CHARACTER TABLE OF THE MAXIMAL SUBGROUP 2^7:G_2(2) OF THE AFFINE SUBGROUP Sp_8(2) BY FISCHER-CLIFFORD MATRICES

Thesis title: CHARACTER TABLE OF THE MAXIMAL SUBGROUP 2^7:G_2(2) OF THE AFFINE SUBGROUP Sp_8(2) BY FISCHER-CLIFFORD MATRICES.

Student’s name: SIKOLIA MURUNGA JACINTA.

Supervisors’ Names:

1. Dr . LUCY CHIKAMAI
2. VINCENT MARANI
3. A.L. PRINS

Abstract:

The group 2^7: G_2(2), is an extension of an elementary abelian group of order 2^7 by the exceptional group G_2(2). There are eleven maximal subgroup conjugacy classes in the simple symplectic group Sp_8(2). The split extension group of the 2^7: G_2(2) of order 1548288 is one of the maximal subgroups Sp_8(2). In this study, the theory of Fischer-Clifford matrices to construct the character table of the maximal subgroup 2^7: G_2(2) of the affine subgroup Sp_8(2) is used. The Fischer-Clifford matrices technique is based on Clifford theorem. To use this technique, the Fischer matrices, the conjugacy classes, the character tables of the inertia factor groups and the fusion maps of the inertia factor groups into G are needed. When the character of 2^7: G_2(2) has been obtained, finally we fuse it into Sp_8(2). Four inertia factor groups for 2^7: G_2(2) were identified H_1= G_2(2), H_2=3^ (1+2):8:2, H_3=L_2(7):2 and H_4=4^2: D_12. The Fischer matrices are all integer-valued matrices with sizes ranging from -48 to 63, whereas the character table of 2^7: G_2(2) is 53 ×53. The character table of finite group provides considerable amount of information about the group and hence is of great importance in group theory, dealing with the symmetry of objects or variables.