New techniques, both theoretical and practical, are presented for constructing permutation representations for computing with matrix groups defined over finite fields. The permutation representation is constructed on a conjugacy class of subgroups of prime order. We construct a base for the permutation representation, which in turn simplifies the computation of a strong generating set. In addition, we present an elementary test for checking the simplicity of the permutation image.The theory has been successfully tested on a representation of the sporadic simple group Ly, discovered by Lyons (1972). With no a priori assumptions, we find a permutation representation of degree 9 606 125 on a conjugacy class of subgroups of order 3, find the order of the resulting permutation group, and verify simplicity. A Monte Carlo variation of the algorithm was used to achieve better space and time efficiency. The construction of the permutation representation required four CPU days on a SPARCserver 670MP with 64 MB. The permutation representation was used implicitly in the sense that the group element was stored as a matrix, and its permutation action on a "point" was determined using a pre-computed data structure. Thus, additional computations required little additional space. The algorithm has also been implemented using the MasPar MP-1 SIMD parallel computer and 8 SPARC-2's running under MPI. The results of those parallel experiments are briefly reviewed.
Utilizing bicrystallography in two dimensions (2D), the symmetries of migration related segments of Coincidence Site Lattice (CSL) boundaries are predicted for projections along their [001] tilt axis in grain boundaries of crystalline materials that possess the holohedral point symmetry of the cubic system (i.e. m3m). These kinds of "edge-on" projections are typical for atomic resolution imaging of such tilt boundaries with Transmission Electron Microscopes (TEM). Such images from a recently published aberration-corrected Z-contrast scanning TEM investigation [H. Yang et al., Phil. Mag. 93 (2013) 1219 and other studies facilitate the direct visual confirmation of our frieze symmetry predictions with experimental results.
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