{"group": "H_k=<A_t,B_t,C_t>", "order": 2, "log2": 1, "involutions": 1, "d": 1, "abelianization_invariants": [2], "num_gens_used": 3, "k": 1}
{"group": "Q_k (claimed closure)", "order": 2, "log2": 1, "involutions": 1, "d": 1, "abelianization_invariants": [2], "num_gens_used": 3, "k": 1}
{"group": "check H_k == Q_k", "equal": true, "H_subset_Q": true, "k": 1}
{"group": "P_k = image of SL_2-Sylow in PGL_2", "order": 2, "log2": 1, "involutions": 1, "d": 1, "abelianization_invariants": [2], "num_gens_used": 3, "k": 1}
{"group": "I_k = PGL_2-Sylow (Iwahori)", "order": 2, "log2": 1, "involutions": 1, "d": 1, "abelianization_invariants": [2], "num_gens_used": 3, "k": 1}
{"group": "S_k = SL_2-Sylow", "order": 2, "log2": 1, "involutions": 1, "d": 1, "abelianization_invariants": [2], "num_gens_used": 3, "k": 1}
{"group": "S_k/{+-I}", "order": 2, "log2": 1, "involutions": 1, "d": 1, "abelianization_invariants": [2], "num_gens_used": 3, "k": 1}
{"group": "H_k=<A_t,B_t,C_t>", "order": 8, "log2": 3, "involutions": 5, "d": 2, "abelianization_invariants": [2, 2], "num_gens_used": 3, "k": 2}
{"group": "Q_k (claimed closure)", "order": 8, "log2": 3, "involutions": 5, "d": 2, "abelianization_invariants": [2, 2], "num_gens_used": 3, "k": 2}
{"group": "check H_k == Q_k", "equal": true, "H_subset_Q": true, "k": 2}
{"group": "P_k = image of SL_2-Sylow in PGL_2", "order": 8, "log2": 3, "involutions": 5, "d": 2, "abelianization_invariants": [2, 2], "num_gens_used": 2, "k": 2}
{"group": "I_k = PGL_2-Sylow (Iwahori)", "order": 16, "log2": 4, "involutions": 11, "d": 3, "abelianization_invariants": [2, 2, 2], "num_gens_used": 4, "k": 2}
{"group": "S_k = SL_2-Sylow", "order": 16, "log2": 4, "involutions": 7, "d": 2, "abelianization_invariants": [2, 4], "num_gens_used": 4, "k": 2}
{"group": "S_k/{+-I}", "order": 8, "log2": 3, "involutions": 5, "d": 2, "abelianization_invariants": [2, 2], "num_gens_used": 2, "k": 2}
{"group": "H_k=<A_t,B_t,C_t>", "order": 32, "log2": 5, "involutions": 15, "d": 3, "abelianization_invariants": [2, 2, 2], "num_gens_used": 3, "k": 3}
{"group": "Q_k (claimed closure)", "order": 32, "log2": 5, "involutions": 15, "d": 3, "abelianization_invariants": [2, 2, 2], "num_gens_used": 5, "k": 3}
{"group": "check H_k == Q_k", "equal": true, "H_subset_Q": true, "k": 3}
{"group": "P_k = image of SL_2-Sylow in PGL_2", "order": 32, "log2": 5, "involutions": 7, "d": 2, "abelianization_invariants": [2, 4], "num_gens_used": 3, "k": 3}
{"group": "I_k = PGL_2-Sylow (Iwahori)", "order": 128, "log2": 7, "involutions": 55, "d": 4, "abelianization_invariants": [2, 2, 2, 2], "num_gens_used": 5, "k": 3}
{"group": "S_k = SL_2-Sylow", "order": 128, "log2": 7, "involutions": 15, "d": 2, "abelianization_invariants": [4, 4], "num_gens_used": 4, "k": 3}
{"group": "S_k/{+-I}", "order": 64, "log2": 6, "involutions": 15, "d": 2, "abelianization_invariants": [2, 4], "num_gens_used": 4, "k": 3}
{"group": "H_k=<A_t,B_t,C_t>", "order": 256, "log2": 8, "involutions": 87, "d": 3, "abelianization_invariants": [2, 2, 2], "num_gens_used": 3, "k": 4}
{"group": "Q_k (claimed closure)", "order": 256, "log2": 8, "involutions": 87, "d": 3, "abelianization_invariants": [2, 2, 2], "num_gens_used": 5, "k": 4}
{"group": "check H_k == Q_k", "equal": true, "H_subset_Q": true, "k": 4}
{"group": "P_k = image of SL_2-Sylow in PGL_2", "order": 256, "log2": 8, "involutions": 23, "d": 2, "abelianization_invariants": [2, 8], "num_gens_used": 6, "k": 4}
