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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '25992.bj', 'ambient_counter': 36, 'ambient_order': 25992, 'ambient_tex': 'D_{19}^2:C_{18}', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': True, 'core_order': 1444, 'counter': 19, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '25992.bj.18.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '18.b1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '18.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 18, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{18}', 'simple': False, 'solvable': True, 'special_labels': ['C3'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '1444.8', 'subgroup_hash': 8, 'subgroup_order': 1444, 'subgroup_tex': 'C_{19}^2:C_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '25992.bj', 'aut_centralizer_order': None, 'aut_label': '18.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '25992.a1.a1', 'complements': ['1444.a1.b1', '1444.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['6.b1.a1', '9.a1.a1'], 'contains': ['36.a1.a1', '6498.b1.a1'], 'core': '18.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [464, 5485, 887, 6429, 9716, 5495, 8674, 4784], 'generators': [18, 36, 1368, 20592], 'label': '25992.bj.18.b1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '18.b1.a1', 'normal_contained_in': ['6.b1.a1', '9.a1.a1'], 'normal_contains': ['36.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '18.b1.a1', 'projective_image': '25992.bj', 'quotient_action_image': '18.2', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '18.b1.a1', 'subgroup_fusion': None, 'weyl_group': '25992.bj'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 6840, 'aut_gen_orders': [18, 36, 36], 'aut_gens': [[1, 4, 76], [1131, 20, 1092], [855, 132, 440], [87, 1104, 756]], 'aut_group': None, 'aut_hash': 5898335423961875981, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 259920, 'aut_permdeg': 722, 'aut_perms': 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'aut_phi_ratio': 380.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 361, 1, 1], [4, 361, 2, 1], [19, 4, 90, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{19}^2.C_{360}.C_2', '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': 6840, 'autcentquo_group': None, 'autcentquo_hash': 5898335423961875981, 'autcentquo_nilpotent': False, 'autcentquo_order': 259920, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_{19}^2.C_{60}.C_6.C_2', 'cc_stats': [[1, 1, 1], [2, 361, 1], [4, 361, 2], [19, 4, 90]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1444.8', 'commutator_count': 1, 'commutator_label': '361.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '19.1', '19.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 8, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 361, 1, 1], [4, 361, 2, 1], [19, 4, 9, 10]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 76, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 3, 5, 19], 'factors_of_order': [2, 19], 'faithful_reps': [[4, 1, 90]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1444.8', 'hash': 8, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 76, 'inner_gen_orders': [4, 19, 19], 'inner_gens': [[1, 404, 1004], [1121, 4, 76], [593, 4, 76]], 'inner_hash': 8, 'inner_nilpotent': False, 'inner_order': 1444, 'inner_split': False, 'inner_tex': 'C_{19}^2:C_4', 'inner_used': [1, 2], 'irrC_degree': 4, 'irrQ_degree': 36, 'irrQ_dim': 36, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [4, 90]], 'label': '1444.8', 'linC_count': 90, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 36, 'linQ_degree_count': 10, 'linQ_dim': 36, 'linQ_dim_count': 10, 'linR_count': 90, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C19^2:C4', '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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 94, 'number_divisions': 13, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 26, 'number_subgroups': 1126, 'old_label': None, 'order': 1444, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 361], [4, 722], [19, 360]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 90, 'outer_gen_orders': [2, 90], 'outer_gen_pows': [0, 3], 'outer_gens': [[3, 404, 1036], [1, 468, 4]], 'outer_group': '180.9', 'outer_hash': 9, 'outer_nilpotent': False, 'outer_order': 180, 'outer_permdeg': 16, 'outer_perms': [7, 1401602631172], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_9\\times D_{10}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 38, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [36, 10]], 'representations': {'PC': {'code': 4024907219024219023955, 'gens': [1, 3, 4], 'pres': [4, -2, -2, -19, 19, 8, 4850, 438, 16067, 10951]}, 'GLFq': {'d': 2, 'q': 361, 'gens': [13080579430, 16108897606, 861943095, 15342631460]}, 'Perm': {'d': 38, 'gens': [275275061824526900926319591509972947591168000, 6493770503706733535750496841697542299647999, 226337273055722118539418330971692413545414400, 247747555642074210833687632522699327410739200]}}, 'schur_multiplier': [19], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{19}^2:C_4', 'transitive_degree': 38, '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': '36.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 1368, 'aut_gen_orders': [72, 72, 18], 'aut_gens': [[1, 18, 72, 1368], [17443, 8874, 23256, 24984], [24355, 21762, 20520, 23976], [1981, 4446, 648, 14688]], 'aut_group': None, 'aut_hash': 6148826751490991875, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 51984, 'aut_permdeg': 437, 'aut_perms': [408544647600404212256818943708568826830909134353352796093806874356933065903780751693868076991804389662423853780276910719388098822194417166243278187358100799159135896664095187585010417198584917832668772162596209541865913693204488834586017797813294359179028353439453918008915011473480752692310661894181807939688715200668381591933299967624071839809981573058917910638425208010090878768314083487416921969046516144502766553641466977239505063933929561801544084170563725776884225437635904697027237758054983626664454641870854943387375868726504728796034522998432552617588466290003703105905237872444856102137661141558723689207655523847523433198958756903737671976814152683658767962390461819887436855248040568411679631539243693739899355807549577093141668122399965814011785867727162073467976862837651574173186819977130457059500087421593438365856640201266960748993876511944950454034556031417080451514844419537913172387751519689440127581327993024557237709595506910511333466976788075, 