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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1660.4', 'ambient_counter': 4, 'ambient_order': 1660, 'ambient_tex': 'C_{1660}', 'central': True, 'central_factor': False, 'centralizer_order': 1660, 'characteristic': True, 'core_order': 83, 'counter': 6, 'cyclic': True, 'direct': True, 'hall': 83, 'label': '1660.4.20.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '20.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '20.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 20, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{20}', 'simple': True, 'solvable': True, 'special_labels': ['C3'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '83.1', 'subgroup_hash': 1, 'subgroup_order': 83, 'subgroup_tex': 'C_{83}', 'supersolvable': True, 'sylow': 83}
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gps_subgroup_data • Show schema
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{'ambient': '1660.4', 'aut_centralizer_order': 8, 'aut_label': '20.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '82.2', 'aut_weyl_index': 8, 'centralizer': '1.a1.a1', 'complements': ['83.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '10.a1.a1'], 'contains': ['1660.a1.a1'], 'core': '20.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [9055, 9055, 9055, 9055, 2250, 2250, 2250, 2250], 'generators': [20], 'label': '1660.4.20.a1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '20.a1.a1', 'normal_contained_in': ['4.a1.a1', '10.a1.a1'], 'normal_contains': ['1660.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '20.a1.a1', 'projective_image': '20.2', 'quotient_action_image': '1.1', 'quotient_action_kernel': '20.2', 'quotient_action_kernel_order': 20, 'quotient_fusion': None, 'short_label': '20.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '83.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 82, 'aut_gen_orders': [82], 'aut_gens': [[1], [42]], 'aut_group': '82.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 82, 'aut_permdeg': 43, 'aut_perms': [1437642729082795807916967057028356205707264000000000], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [83, 1, 82, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{82}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 82, 'autcent_group': '82.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 82, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{82}', '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], [83, 1, 82]], 'center_label': '83.1', 'center_order': 83, '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': ['83.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [83, 1, 82, 1]], 'element_repr_type': 'PC', 'elementary': 83, 'eulerian_function': 1, 'exponent': 83, 'exponents_of_order': [1], 'factors_of_aut_order': [2, 41], 'factors_of_order': [83], 'faithful_reps': [[1, 0, 82]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '83.1', 'hash': 1, 'hyperelementary': 83, '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': 82, 'irrQ_dim': 82, 'irrR_degree': 2, 'irrep_stats': [[1, 83]], 'label': '83.1', 'linC_count': 82, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 82, 'linQ_degree_count': 1, 'linQ_dim': 82, 'linQ_dim_count': 1, 'linR_count': 41, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C83', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 83, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 83, 'order_factorization_type': 1, 'order_stats': [[1, 1], [83, 82]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 82, 'outer_gen_orders': [82], 'outer_gen_pows': [0], 'outer_gens': [[42]], 'outer_group': '82.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 82, 'outer_permdeg': 43, 'outer_perms': [1437642729082795807916967057028356205707264000000000], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{82}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 83, 'pgroup': 83, 'primary_abelian_invariants': [83], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [82, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -83]}, 'Lie': [{'d': 1, 'q': 83, 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 83, 'gens': [571871]}, 'Perm': {'d': 83, 'gens': [38979875363505302337055332973131205624792690113575452946607196594293007568261730186266584808295291703787520000000000000000000]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [83], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{83}', 'transitive_degree': 83, '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': '1660.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 164, 'aut_gen_orders': [2, 2, 164], 'aut_gens': [[1], [1161], [831], [1037]], 'aut_group': '656.46', 'aut_hash': 46, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 656, 'aut_permdeg': 88, 'aut_perms': [240653664262646398824364291566343783056841073003064150375214511720418715778479274250420219239686814127869506093056000000000, 2107757298379527717213600518699389595229783738061356212322972511214654115727593174080683423236414793504734471782400000000000000000000, 285064341596380282813238570119582695616465865745660975498642690668824353012129487853579065019958376597791165375861227520000000000], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [5, 1, 4, 1], [10, 1, 4, 1], [20, 1, 8, 1], [83, 1, 82, 1], [166, 1, 82, 1], [332, 1, 164, 1], [415, 1, 328, 1], [830, 1, 328, 1], [1660, 1, 656, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times C_{164}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 164, 'autcent_group': '656.46', 'autcent_hash': 46, 'autcent_nilpotent': True, 'autcent_order': 656, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_{164}', '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], [5, 1, 4], [10, 1, 4], [20, 1, 8], [83, 1, 82], [166, 1, 82], [332, 1, 164], [415, 1, 328], [830, 1, 328], [1660, 1, 656]], 'center_label': '1660.4', 'center_order': 1660, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '5.1', '83.