Formats: - HTML - YAML - JSON - 2026-07-19T03:00:48.874433
  • gps_subgroup_search • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '11337408.lq', 'ambient_counter': 303, 'ambient_order': 11337408, 'ambient_tex': 'C_3^7.S_3\\wr C_2^2', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': True, 'core_order': 708588, 'counter': 112, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '11337408.lq.16.A', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '16.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '16.11', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 11, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 16, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times D_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': None, 'subgroup_hash': 4131991524667752810, 'subgroup_order': 708588, 'subgroup_tex': 'C_9^4.C_3.C_6^2', 'supersolvable': True, 'sylow': 0}
  • gps_subgroup_data • Show schema
    {'ambient': '11337408.lq', 'aut_centralizer_order': None, 'aut_label': '16.A', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': None, 'complements': None, 'conjugacy_class_count': 1, 'contained_in': None, 'contains': None, 'core': '16.A', 'coset_action_label': None, 'count': 1, 'diagramx': [164, 1841, 1873, 4949], 'generators': [1759320, 2717388, 419904, 4219572, 11111904, 8298720, 887760, 2, 3779136, 8001504, 4939560, 5040144, 4385664], 'label': '11337408.lq.16.A', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '16.A', 'normal_contained_in': [], 'normal_contains': [], 'normalizer': '1.a1', 'old_label': '16.a1', 'projective_image': '11337408.lq', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '16.A', 'subgroup_fusion': None, 'weyl_group': '11337408.lq'}
  • label None does not appear in gps_groups
  • gps_groups • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '24.15', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 72, 'aut_gen_orders': [18, 18, 36], 'aut_gens': [[1, 6, 36, 216, 3888, 69984, 1259712], [1925254, 4750095, 7341972, 1811916, 10100160, 5062932, 3456], [4369621, 9408822, 9272196, 4949532, 2296404, 7261704, 1275264], [10298086, 8352771, 4959696, 9571932, 1924128, 962280, 1275264]], 'aut_group': '22674816.mj', 'aut_hash': 8826279076269810851, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 22674816, 'aut_permdeg': 2016, 'aut_perms': 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69984, 8], [18, 104976, 4], [36, 104976, 18], [36, 314928, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '11337408.lq', 'commutator_count': 1, 'commutator_label': None, 'complements_known': False, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 17, 'conjugacy_classes_known': False, 'counter': 303, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 36, 1, 1], [2, 162, 1, 3], [2, 972, 1, 2], [2, 2916, 1, 2], [2, 6561, 1, 1], [3, 8, 1, 4], [3, 16, 1, 1], [3, 18, 2, 2], [3, 32, 1, 1], [3, 36, 1, 1], [3, 72, 2, 5], [3, 81, 2, 1], [3, 108, 2, 1], [3, 144, 1, 1], [3, 162, 1, 2], [3, 216, 2, 1], [4, 17496, 1, 2], [4, 52488, 1, 1], [4, 78732, 1, 2], [4, 236196, 1, 1], [6, 72, 1, 3], [6, 108, 2, 3], [6, 144, 1, 3], [6, 162, 2, 2], [6, 216, 2, 6], [6, 288, 1, 1], [6, 324, 1, 1], [6, 324, 2, 6], [6, 432, 2, 3], [6, 648, 1, 6], [6, 648, 2, 11], [6, 972, 2, 13], [6, 1296, 1, 1], [6, 1458, 2, 7], [6, 1944, 2, 9], [6, 2916, 1, 5], [6, 2916, 2, 7], [6, 3888, 1, 4], [6, 5832, 1, 1], [6, 5832, 2, 6], [6, 6561, 2, 1], [6, 8748, 2, 11], [6, 11664, 1, 2], [6, 11664, 2, 2], [6, 13122, 1, 2], [6, 13122, 2, 2], [6, 17496, 1, 2], [6, 26244, 1, 1], [6, 26244, 2, 2], [9, 24, 1, 1], [9, 48, 1, 3], [9, 72, 1, 3], [9, 72, 2, 5], [9, 96, 1, 3], [9, 144, 1, 1], [9, 144, 2, 6], [9, 192, 1, 1], [9, 216, 2, 6], [9, 288, 1, 6], [9, 288, 2, 8], [9, 432, 1, 1], [9, 432, 2, 15], [9, 576, 1, 1], [9, 648, 2, 11], [9, 864, 1, 2], [9, 864, 2, 13], [9, 1296, 1, 5], [9, 1296, 2, 4], [9, 1728, 1, 1], [9, 