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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '11664.lh', 'ambient_counter': 294, 'ambient_order': 11664, 'ambient_tex': '(A_4\\times C_3^3).S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 54, 'characteristic': False, 'core_order': 12, 'counter': 933, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '11664.lh.324.dl1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '324.dl1', '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': 324, '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': '36.11', 'subgroup_hash': 11, 'subgroup_order': 36, 'subgroup_tex': 'C_3\\times A_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '11664.lh', 'aut_centralizer_order': None, 'aut_label': '324.dl1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '216.q1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['108.cz1', '108.dx1', '108.dx2', '162.bg1'], 'contains': ['972.b1', '972.bs1', '972.by1', '1296.bl1'], 'core': '972.b1', 'coset_action_label': None, 'count': 18, 'diagramx': [723, -1, 8277, -1], 'generators': [5438, 1776, 6156, 5832], 'label': '11664.lh.324.dl1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '12.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '18.r1', 'old_label': '324.dl1', 'projective_image': '11664.lh', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '324.dl1', 'subgroup_fusion': None, 'weyl_group': '12.3'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '9.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 3, 3, 2, 2], 'aut_gens': [[1, 3, 6], [13, 3, 30], [2, 18, 15], [1, 21, 27], [25, 3, 6], [22, 3, 6], [19, 3, 6]], 'aut_group': '144.183', 'aut_hash': 183, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 144, 'aut_permdeg': 7, 'aut_perms': [1, 120, 144, 3, 744, 1680], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 1], [3, 4, 6, 1], [6, 3, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 3, 1], [3, 1, 2], [3, 4, 6], [6, 3, 2]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '12.3', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 11, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 1], [3, 4, 2, 3], [6, 3, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 4, 'exponent': 6, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[3, 0, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '36.11', 'hash': 11, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 2, 2], 'inner_gens': [[1, 21, 27], [19, 3, 6], [22, 3, 6]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'A_4', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 6, 'irrep_stats': [[1, 9], [3, 3]], 'label': '36.11', 'linC_count': 2, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 3, 'linQ_dim': 5, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3*A4', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 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': 12, 'number_divisions': 7, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 10, 'number_subgroup_classes': 14, 'number_subgroups': 30, 'old_label': None, 'order': 36, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [3, 26], [6, 6]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[2, 18, 27], [26, 18, 15]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [7, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4], [3, 1], [6, 1]], 'representations': {'PC': {'code': 107356617, 'gens': [1, 2, 3], 'pres': [4, -3, -2, 2, -3, 169, 326, 34]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [706461792882, 48233911639]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [16247572, 34591248, 17540971, 23068812]}, 'Perm': {'d': 7, 'gens': [147, 3, 744, 1680]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times A_4', 'transitive_degree': 12, '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': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 36, 'aut_gen_orders': [12, 12, 6, 6], 'aut_gens': [[1, 6, 36, 108, 648], [5809, 7950, 7812, 9936, 9468], [9757, 8670, 36, 5940, 4968], [8497, 9222, 252, 540, 7200], [10357, 3462, 288, 10152, 7236]], 'aut_group': None, 'aut_hash': 6340318746942157276, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 23328, 'aut_permdeg': 84, 'aut_perms': [1447821512450998190763138844660408238226268841910772118223815936672808325505009340808162201531603916148323272997449641762872115, 2710722985557940923777116139195688749718885827282592198483848858207493416689872240560719109390946560420858874364151765438881827, 645821282143115917807825461374464059590105826375714923619517493091947900552510202048503364839305135789367453510024440441937787, 603363721986747111848936351771124935557981845068750075612808054436890868120890512923818345863203236409590138710345164273427654], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 27, 1, 1], [2, 54, 1, 1], [2, 81, 1, 1], [2, 162, 1, 1], [3, 2, 1, 