Talk:Conway group Co3

Representation

The Mathieu group M₂₃ occurs as a maximal subgroup of Co₃. One permutation matrix representation of M₂₃ fixes the type 3 vector (5,1²³). A non-monomial matrix is needed to complete generation of a representation of Co₃. Scott Tillinghast, Houston TX (talk) 15:52, 21 December 2015 (UTC)

A convenient non-monomial matrix would be an involution of trace 0 fixing both vectors (5,1²³) and (1,5,1²²). This would would also serve as a generator for a Higman-Sims group; it should move all 100 points of a Higman-Sims graqph. Clearly no HS group is a subgroup of the monomial subgroup of Co₀ becaue HS contains a Heisenberg group of order 125. I have not seen this problem addressed in the literature; the group theorists do not seem to prefer constructive proofs. Would this matrix have entries with denominator 4 or 6? I have seen no example of this. Scott Tillinghast, Houston TX (talk) 18:54, 20 May 2018 (UTC)

Simple subgroup of order 504?

That would be PSL(2,8). The Atlas of Finite Group Representations says that an element in Co₃ of class 3C has a centralizer of order 4536. That seems like 3 × PSL(2,8):3, which includes elements of order 9, not to be found in the monomial group 2¹²:M₂₄. An easiest representation may be one fixing the type 3 vector (-3,1⁷,-3,1⁷,-3,1⁷). Scott Tillinghast, Houston TX (talk) 22:01, 21 May 2018 (UTC)