{"group": "I_k = PGL_2-Sylow (Iwahori)", "order": 1024, "log2": 10, "involutions": 199, "d": 4, "abelianization_invariants": [2, 2, 2, 2], "num_gens_used": 4, "k": 4}
{"group": "S_k = SL_2-Sylow", "order": 1024, "log2": 10, "involutions": 15, "d": 2, "abelianization_invariants": [4, 8], "num_gens_used": 3, "k": 4}
{"group": "S_k/{+-I}", "order": 512, "log2": 9, "involutions": 39, "d": 2, "abelianization_invariants": [2, 8], "num_gens_used": 3, "k": 4}
{"group": "H_k=<A_t,B_t,C_t>", "order": 2048, "log2": 11, "involutions": 327, "d": 3, "abelianization_invariants": [2, 2, 2], "num_gens_used": 3, "k": 5}
{"group": "Q_k (claimed closure)", "order": 2048, "log2": 11, "involutions": 327, "d": 3, "abelianization_invariants": [2, 2, 2], "num_gens_used": 4, "k": 5}
{"group": "check H_k == Q_k", "equal": true, "H_subset_Q": true, "k": 5}
{"group": "P_k = image of SL_2-Sylow in PGL_2", "order": 2048, "log2": 11, "involutions": 71, "d": 2, "abelianization_invariants": [2, 8], "num_gens_used": 7, "k": 5}
{"group": "I_k = PGL_2-Sylow (Iwahori)", "order": 8192, "log2": 13, "involutions": 775, "d": 4, "abelianization_invariants": [2, 2, 2, 2], "num_gens_used": 4, "k": 5}
{"group": "S_k = SL_2-Sylow", "order": 8192, "log2": 13, "involutions": 15, "d": 2, "abelianization_invariants": [4, 8], "num_gens_used": 2, "k": 5}
{"group": "S_k/{+-I}", "order": 4096, "log2": 12, "involutions": 135, "d": 2, "abelianization_invariants": [2, 8], "num_gens_used": 5, "k": 5}
{"group": "H_k=<A_t,B_t,C_t>", "order": 16384, "log2": 14, "involutions": 1287, "d": 3, "abelianization_invariants": [2, 2, 2], "num_gens_used": 3, "k": 6}
{"group": "Q_k (claimed closure)", "order": 16384, "log2": 14, "involutions": 1287, "d": 3, "abelianization_invariants": [2, 2, 2], "num_gens_used": 5, "k": 6}
{"group": "check H_k == Q_k", "equal": true, "H_subset_Q": true, "k": 6}
{"group": "P_k = image of SL_2-Sylow in PGL_2", "order": 16384, "log2": 14, "involutions": 263, "d": 2, "abelianization_invariants": [2, 8], "num_gens_used": 3, "k": 6}
{"group": "I_k = PGL_2-Sylow (Iwahori)", "order": 65536, "log2": 16, "involutions": 3079, "d": 4, "abelianization_invariants": [2, 2, 2, 2], "num_gens_used": 7, "k": 6}
{"group": "S_k = SL_2-Sylow", "order": 65536, "log2": 16, "involutions": 15, "d": 2, "abelianization_invariants": [4, 8], "num_gens_used": 5, "k": 6}
{"group": "S_k/{+-I}", "order": 32768, "log2": 15, "involutions": 519, "d": 2, "abelianization_invariants": [2, 8], "num_gens_used": 6, "k": 6}
{"group": "H_k=<A_t,B_t,C_t>", "order": 131072, "log2": 17, "involutions": 5127, "d": 3, "abelianization_invariants": [2, 2, 2], "num_gens_used": 3, "k": 7}
{"group": "Q_k (claimed closure)", "order": 131072, "log2": 17, "involutions": 5127, "d": 3, "abelianization_invariants": [2, 2, 2], "num_gens_used": 4, "k": 7}
{"group": "check H_k == Q_k", "equal": true, "H_subset_Q": true, "k": 7}
{"group": "P_k = image of SL_2-Sylow in PGL_2", "order": 131072, "log2": 17, "involutions": 1031, "d": 2, "abelianization_invariants": [2, 8], "num_gens_used": 4, "k": 7}
{"group": "I_k = PGL_2-Sylow (Iwahori)", "order": 524288, "log2": 19, "involutions": 12295, "d": 4, "abelianization_invariants": [2, 2, 2, 2], "num_gens_used": 6, "k": 7}
{"group": "S_k = SL_2-Sylow", "order": 524288, "log2": 19, "involutions": 15, "d": 2, "abelianization_invariants": [4, 8], "num_gens_used": 2, "k": 7}
{"group": "S_k/{+-I}", "order": 262144, "log2": 18, "involutions": 2055, "d": 2, "abelianization_invariants": [2, 8], "num_gens_used": 3, "k": 7}