1728, 2, 4], [9, 2592, 1, 2], [9, 2592, 2, 11], [9, 5184, 1, 1], [9, 5184, 2, 1], [12, 17496, 2, 2], [12, 34992, 1, 2], [12, 34992, 2, 2], [12, 52488, 2, 6], [12, 78732, 2, 2], [12, 104976, 1, 1], [12, 157464, 1, 2], [12, 157464, 2, 3], [12, 236196, 2, 1], [18, 216, 1, 3], [18, 216, 2, 3], [18, 432, 1, 6], [18, 432, 2, 6], [18, 648, 2, 18], [18, 864, 1, 3], [18, 864, 2, 3], [18, 1296, 1, 3], [18, 1296, 2, 30], [18, 1944, 1, 3], [18, 1944, 2, 30], [18, 2592, 1, 3], [18, 2592, 2, 12], [18, 3888, 1, 3], [18, 3888, 2, 39], [18, 5832, 1, 3], [18, 5832, 2, 36], [18, 7776, 1, 1], [18, 7776, 2, 13], [18, 11664, 1, 8], [18, 11664, 2, 14], [18, 17496, 1, 1], [18, 17496, 2, 13], [18, 23328, 1, 2], [18, 23328, 2, 2], [18, 34992, 1, 4], [18, 34992, 2, 8], [18, 69984, 1, 4], [18, 69984, 2, 2], [18, 104976, 2, 2], [36, 104976, 1, 2], [36, 104976, 2, 8], [36, 314928, 1, 1], [36, 314928, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': None, 'exponent': 36, 'exponents_of_order': [11, 6], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[24, 0, 64], [24, 1, 8], [48, 0, 96], [48, 1, 12], [72, 0, 16], [72, 1, 12], [96, 0, 48], [96, 1, 6], [144, 0, 6], [144, 1, 5], [192, 0, 8], [192, 1, 1], [288, 0, 20], [288, 1, 12], [432, 1, 4], [576, 1, 1], [864, 1, 4], [1728, 1, 1]], 'familial': False, 'frattini_label': '81.15', 'frattini_quotient': '139968.bv', 'hash': 1146316497117696609, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [18, 36, 6, 18, 18, 18, 9], 'inner_gens': [[1, 10612614, 4082436, 232740, 1331424, 11092464, 979776], [817717, 6, 1875492, 4951692, 7312788, 7259112, 8856864], [8584705, 11262318, 36, 3688200, 7601040, 2869344, 6298560], [10473949, 8000106, 8983044, 216, 625968, 4968864, 10077696], [6374593, 4081866, 3810276, 707832, 3888, 6228576, 1259712], [11162449, 6072846, 9797796, 7698456, 6442416, 69984, 10077696], [1539649, 3810246, 6298596, 2519640, 3888, 2589408, 1259712]], 'inner_hash': 1146316497117696609, 'inner_nilpotent': False, 'inner_order': 11337408, 'inner_split': True, 'inner_tex': 'C_3^7.S_3\\wr C_2^2', 'inner_used': [1, 2, 4], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 24], [2, 66], [4, 90], [8, 300], [16, 153], [24, 72], [32, 66], [48, 108], [64, 3], [72, 28], [96, 54], [144, 11], [192, 9], [288, 32], [432, 4], [576, 1], [864, 4], [1728, 1]], 'label': '11337408.lq', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': None, 'name': 'C3^7.S3wrC2^2', 'ngens': 17, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 611, 'number_characteristic_subgroups': 41, 'number_conjugacy_classes': 1026, 'number_divisions': 585, 'number_normal_subgroups': 125, 'number_subgroup_autclasses': None, 'number_subgroup_classes': None, 'number_subgroups': None, 'old_label': None, 'order': 11337408, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 14859], [3, 2186], [4, 481140], [6, 779022], [9, 174960], [12, 3061800], [18, 3989088], [36, 2834352]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[374290, 8228811, 4221408, 8001720, 5697864, 5676480, 1259712]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': None, 'perfect': False, 'permutation_degree': 36, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 8], [2, 22], [4, 36], [8, 60], [16, 159], [24, 8], [32, 73], [48, 44], [64, 31], [72, 12], [96, 54], [128, 1], [144, 13], [192, 25], [288, 15], [384, 4], [432, 4], [576, 11], [864, 4], [1728, 1]], 'representations': {'PC': {'code': '87872021495828688998845464471946453170743544580200687724041557926678090147533058938370075103282472426752011398322343941375849994268077370792691399659604557553271797644797183872561912330634779755905580752900544622819288201446173493667564357237939426701114782798644598183272859340046703303323414893067972150589164794843373266963256863285635061413681844681037146286593101370753082460079846146969274826403436353737626857834430917547415977838946158467805914632885268424697742963910706132728491397857406195133296551519178140212735635280499538347996054477054450253973625493899694850593982249555725446052258577497406711096890415969988284385059611517223367433432369834962672448183645954496997780597058292927105510255', 