Character Table of The Subgroup 2^7: G_2(2) of The Automorphism Group 〖Fi〗_22 by Fischer Clifford Matrices

The subgroup  : of the Automorphism group   is a split extension group where the two component groups include the abelian group  with order 2 and dimension 7 and the Chevalley group    over the finite field with 2 elements. From the ATLAS of Finite Groups, the order of the subgroup    was calculated as 1,548,288. The character table for the group  was determined in the same ATLAS but the character table for  was not computed. The general objective of this research was to construct the Character table of the subgroup  of the Automorphism Group  using Fischer Clifford Matrices. The method employed was Fischer Clifford Matrices technique where the properties of Fischer Clifford Matrices were used to find the entries of the Fischer Clifford Matrices M for each class representative   G = . The character table of  =: was constructed from the Fischer Clifford Matrices of  :  and the ordinary characters of the inertia factor groups . Therefore, the conjugacy classes of the subgroup :  were obtained and the Fischer Clifford Matrices were computed. Eventually, the ordinary Character table of  (2) associated with these matrices that contains information about the irreducible representations of the group, their dimensions and their characters was constructed. It was realized that the subgroup :(2) has 60 Conjugacy Classes, 16 sets of Fischer Clifford Matrices of dimensions varying between 2 and 8 and a character table which is a square matrix of order 60. The character table for the subgroup   = : would be important in coding theory/error correcting codes (communication channel), crystallography and cryptography. It was recommended that the analysis of error-correcting codes based on the structure of :  be developed, possibility of using  (2) in new cryptographic and cryptollographic protocols be investigated and the methods used here be extended to the study of other maximal subgroups of Aut().