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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '410.1', 'ambient_counter': 1, 'ambient_order': 410, 'ambient_tex': 'C_{41}:C_{10}', 'central': False, 'central_factor': False, 'centralizer_order': 41, 'characteristic': True, 'core_order': 41, 'counter': 4, 'cyclic': True, 'direct': False, 'hall': 41, 'label': '410.1.10.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '10.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '10.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 10, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{10}', 'simple': True, 'solvable': True, 'special_labels': ['D', 'F', 'S', 'L1', 'D1', 'C2'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '41.1', 'subgroup_hash': 1, 'subgroup_order': 41, 'subgroup_tex': 'C_{41}', 'supersolvable': True, 'sylow': 41}
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gps_subgroup_data • Show schema
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{'ambient': '410.1', 'aut_centralizer_order': 41, 'aut_label': '10.a1', 'aut_quo_index': 4, 'aut_stab_index': 1, 'aut_weyl_group': '40.2', 'aut_weyl_index': 41, 'centralizer': '10.a1.a1', 'complements': ['41.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1', '5.a1.a1'], 'contains': ['410.a1.a1'], 'core': '10.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3475, 5312, 3475, 5312, 5251, 5395, 5251, 5395], 'generators': [10], 'label': '410.1.10.a1.a1', 'mobius_quo': -1, 'mobius_sub': 1, 'normal_closure': '10.a1.a1', 'normal_contained_in': ['2.a1.a1', '5.a1.a1'], 'normal_contains': ['410.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '10.a1.a1', 'projective_image': '410.1', 'quotient_action_image': '10.2', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '10.a1.a1', 'subgroup_fusion': None, 'weyl_group': '10.2'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '41.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 40, 'aut_gen_orders': [40], 'aut_gens': [[1], [6]], 'aut_group': '40.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 40, 'aut_permdeg': 13, 'aut_perms': [522956193], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [41, 1, 40, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{40}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 40, 'autcent_group': '40.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 40, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{40}', '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], [41, 1, 40]], 'center_label': '41.1', 'center_order': 41, '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': ['41.1'], '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], [41, 1, 40, 1]], 'element_repr_type': 'PC', 'elementary': 41, 'eulerian_function': 1, 'exponent': 41, 'exponents_of_order': [1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [41], 'faithful_reps': [[1, 0, 40]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '41.1', 'hash': 1, 'hyperelementary': 41, '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': 40, 'irrQ_dim': 40, 'irrR_degree': 2, 'irrep_stats': [[1, 41]], 'label': '41.1', 'linC_count': 40, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 40, 'linQ_degree_count': 1, 'linQ_dim': 40, 'linQ_dim_count': 1, 'linR_count': 20, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C41', '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': 41, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 41, 'order_factorization_type': 1, 'order_stats': [[1, 1], [41, 40]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 40, 'outer_gen_orders': [40], 'outer_gen_pows': [0], 'outer_gens': [[6]], 'outer_group': '40.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 40, 'outer_permdeg': 13, 'outer_perms': [522956193], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{40}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 41, 'pgroup': 41, 'primary_abelian_invariants': [41], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [40, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -41]}, 'Lie': [{'d': 1, 'q': 41, 'gens': [836850334330315506193242641144055892504420940313], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 41, 'gens': [68963]}, 'Perm': {'d': 41, 'gens': [32636611329915909373824450783844635770880000000000]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [41], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{41}', 'transitive_degree': 41, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '10.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 1640, 'aut_gen_orders': [40, 41], 'aut_gens': [[1, 10], [1, 220], [131, 10]], 'aut_group': '1640.47', 'aut_hash': 47, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1640, 'aut_permdeg': 41, 'aut_perms': [25407499915471720974798984799219353375680930211593, 33310278938561551556550945142889818316561667470070], 'aut_phi_ratio': 10.25, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 41, 1, 1], [5, 41, 1, 4], [10, 41, 1, 4], [41, 10, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'F_{41}', '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': 1640, 'autcentquo_group': '1640.47', 'autcentquo_hash': 47, 'autcentquo_nilpotent': False, 'autcentquo_order': 1640, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{41}', 'cc_stats': [[1, 1, 1], [2, 41, 1], [5, 41, 4], [10, 41, 4], [41, 10, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '410.1', 'commutator_count': 1, 'commutator_label': '41.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '5.1', '41.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 41, 1, 1], [5, 41, 4, 1], [10, 41, 4, 1], [41, 10, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72, 'exponent': 410, 'exponents_of_order': [1, 1, 1], 'factors_of_aut_order': [2, 5, 41], 'factors_of_order': [2, 5, 41], 'faithful_reps': [[10, 1, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '410.1', 'hash': 1, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 410, 'inner_gen_orders': [10, 41], 'inner_gens': [[1, 310], [111, 10]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 410, 'inner_split': False, 'inner_tex': 'C_{41}:C_{10}', 'inner_used': [1, 2], 'irrC_degree': 10, 'irrQ_degree': 40, 'irrQ_dim': 40, 'irrR_degree': 10, 'irrep_stats': [[1, 10], [10, 4]], 'label': '410.1', 'linC_count': 4, 'linC_degree': 10, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 40, 'linQ_degree_count': 1, 'linQ_dim': 40, 'linQ_dim_count': 1, 'linR_count': 4, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C41:C10', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 11, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 14, 'number_divisions': 5, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 128, 'old_label': None, 'order': 410, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 41], [5, 164], [10, 164], [41, 40]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [7], 'outer_gens': [[1, 220]], 'outer_group': '4.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 41, 'pgroup': 0, 'primary_abelian_invariants': [2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 2], [40, 1]], 'representations': {'PC': {'code': 5877845568159, 'gens': [1, 3], 'pres': [3, -2, -5, -41, 6, 2792, 815]}, 'GLFp': {'d': 2, 'p': 41, 'gens': [68963, 1102776]}, 'Perm': {'d': 41, 'gens': [815915283247897734345611269596115894271999999999, 193380446544193151050670286765040239042532465176, 32636611329915909373824450783844635770880000000000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [10], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{41}:C_{10}', 'transitive_degree': 41, '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': '10.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [4], 'aut_gens': [[1], [7]], 'aut_group': '4.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [5, 1, 4, 1], [10, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': '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], [5, 1, 4], [10, 1, 4]], 'center_label': '10.2', 'center_order': 10, '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', '5.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [5, 1, 4, 1], [10, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 10, 'eulerian_function': 1, 'exponent': 10, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 5], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '1.1', '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': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 10]], 'label': '10.2', 'linC_count': 4, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C10', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 10, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 10, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [5, 4], [10, 4]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [0], 'outer_gens': [[7]], 'outer_group': '4.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 2]], 'representations': {'PC': {'code': 83, 'gens': [1], 'pres': [2, -2, -5, 4]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16717348]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 504]}, 'Perm': {'d': 7, 'gens': [720, 96]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [10], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}', 'transitive_degree': 10, 'wreath_data': None, 'wreath_product': False}