1], [3, 3, 1, 2], [3, 6, 1, 3], [3, 8, 1, 1], [3, 16, 1, 1], [3, 18, 1, 6], [3, 24, 1, 2], [3, 48, 1, 3], [3, 72, 2, 3], [3, 144, 1, 3], [4, 54, 1, 1], [4, 162, 1, 1], [6, 6, 1, 1], [6, 9, 1, 2], [6, 18, 1, 3], [6, 27, 1, 2], [6, 54, 1, 9], [6, 81, 1, 2], [6, 108, 1, 1], [6, 162, 1, 7], [6, 216, 1, 3], [6, 216, 2, 3], [6, 324, 1, 6], [9, 36, 1, 3], [9, 144, 2, 3], [12, 108, 1, 1], [12, 162, 1, 4], [12, 324, 1, 6], [18, 108, 1, 3]], 'aut_supersolvable': False, 'aut_tex': 'C_6^2.C_3^4.C_2^3', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 36, 'autcentquo_group': None, 'autcentquo_hash': 6340318746942157276, 'autcentquo_nilpotent': False, 'autcentquo_order': 23328, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_6^2.C_3^4.C_2^3', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 27, 1], [2, 54, 1], [2, 81, 1], [2, 162, 1], [3, 2, 1], [3, 3, 2], [3, 6, 3], [3, 8, 1], [3, 16, 1], [3, 18, 6], [3, 24, 2], [3, 48, 3], [3, 72, 6], [3, 144, 3], [4, 54, 1], [4, 162, 1], [6, 6, 1], [6, 9, 2], [6, 18, 3], [6, 27, 2], [6, 54, 9], [6, 81, 2], [6, 108, 1], [6, 162, 7], [6, 216, 9], [6, 324, 6], [9, 36, 3], [9, 144, 6], [12, 108, 1], [12, 162, 4], [12, 324, 6], [18, 108, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '11664.lh', 'commutator_count': 1, 'commutator_label': '972.525', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 294, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 27, 1, 1], [2, 54, 1, 1], [2, 81, 1, 1], [2, 162, 1, 1], [3, 2, 1, 1], [3, 3, 2, 1], [3, 6, 1, 1], [3, 6, 2, 1], [3, 8, 1, 1], [3, 16, 1, 1], [3, 18, 1, 2], [3, 18, 2, 2], [3, 24, 2, 1], [3, 48, 1, 1], [3, 48, 2, 1], [3, 72, 1, 2], [3, 72, 2, 2], [3, 144, 1, 1], [3, 144, 2, 1], [4, 54, 1, 1], [4, 162, 1, 1], [6, 6, 1, 1], [6, 9, 2, 1], [6, 18, 1, 1], [6, 18, 2, 1], [6, 27, 2, 1], [6, 54, 1, 3], [6, 54, 2, 3], [6, 81, 2, 1], [6, 108, 1, 1], [6, 162, 1, 1], [6, 162, 2, 3], [6, 216, 1, 3], [6, 216, 2, 3], [6, 324, 1, 2], [6, 324, 2, 2], [9, 36, 1, 1], [9, 36, 2, 1], [9, 144, 1, 2], [9, 144, 2, 2], [12, 108, 1, 1], [12, 162, 2, 2], [12, 324, 1, 2], [12, 324, 2, 2], [18, 108, 1, 1], [18, 108, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 21228480, 'exponent': 36, 'exponents_of_order': [6, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[54, 1, 2]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '1296.3547', 'hash': 7205121969201056384, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [6, 6, 3, 6, 18], 'inner_gens': [[1, 30, 7848, 2052, 3672], [13, 6, 504, 2484, 6624], [3961, 294, 36, 108, 864], [9721, 9942, 36, 108, 4536], [5401, 2130, 8244, 7884, 648]], 'inner_hash': 7205121969201056384, 'inner_nilpotent': False, 'inner_order': 11664, 'inner_split': True, 'inner_tex': '(A_4\\times C_3^3).S_3^2', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': 54, 'irrQ_degree': 54, 'irrQ_dim': 54, 'irrR_degree': 54, 'irrep_stats': [[1, 12], [2, 30], [3, 12], [4, 12], [6, 18], [12, 6], [18, 8], [36, 1], [54, 2]], 'label': '11664.lh', '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': True, 'name': '(A4*C3^3).S3^2', 'ngens': 10, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 92, 'number_characteristic_subgroups': 53, 'number_conjugacy_classes': 101, 'number_divisions': 69, 'number_normal_subgroups': 65, 'number_subgroup_autclasses': 1377, 'number_subgroup_classes': 1585, 'number_subgroups': 64620, 'old_label': None, 'order': 11664, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 327], [3, 1214], [4, 216], [6, 5910], [9, 972], [12, 2700], [18, 324]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[1, 1770, 36, 4428, 7632]], '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': 5, 'perfect': False, 'permutation_degree': 31, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 14], [3, 4], [4, 14], [6, 10], [8, 4], [12, 8], [18, 4], [24, 2], [36, 3], [54, 2]], 'representations': {'PC': {'code': '254164295204174312159417022600326061963595825288706024822308677655702546108419280246034515908061710762069538847573280951312457127623835180947605984619', 'gens': [1, 3, 5, 6, 8], 'pres': [10, 2, 3, 2, 3, 3, 2, 3, 2, 3, 3, 20, 902, 82, 963, 392404, 195314, 4224, 34384, 123125, 24865, 11915, 175, 287286, 50426, 23976, 293767, 336977, 88347, 56917, 1967, 3417, 3187, 237, 252728, 233298, 85348, 69698, 23808, 7618, 2228, 358, 777609, 129629]}, 'Perm': {'d': 31, 'gens': [44529263917233201152911382152440, 292725650025632391666438937448880, 38134660218354381718469509833000, 29084237993223704155593274163280, 842833822099010828689749882719520, 44529263917233201152911382152447, 568404082391255487669183792536880, 73908593275086190435173786216736, 56221120753824761726941184957284, 65371314463054345480021120446602]}}, 'schur_multiplier': [2, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '(A_4\\times C_3^3).S_3^2', 'transitive_degree': 108, 'wreath_data': None, 'wreath_product': False}
