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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '2000.505', 'ambient_counter': 505, 'ambient_order': 2000, 'ambient_tex': 'C_{10}^2.D_{10}', 'central': False, 'central_factor': False, 'centralizer_order': 200, 'characteristic': False, 'core_order': 5, 'counter': 34, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '2000.505.80.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '80.c1', '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': 80, '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': '25.1', 'subgroup_hash': 1, 'subgroup_order': 25, 'subgroup_tex': 'C_{25}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '2000.505', 'aut_centralizer_order': 33600, 'aut_label': '80.c1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '5.1', 'aut_weyl_index': 134400, 'centralizer': '10.e1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['16.a1', '40.d1'], 'contains': ['400.a1'], 'core': '400.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [908, -1, 2534, -1], 'generators': [88], 'label': '2000.505.80.c1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '16.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.b1', 'old_label': '80.c1', 'projective_image': '2000.505', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '80.c1', 'subgroup_fusion': None, 'weyl_group': '5.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '25.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 20, 'aut_gen_orders': [20], 'aut_gens': [[1], [2]], 'aut_group': '20.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 20, 'aut_permdeg': 9, 'aut_perms': [46121], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [5, 1, 4, 1], [25, 1, 20, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{20}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 20, 'autcent_group': '20.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 20, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{20}', '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], [5, 1, 4], [25, 1, 20]], 'center_label': '25.1', 'center_order': 25, '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': ['5.1', '5.1'], '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], [5, 1, 4, 1], [25, 1, 20, 1]], 'element_repr_type': 'PC', 'elementary': 5, 'eulerian_function': 1, 'exponent': 25, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 5], 'factors_of_order': [5], 'faithful_reps': [[1, 0, 20]], 'familial': True, 'frattini_label': '5.1', 'frattini_quotient': '5.1', 'hash': 1, 'hyperelementary': 5, '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': 20, 'irrQ_dim': 20, 'irrR_degree': 2, 'irrep_stats': [[1, 25]], 'label': '25.1', 'linC_count': 20, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 20, 'linQ_degree_count': 1, 'linQ_dim': 20, 'linQ_dim_count': 1, 'linR_count': 10, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C25', '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': 25, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 25, 'order_factorization_type': 2, 'order_stats': [[1, 1], [5, 4], [25, 20]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 20, 'outer_gen_orders': [20], 'outer_gen_pows': [0], 'outer_gens': [[2]], 'outer_group': '20.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 20, 'outer_permdeg': 9, 'outer_perms': [46121], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{20}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 25, 'pgroup': 5, 'primary_abelian_invariants': [25], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 1], [20, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -5, -5, 10]}, 'GLFp': {'d': 2, 'p': 101, 'gens': [16484817]}, 'Perm': {'d': 25, 'gens': [15431352467814595952640000, 2481794093513708107384416]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [25], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{25}', 'transitive_degree': 25, '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': '80.52', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 2100, 'aut_gen_orders': [175, 60], 'aut_gens': [[1, 2, 4, 40], [1001, 3, 1724, 43], [3, 1000, 1124, 1482]], 'aut_group': None, 'aut_hash': 8170964205122769949, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 672000, 'aut_permdeg': 200, 'aut_perms': [509328571186488663586586724390100968431572890160164745536110617934342291338270679840180448207112096501820970852561482357964627702581739715839903834970511768068848763211585747322535907866163681013929694410078809223558278497191125857096107051300814550862150278553279845711996594242632276907424831964034374318350248468422335407140322619002082606360320197080275339735019863739210, 386947004043177170201324490468652072385135356908854965589296348505776793385652732648379448239177925250701210139506178688835247572798536290399359091815580735619398804572877298996667963128185252181516774027220610720304108139014200221780215377694268681964470253563546518071632504142992387590872676344390768353976667163075398072773017881255172399022859678806148991831274513948785], 'aut_phi_ratio': 840.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1], [2, 25, 8, 1], [5, 2, 2, 1], [5, 5, 1, 4], [10, 2, 14, 1], [10, 5, 7, 4], [10, 25, 8, 4], [25, 10, 2, 5], [50, 10, 14, 5]], 'aut_supersolvable': False, 'aut_tex': 'D_{25}.C_{10}\\times C_2^3.\\PSL(2,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '1344.11686', 'autcent_hash': 11686, 'autcent_nilpotent': False, 'autcent_order': 1344, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^3:\\GL(3,2)', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 100, 'autcentquo_group': '500.18', 'autcentquo_hash': 18, 'autcentquo_nilpotent': False, 'autcentquo_order': 500, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{25}:C_{20}', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 25, 8], [5, 2, 2], [5, 5, 4], [10, 2, 14], [10, 5, 28], [10, 25, 32], [25, 10, 10], [50, 10, 70]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '250.6', 'commutator_count': 1, 'commutator_label': '25.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '5.1', '5.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 505, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['250.6', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 25, 1, 8], [5, 2, 2, 1], [5, 5, 4, 1], [10, 2, 2, 7], [10, 5, 4, 7], [10, 25, 4, 8], [25, 10, 2, 1], [25, 10, 4, 2], [50, 10, 2, 7], [50, 10, 4, 14]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 7254000, 'exponent': 50, 'exponents_of_order': [4, 3], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '5.1', 'frattini_quotient': '400.219', 'hash': 505, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 50, 'inner_gen_orders': [1, 1, 10, 25], 'inner_gens': [[1, 2, 4, 40], [1, 2, 4, 40], [1, 2, 4, 1160], [1, 2, 884, 40]], 'inner_hash': 6, 'inner_nilpotent': False, 'inner_order': 250, 'inner_split': False, 'inner_tex': 'C_{25}:C_{10}', 'inner_used': [3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 80], [2, 80], [10, 16]], 'label': '2000.505', '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': 'C10^2.D10', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 27, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 176, 'number_divisions': 64, 'number_normal_subgroups': 182, 'number_subgroup_autclasses': 52, 'number_subgroup_classes': 434, 'number_subgroups': 3386, 'old_label': None, 'order': 2000, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 207], [5, 24], [10, 968], [25, 100], [50, 700]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 84, 'outer_gen_orders': [3, 2, 2, 7], 'outer_gen_pows': [0, 4, 0, 0], 'outer_gens': [[1, 1002, 1006, 1042], [1, 2, 5, 1080], [1, 2, 6, 42], [1001, 1003, 6, 42]], 'outer_group': '2688.cd', 'outer_hash': 8270626938983654557, 'outer_nilpotent': False, 'outer_order': 2688, 'outer_permdeg': 10, 'outer_perms': [403470, 2442529, 362886, 2636664], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^4:\\GL(3,2)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 31, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 5], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [4, 24], [8, 16], [20, 8]], 'representations': {'PC': {'code': 844988091719774463235650491071135951, 'gens': [1, 2, 3, 5], 'pres': [7, -2, -2, -2, -5, -2, -5, -5, 58, 10168, 7200, 102, 3379, 6746, 250, 19620]}, 'GLZN': {'d': 2, 'p': 75, 'gens': [10968776, 19663201, 421924, 20671924, 423001, 10267524, 6750061]}, 'Perm': {'d': 31, 'gens': [18904799343284619429940398633007, 289, 415, 414, 26887164460621422987184866910080, 301537541203511656065792465795120, 558623293595663863325335335400320]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}^2.D_{10}', 'transitive_degree': 200, 'wreath_data': None, 'wreath_product': False}