'gens': [1, 3, 5, 7, 10, 13, 16], 'pres': [17, 2, 3, 2, 3, 2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 3, 3, 34, 71383698, 541243316, 208492660, 138, 480615027, 343435244, 49421502, 347007064, 476994351, 26569508, 36980410, 242, 42837557, 214154734, 7934007, 39669284, 27696066, 167962811, 98208598, 47593155, 12191624, 6944812, 346, 693743623, 22208297, 34330, 3339, 3356, 534, 585346184, 10707594, 11092, 226342089, 702270026, 207195703, 8206980, 35893877, 17965354, 492771, 502, 26820298, 354076299, 79006796, 605941, 19711374, 9896135, 53968, 741, 86189195, 258567597, 793214, 264463, 44177, 2451434556, 796522925, 267377338, 194619735, 17614664, 13103629, 5083998, 354207, 288587, 72466, 658, 2098680205, 66624798, 17183, 19432288, 4626801, 3238802, 1233907, 68710, 948, 963679694, 55128, 5948722, 991556, 55247, 266499087, 685283360, 401511217, 142767426, 47589203, 33312484, 12690549, 39387, 1086, 121352272, 184275698, 182028451, 60676212, 10112806, 31432]}, 'Perm': {'d': 36, 'gens': [159556224623760164928697900143247054898163, 265626768539161335327960573900821253882238, 351025007822651663201027327447747742014012]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 64, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^7.S_3\\wr C_2^2', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
  • gps_groups • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 2, 2, 2, 2, 2], 'aut_gens': [[126, 55, 289, 288], [127, 55, 289, 288], [126, 265, 1, 288], [54, 127, 289, 288], [414, 55, 289, 288], [127, 54, 289, 288], [414, 265, 289, 288]], 'aut_group': '64.138', 'aut_hash': 138, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 64, 'aut_permdeg': 8, 'aut_perms': [2309, 526, 5329, 3043, 12316, 18498], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 4, 1], [4, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\wr C_2^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr C_2', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 4], [4, 2, 2]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 11, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 4], [4, 2, 1, 2]], 'element_repr_type': 'Perm', 'elementary': 2, 'eulerian_function': 21, 'exponent': 4, 'exponents_of_order': [4], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '8.5', 'hash': 11, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 1, 1], 'inner_gens': [[126, 265, 289, 288], [414, 55, 289, 288], [126, 55, 289, 288], [126, 55, 289, 288]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 2]], 'label': '16.11', 'linC_count': 8, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 8, 'linQ_dim': 3, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*D4', 'ngens': 4, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 10, 'number_divisions': 10, 'number_normal_subgroups': 19, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 27, 'number_subgroups': 35, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 11], [4, 4]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 151, 0], 'outer_gens': [[54, 127, 289, 288], [127, 55, 289, 288], [415, 54, 1, 288], [127, 54, 289, 288]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [126, 55, 289, 288], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2]], 'representations': {'PC': {'code': 8772, 'gens': [1, 2, 3], 'pres': [4, -2, 2, 2, -2, 78, 34]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [16322, 16432, 3198]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [8912, 8156, 13286, 14044]}, 'Perm': {'d': 6, 'gens': [126, 55, 289, 288]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times D_4', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}