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6.333333333333333, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 38, 2, 1], [2, 361, 1, 1], [3, 361, 1, 2], [4, 722, 1, 1], [6, 361, 1, 2], [6, 722, 2, 2], [9, 361, 1, 6], [12, 722, 1, 2], [18, 361, 1, 6], [18, 722, 2, 6], [19, 36, 2, 1], [19, 72, 2, 2], [36, 722, 1, 6], [38, 684, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{19}^2.C_{36}.C_2^2', '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': 1368, 'autcentquo_group': None, 'autcentquo_hash': 6148826751490991875, 'autcentquo_nilpotent': False, 'autcentquo_order': 51984, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_{19}^2.C_{36}.C_2^2', 'cc_stats': [[1, 1, 1], [2, 38, 2], [2, 361, 1], [3, 361, 2], [4, 722, 1], [6, 361, 2], [6, 722, 4], [9, 361, 6], [12, 722, 2], [18, 361, 6], [18, 722, 12], [19, 36, 2], [19, 72, 4], [36, 722, 6], [38, 684, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '25992.bj', 'commutator_count': 1, 'commutator_label': '722.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '19.1', '19.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 36, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 38, 1, 2], [2, 361, 1, 1], [3, 361, 2, 1], [4, 722, 1, 1], [6, 361, 2, 1], [6, 722, 2, 2], [9, 361, 6, 1], [12, 722, 2, 1], [18, 361, 6, 1], [18, 722, 6, 2], [19, 36, 1, 2], [19, 72, 1, 4], [36, 722, 6, 1], [38, 684, 1, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 4320, 'exponent': 684, 'exponents_of_order': [3, 2, 2], 'factors_of_aut_order': [2, 3, 19], 'factors_of_order': [2, 3, 19], 'faithful_reps': [[36, 1, 4], [72, 1, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '25992.bj', 'hash': 3331103429084464448, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 684, 'inner_gen_orders': [18, 4, 19, 19], 'inner_gens': [[1, 24246, 19584, 17784], [24229, 18, 6768, 2016], [7849, 20682, 72, 1368], [9577, 738, 72, 1368]], 'inner_hash': 3331103429084464448, 'inner_nilpotent': False, 'inner_order': 25992, 'inner_split': False, 'inner_tex': 'D_{19}^2:C_{18}', 'inner_used': [1, 2], 'irrC_degree': 36, 'irrQ_degree': 36, 'irrQ_dim': 36, 'irrR_degree': 36, 'irrep_stats': [[1, 36], [2, 9], [36, 4], [72, 4]], 'label': '25992.bj', 'linC_count': 4, 'linC_degree': 36, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 36, 'linQ_degree_count': 4, 'linQ_dim': 36, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 36, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'D19^2:C18', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 40, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 53, 'number_divisions': 23, 'number_normal_subgroups': 19, 'number_subgroup_autclasses': 63, 'number_subgroup_classes': 102, 'number_subgroups': 11664, 'old_label': None, 'order': 25992, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 437], [3, 722], [4, 722], [6, 3610], [9, 2166], [12, 1444], [18, 10830], [19, 360], [36, 4332], [38, 1368]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [22162], 'outer_gens': [[14491, 8442, 6840, 17568]], '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': 4, 'perfect': False, 'permutation_degree': 38, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 9], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 5], [4, 1], [6, 4], [12, 1], [36, 4], [72, 4]], 'representations': {'PC': {'code': '40982482990919089663600766416184331648319524383722826096904233817165304008350517658522744464148979', 'gens': [1, 4, 6, 7], 'pres': [7, -2, -3, -3, -2, -2, -19, 19, 14, 50, 678891, 225046, 90821, 80, 255784, 53561, 53148, 822533, 25716, 5563, 15818, 1545, 871422, 569785, 122912, 5515, 33550]}, 'Perm': {'d': 38, 'gens': [13933637039504757885110485832213800225661400, 285677560510209893698935868588048431477329440]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 18], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'D_{19}^2:C_{18}', 'transitive_degree': 38, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '18.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 6, 'aut_gen_orders': [6], 'aut_gens': [[1], [11]], 'aut_group': '6.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 6, 'aut_permdeg': 5, 'aut_perms': [27], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1], [9, 1, 6, 1], [18, 1, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_6', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 6, 'autcent_group': '6.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_6', '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], [3, 1, 2], [6, 1, 2], [9, 1, 6], [18, 1, 6]], 'center_label': '18.2', 'center_order': 18, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1], [9, 1, 6, 1], [18, 1, 6, 1]], 'element_repr_type': 'PC', 'elementary': 6, 'eulerian_function': 1, 'exponent': 18, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[1, 0, 6]], 'familial': True, 'frattini_label': '3.1', 'frattini_quotient': '6.2', 'hash': 2, 'hyperelementary': 6, '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': 6, 'irrQ_dim': 6, 'irrR_degree': 2, 'irrep_stats': [[1, 18]], 'label': '18.2', 'linC_count': 6, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C18', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 18, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 18, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [3, 2], [6, 2], [9, 6], [18, 6]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[11]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 9], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2], [6, 2]], 'representations': {'PC': {'code': 5403, 'gens': [1], 'pres': [3, -2, -3, -3, 6, 22]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [71343819913236817]}, 'GLFp': {'d': 2, 'p': 17, 'gens': [26270]}, 'Perm': {'d': 11, 'gens': [3628800, 357120, 80884]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [18], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{18}', 'transitive_degree': 18, 'wreath_data': None, 'wreath_product': False}
