One method to generate random permutations involves using Gaussian elimination with partial pivoting (GEPP) on a random matrix A and storing the permutation matrix factor P from the resulting GEPP factorization PA=LU. We are interested in exploring properties of random butterfly permutations, which are generated using GEPP on specific random butterfly matrices. We address the questions of the longest increasing subsequence (LIS) and number of cycles for particular uniform butterfly permutations, with full distributional descriptions and limit theorems for simple butterfly permutations. We also establish scaling limit results and limit theorems for nonsimple butterfly permutations, which include certain p-Sylow subgroups of the symmetric group of N=pⁿ elements. For the LIS, we establish power-law bounds on the first moment N^αₚ ≤ E(LIS) ≤ N^βₚ where 1/2 < αₚ < βₚ < 1 and αₚ = 1 - oₚ(1), showing distinction from the typical O(N¹ᐟ²) expected LIS frequently encountered in the study of random permutations (e.g., uniform permutations, colored permutations, recent wreath product permutations studied by Diaconis). For the number of cycles scaled by (2-1/p)ⁿ, we establish a full CLT to a new limiting distribution depending on p with positive support we introduce that is uniquely determined by its positive moments that satisfy explicit recursive formulas; this thus determines a number of cycles CLT for any uniform p-Sylow subgroups of Sₚn. This work is joint with Chenyang Zhong, and continues from a UCSD probability seminar talk I gave last May to now include novel results and proof techniques from our recent preprint arXiv:2410.20952.