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['5.1', 1], ['83.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [5, 1, 4, 1], [10, 1, 4, 1], [20, 1, 8, 1], [83, 1, 82, 1], [166, 1, 82, 1], [332, 1, 164, 1], [415, 1, 328, 1], [830, 1, 328, 1], [1660, 1, 656, 1]], 'element_repr_type': 'PC', 'elementary': 830, 'eulerian_function': 1, 'exponent': 1660, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 41], 'factors_of_order': [2, 5, 83], 'faithful_reps': [[1, 0, 656]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '830.4', 'hash': 4, 'hyperelementary': 830, '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': 656, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 1660]], 'label': '1660.4', 'linC_count': None, 'linC_degree': 1, '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': True, 'monomial': True, 'name': 'C1660', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 12, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 1660, 'number_divisions': 12, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 12, 'number_subgroups': 12, 'old_label': None, 'order': 1660, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 1], [4, 2], [5, 4], [10, 4], [20, 8], [83, 82], [166, 82], [332, 164], [415, 328], [830, 328], [1660, 656]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 164, 'outer_gen_orders': [2, 2, 164], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1161], [831], [1037]], 'outer_group': '656.46', 'outer_hash': 46, 'outer_nilpotent': True, 'outer_order': 656, 'outer_permdeg': 88, 'outer_perms': [240653664262646398824364291566343783056841073003064150375214511720418715778479274250420219239686814127869506093056000000000, 2107757298379527717213600518699389595229783738061356212322972511214654115727593174080683423236414793504734471782400000000000000000000, 285064341596380282813238570119582695616465865745660975498642690668824353012129487853579065019958376597791165375861227520000000000], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{164}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 92, 'pgroup': 0, 'primary_abelian_invariants': [4, 5, 83], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 2], [2, 1], [4, 2], [8, 1], [82, 2], [164, 1], [328, 2], [656, 1]], 'representations': {'PC': {'code': 66987401966620599, 'gens': [1], 'pres': [4, -2, -2, -5, -83, 8, 21, 70]}, 'GLFp': {'d': 2, 'p': 331, 'gens': [11925554974]}, 'Perm': {'d': 92, 'gens': [408571890232484411760118751745727077918531739353289410909109637518870983100868116050147836126993116402544739479243980800000000000000000000000, 135216660723001204716247785678757595766896706143143174333596648117861503942594373275424683867321237028413897417151007948800000000000000000000, 8431029193518110868854402074797558380919134952245424849291890044858616462910372696322733692945659174018937887129600000000000000000000, 38979875363505302337055332973131205624792690113575452946607196594293007568261730186266584808295291703787520000000000000000000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [1660], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{1660}', 'transitive_degree': 1660, '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': '20.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1], [11], [17]], 'aut_group': '8.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 6, 'aut_perms': [120, 9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [5, 1, 4, 1], [10, 1, 4, 1], [20, 1, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_4', '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], [5, 1, 4], [10, 1, 4], [20, 1, 8]], 'center_label': '20.2', 'center_order': 20, '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', '2.1', '5.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [5, 1, 4, 1], [10, 1, 4, 1], [20, 1, 8, 1]], 'element_repr_type': 'PC', 'elementary': 10, 'eulerian_function': 1, 'exponent': 20, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 5], 'faithful_reps': [[1, 0, 8]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '10.2', 'hash': 2, 'hyperelementary': 10, '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': 8, 'irrQ_dim': 8, 'irrR_degree': 2, 'irrep_stats': [[1, 20]], 'label': '20.2', 'linC_count': 8, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 2, 'linQ_dim': 6, 'linQ_dim_count': 2, 'linR_count': 4, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C20', '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': 20, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 20, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [4, 2], [5, 4], [10, 4], [20, 8]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[11], [17]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [4, 5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 2], [8, 1]], 'representations': {'PC': {'code': 51395, 'gens': [1], 'pres': [3, -2, -2, -5, 6, 16]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [58438750843733785]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 504, 252]}, 'Perm': {'d': 9, 'gens': [131040, 96, 41040]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [20], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{20}', 'transitive_degree': 20, 'wreath_data': None, 'wreath_product': False}
