a permutation representation of the group of the bitangents . W L Edge . 1 . the group G of the bitangents has been studied in two recent papers ( ( 3 ) and ( 4 ) ) . 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) . the culmination of ( 4 ) is the compilation , for the first time , of the complete table of characters of G , and Frame uses this table to suggest possible degrees for permutation representations . 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 . as there is no guarantee that the set of necessary conditions is sufficient , and as no representation of G 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 ) . 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 . 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 . 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 . 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 k = l = 2 in 8.2.4 of ( 5 ) ; 8.2.6 then says that , of 63 m , 32 lie outside the tangent prime T0 to Q at a given point m0 while 8.2.5 says that there are 63 rays , or fixed lines in Tits &apos; phraseology . 2 . let d , d&amp;prime; 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 d passes another plane of K which , belonging to the opposite system to d , is in the same system as d&amp;prime; and so meets d&amp;prime; at a point m&amp;prime; ; moreover , the points m&amp;prime; so arising from g in d concurrent at m lie on g&amp;prime; , the line of intersection of d&amp;prime; with the tangent space ( dg&amp;prime; ) of K at m . the plane , other than d&amp;prime; , on K that contains g&amp;prime; is ( mg&amp;prime; ) . so there is set up a correlation between d and d&amp;prime; ; each point of either is correlative to a line of the other . if m in d and m&amp;prime; in d&amp;prime; each lie on the line correlative to the other their join is on K . there are 21 such joins ; through each point m of d there pass three , lying in the plane joining m to its correlative g&amp;prime; , and likewise there pass three coplanar joins through each point m&amp;prime; of d&amp;prime; . since K consists of 35 m there are 21 , which may be labelled temporarily as points m , that lie neither in d nor in d&amp;prime; ; through each m passes one transversal to d and d&amp;prime; ; these 21 lines , one through each m , are the joins mm&amp;prime; of points each on the line correlative to the other . through each point on K pass nine lines lying on K ; if m is in d three of them lie in d while another three join m to the points on its correlative g&amp;prime; ; there remain three others , so that 21 g on K meet d in points and are skew to d&amp;prime; . another 21 meet d&amp;prime; in points and are skew to d . there are also among the 105g on K seven in d , seven in d&amp;prime; , 21 transversal to d and d&amp;prime; ; there remain 28 , which may be labelled g&amp;ast; , skew to both d and d&amp;prime; . these 28 g&amp;ast; may be identified as follows . take any g in d ; the solid that joins it to any g&amp;prime; through its correlative m&amp;prime; in d meets K in two planes through mm&amp;prime; , m being that point on g to which g&amp;prime; is correlative . but there are four lines g&amp;prime; in d&amp;prime; that do not contain m&amp;prime; ; then the solid ( gg&amp;prime; ) meets K in a hyperboloid whereon the regulus that includes g and g&amp;prime; is completed by g&amp;ast; . as there are seven g in d , and four g&amp;prime; in d&amp;prime; not containing the correlative m , the 28 g&amp;ast; are accounted for . there being three m on each g&amp;ast; , but only 21 m in all , one expects there to be four g&amp;ast; through each m ; this is so . for let the transversal from m to d , d&amp;prime; meet d in m , d&amp;prime; in m&amp;prime; ; through m , and in d , are lines g1 , g2 other than the correlative g to m&amp;prime; ; through m&amp;prime; , and in d&amp;prime; , are lines g1&amp;prime; , g2&amp;prime; other than the correlative g&amp;prime; to m ; each solid &amp;formula; meets K in a hyperboloid whereon a regulus is completed by a g&amp;ast; through m . 3 . take , now , one of these g&amp;ast; : the transversals from its three m to d , d&amp;prime; form a regulus whose complement includes g in d and g&amp;prime; in d&amp;prime; , neither g nor g&amp;prime; being correlative to any point on the other . the correlative m in d of g&amp;prime; is conjugate to every point of g and , by the defining property of the correlation , to every point of g&amp;prime; ; so , likewise , is the correlative m&amp;prime; in d&amp;prime; of g . hence the polar plane j0 ( ( 3 ) , &amp;symbol;6 ) of ( gg&amp;prime; ) with respect to Q contains both m and m&amp;prime; ; there is one remaining point m&amp;ast; of Q in j0 , and it lies outside C - for to suppose that it belonged to C would put the whole of j0 in C , whereas the kernel of Q , which is in j0 , is outside C . now there are 63-25 = 28 points m&amp;ast; on Q that are not on K ; thus each m&amp;ast; is linked to a g&amp;ast; , and m&amp;ast;g&amp;ast; is a plane d on Q . there are three planes on Q through any line thereon ; if this line is a transversal mmm&amp;prime; from one of the 21 m to d and d&amp;prime; two of these planes are on K , while the third contains a quadrangle &amp;formula; with its diagonal points at m , m , m&amp;prime; . the tangent prime to Q at any vertex of this quadrangle contains mmm&amp;prime; and meets d , d&amp;prime; in lines belonging to a regulus completed by g&amp;ast; through m . thus four concurrent g&amp;ast; are linked with coplanar m&amp;ast; whose plane , containing the transversal to d and d&amp;prime; from the point of concurrence , lies on Q but not on K . 4 . choose now , from among the 315 g on Q , the 21 transversals of d , d&amp;prime; and those , three through each m&amp;ast; , that join m&amp;ast; to those m on the g&amp;ast; that is linked with it . each such join contains two m&amp;ast; , the g&amp;ast; that are linked therewith both passing through m ; hence , under this second heading , the number of g selected is &amp;formula; . so 63 g are chosen : call them rays . through each m on Q pass three rays , and they are coplanar . if m is m&amp;ast; this is manifest from the prescription of choice , as it is too if m is in d or d&amp;prime; . if m is m the rays are , say , &amp;formula; and lie in that d through mmm&amp;prime; that is not on K . so 63 d are chosen from among the 135 on Q ; each contains three concurrent rays . 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 . 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&amp;prime; in F are such that the focus of d&amp;prime; is in d then the focus of d is in d&amp;prime; . whenever two rays meet the third line through their intersection and lying in their plane is a ray too . 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 . 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 d ; the planes above denominated by d and d&amp;prime; are in this category . no g in d 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 . 5 . label the m in any of the 72 d by &amp;formula; they lie on g that can be taken as &amp;formula; . through each such g there is a single d belonging to F ; label the foci of these d , none of which can lie in d , respectively &amp;formula; . then those d whose foci are in d join its points to the respective triads &amp;formula; . thus the join of every pair of points 1 is on Q and , there being no solid on Q , the points 1 lie in a plane d&amp;prime; whose lines consist of the triads 2 . each of the 72 d has , it is now clear , a twin d&amp;prime; coupled with it by F . the correlation between d and d&amp;prime; is shown by 1 and 2 or , alternatively , by 1 and 2 . those d that pass one through each line of d&amp;prime; have for their foci the points of d correlative to these lines ; if d passes , say , through 1 3 5 its focus is the point 5 common to those d whose foci are 1 , 3 , 5 . since , by the construction in &amp;symbol;4 , d and d&amp;prime; 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 d0 in points only , that the 15 d through a point m of d0 project , from m , the figure of 15 g in ( 4 ) passing three by three through 15 points ( ( 2 ) , &amp;symbol;&amp;symbol;13-15 ) . 