m in any of the 72 5d by they lie on g that can be taken as Through each such g there is a single d belonging to F ; label the foci of these d , none of which can lie in 5d , respectively Then those d whose foci are in 5d join its points to the respective triads Thus the join of every pair of points =1@7 is on Q and , there being no solid on Q , the points =1@7 lie in a plane 5d@7 whose lines consist of the triads =2@7 .sx Each of the 72 5d has , it is now clear , a twin 5d@7 coupled with it by F. The correlation between 5d and 5d@7 is shown by =1 and =2@7 or , alternatively , by =1@7 and =2 .sx Those d that pass one through each line of 5d@7 have for their foci the points of 5d correlative to these lines ; if d passes , say , through 1@7 3@7 5@7 its focus is the point 5 common to those d whose foci are 1@7 , 3@7 , 5@7 .sx Since , by the construction in @134 , 5d and 5d@7 determine F uniquely there are x/36 figures F where x is the number of pairs of skew planes on Q. To calculate x note , in the first place ( using d now to signify a plane on Q whether it be in F or extraneous thereto ) , that each d is met in lines by 14 others , two passing through each g in d. Note next , to ascertain how many d meet a given d;0 ; in points only , that the 15 d through a point m of d;0 ; project , from m , the figure of 15 g in [4] passing three by three through 15 points ( [2] , @13@1313-15) .sx