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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1728.47271', 'ambient_counter': 47271, 'ambient_order': 1728, 'ambient_tex': 'C_{12}.D_6^2', 'central': False, 'central_factor': False, 'centralizer_order': 192, 'characteristic': False, 'core_order': 2, 'counter': 1088, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '1728.47271.432.j1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '432.j1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 432, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '4.1', 'subgroup_hash': 1, 'subgroup_order': 4, 'subgroup_tex': 'C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1728.47271', 'aut_centralizer_order': None, 'aut_label': '432.j1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '9.c1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['144.bl1', '144.bq1', '144.bu1', '144.bz1', '216.i1', '216.p1', '216.t1', '216.x1', '216.bb1', '216.be1'], 'contains': ['864.a1'], 'core': '864.a1', 'coset_action_label': None, 'count': 18, 'diagramx': None, 'generators': [134], 'label': '1728.47271.432.j1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '48.l1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.c1', 'old_label': '432.j1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '432.j1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [3]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 1, 2]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [2], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4]], 'label': '4.1', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 1], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3]], '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': 1, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -2, -2, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [56, 15], 'family': 'CSOPlus'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [21]}, 'Perm': {'d': 4, 'gens': [22, 7]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [12, 12, 2, 6, 6, 12, 2, 4, 6], 'aut_gens': [[1, 2, 4, 24, 144], [3, 2, 452, 266, 144], [873, 866, 1350, 840, 1584], [9, 866, 502, 842, 720], [11, 2, 100, 698, 146], [873, 2, 100, 1562, 1008], [883, 866, 484, 1706, 1008], [865, 2, 964, 1562, 720], [11, 2, 1316, 1128, 722], [867, 2, 118, 314, 1008]], 'aut_group': '3888.jx', 'aut_hash': 2380107896618026724, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 221184, 'aut_permdeg': 60, 'aut_perms': [676421315749975111006004860413828010942784107729215645085908877637544365363783330, 5648040316680014215177554008590850825896344921789842667382313633059894203620649519, 5370670537292826591087987916482274759432049775417241393755244201891414511087964159, 5422545658177315970397957055722642194209632082231502437160943109215882213069886210, 5681191352793707688097665208657235047274130582456547634063679167305871486900438983, 2044692630326094210782427737369490680069816692353167743338781043687740269613962475, 1243331191135879908901187221706737716550400117174165128766959015379692541154657223, 5528160966470304806964886893156645509898201187077003244385017840337435092289848450, 1800518006822276218289745488797687490925302666724917749609328014645287435733568338], 'aut_phi_ratio': 384.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 3, 4, 1], [2, 6, 4, 1], [2, 18, 2, 1], [2, 18, 4, 1], [2, 54, 2, 1], [3, 2, 1, 3], [3, 4, 1, 3], [3, 8, 1, 1], [4, 2, 2, 1], [4, 6, 2, 1], [4, 6, 4, 1], [4, 9, 4, 1], [4, 18, 4, 1], [4, 27, 4, 1], [6, 2, 1, 3], [6, 2, 2, 3], [6, 4, 1, 3], [6, 4, 2, 3], [6, 6, 4, 2], [6, 8, 1, 1], [6, 8, 2, 1], [6, 12, 4, 3], [6, 24, 4, 1], [6, 36, 2, 1], [6, 36, 4, 1], [12, 4, 2, 3], [12, 4, 4, 1], [12, 8, 2, 2], [12, 8, 4, 1], [12, 12, 2, 2], [12, 12, 4, 3], [12, 18, 4, 1], [12, 24, 4, 1], [12, 36, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3\\wr C_4:D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '512.6249567', 'autcent_hash': 6452065007057432321, 'autcent_nilpotent': True, 'autcent_order': 512, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '432.759', 'autcentquo_hash': 759, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times S_3^3', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 3, 4], [2, 6, 4], [2, 18, 6], [2, 54, 2], [3, 2, 3], [3, 4, 3], [3, 8, 1], [4, 2, 2], [4, 6, 6], [4, 9, 4], [4, 18, 4], [4, 27, 4], [6, 2, 9], [6, 4, 9], [6, 6, 8], [6, 8, 3], [6, 12, 12], [6, 24, 4], [6, 36, 6], [12, 4, 10], [12, 8, 8], [12, 12, 16], [12, 18, 4], [12, 24, 4], [12, 36, 4]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '432.759', 'commutator_count': 1, 'commutator_label': '54.15', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 47271, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['144.139', 1], ['2.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 1, 4], [2, 6, 1, 4], [2, 18, 1, 6], [2, 54, 1, 2], [3, 2, 1, 3], [3, 4, 1, 3], [3, 8, 1, 1], [4, 2, 1, 2], [4, 6, 1, 6], [4, 9, 2, 2], [4, 18, 1, 4], [4, 27, 2, 2], [6, 2, 1, 9], [6, 4, 1, 9], [6, 6, 1, 8], [6, 8, 1, 3], [6, 12, 1, 12], [6, 24, 1, 4], [6, 36, 1, 6], [12, 4, 1, 6], [12, 4, 2, 2], [12, 8, 1, 4], [12, 8, 2, 2], [12, 12, 1, 12], [12, 12, 2, 2], [12, 18, 2, 2], [12, 24, 1, 4], [12, 36, 1, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 19998720000, 'exponent': 12, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '864.4704', 'hash': 47271, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 1, 6, 6, 6], 'inner_gens': [[1, 2, 20, 24, 144], [1, 2, 4, 24, 144], [9, 2, 4, 120, 1008], [1, 2, 916, 24, 1584], [1, 2, 868, 312, 144]], 'inner_hash': 759, 'inner_nilpotent': False, 'inner_order': 432, 'inner_split': False, 'inner_tex': 'C_2\\times S_3^3', 'inner_used': [1, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 56], [4, 44], [8, 12]], 'label': '1728.47271', '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': True, 'metacyclic': False, 'monomial': True, 'name': 'C12.D6^2', 'ngens': 9, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 58, 'number_characteristic_subgroups': 80, 'number_conjugacy_classes': 144, 'number_divisions': 132, 'number_normal_subgroups': 684, 'number_subgroup_autclasses': 1112, 'number_subgroup_classes': 4178, 'number_subgroups': 29236, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 255], [3, 26], [4, 256], [6, 582], [12, 608]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 4, 2, 4, 2, 2, 2], 'outer_gen_pows': [544, 1504, 448, 0, 16, 584, 0], 'outer_gens': [[881, 866, 102, 890, 720], [11, 866, 1398, 1608, 144], [17, 2, 484, 840, 720], [873, 2, 1318, 1704, 722], [17, 2, 6, 890, 144], [873, 2, 100, 1562, 1008], [1, 2, 116, 696, 1584]], 'outer_group': '512.519620', 'outer_hash': 1407549917715648783, 'outer_nilpotent': True, 'outer_order': 512, 'outer_permdeg': 64, 'outer_perms': [82541319774199736642561197913923461666008543466036066100744820741907261478478265279720069, 81452404607988215914817063575214044183643377401668886000803737178122118265581484064557492, 158240884496723156629866695619016410752708522093245659099190981079596438900848341913, 48837402216470427699894866629796210956661730166817764495407620254730257976222048528361692, 14777899825443685722762976026658374950670681437580064045493671204018181565701235368160479, 322841132253921856755165880271971075613384016546792050801744640289478087926183676436033, 32262693616481217654810540052933682036788201221567215445149956197742775908880340147054164], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^6:D_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 48], [4, 36], [8, 14], [16, 2]], 'representations': {'PC': {'code': 6391719674499612757457398147173445738218237978225797161, 'gens': [1, 2, 3, 5, 7], 'pres': [9, -2, -2, -2, -3, -2, -3, -2, -2, -3, 542, 74, 579, 1372, 130, 12983, 1022, 15900, 4200, 186, 4363, 214, 3932]}, 'Perm': {'d': 19, 'gens': [355687431863040, 1313910120264, 1307687276281, 1307678135040, 1307691197544, 3767040, 6706022400, 6758061133824000, 3]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}.D_6^2', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
