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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '800.1168', 'ambient_counter': 1168, 'ambient_order': 800, 'ambient_tex': 'C_{10}^2.C_2^3', 'central': False, 'central_factor': False, 'centralizer_order': 80, 'characteristic': True, 'core_order': 800, 'counter': 1, 'cyclic': False, 'direct': True, 'hall': 10, 'label': '800.1168.1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '1.a1', 'outer_equivalence': True, 'perfect': False, 'proper': False, 'quotient': '1.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 1, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_1', 'simple': False, 'solvable': True, 'special_labels': ['R', 'L0', 'D0', 'C0'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '800.1168', 'subgroup_hash': 1168, 'subgroup_order': 800, 'subgroup_tex': 'C_{10}^2.C_2^3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '800.1168', 'aut_centralizer_order': None, 'aut_label': '1.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '10.b1', 'complements': ['800.a1'], 'conjugacy_class_count': 1, 'contained_in': [], 'contains': ['2.a1', '2.b1', '5.a1', '5.b1'], 'core': '1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5734, 3515, 5380, 5157], 'generators': [1, 4, 40, 400, 2, 160, 16], 'label': '800.1168.1.a1', 'mobius_quo': 0, 'mobius_sub': 1, 'normal_closure': '1.a1', 'normal_contained_in': [], 'normal_contains': ['2.a1', '2.b1', '5.a1'], 'normalizer': '1.a1', 'old_label': '1.a1', 'projective_image': '10.1', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '1.a1', 'subgroup_fusion': None, 'weyl_group': '10.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '160.228', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 840, 'aut_gen_orders': [28, 12, 70], 'aut_gens': [[1, 2, 8, 80], [444, 442, 448, 161], [40, 2, 461, 244], [440, 727, 13, 481]], 'aut_group': None, 'aut_hash': 5428847875685733254, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1720320, 'aut_permdeg': 84, 'aut_perms': [2265954857620450689606847619055453162504586209952055671217166012886308362983585963676631047542063492902219920542037957038358848, 2278180922160465041896163300467684140310960245612017529321737711950167795175408483214599575318104511013699690323620251341353399, 1587521694437335063168396256607585063345659988675559092468325278533770535881677798087084748018091604132349671793193987152563145], 'aut_phi_ratio': 5376.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 14, 1], [4, 5, 16, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 8, 1], [10, 1, 4, 1], [10, 1, 56, 1], [10, 2, 2, 1], [10, 2, 8, 1], [10, 2, 28, 1], [10, 2, 112, 1], [20, 5, 64, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_2^5\\times D_5).C_2^5.\\PSL(2,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 168, 'autcent_group': '86016.bh', 'autcent_hash': 51025793962120521, 'autcent_nilpotent': False, 'autcent_order': 86016, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^2.C_2^7:\\GL(3,2)', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 1, 15], [4, 5, 16], [5, 1, 4], [5, 2, 10], [10, 1, 60], [10, 2, 150], [20, 5, 64]], 'center_label': '80.52', 'center_order': 80, 'central_product': True, 'central_quotient': '10.1', 'commutator_count': 1, 'commutator_label': '5.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1168, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['20.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15], [4, 5, 2, 8], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 4, 2], [10, 1, 4, 15], [10, 2, 2, 15], [10, 2, 4, 30], [20, 5, 8, 8]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72540, 'exponent': 20, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '400.219', 'hash': 1168, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [1, 2, 1, 5], 'inner_gens': [[1, 2, 8, 80], [1, 2, 8, 720], [1, 2, 8, 80], [1, 162, 8, 80]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 10, 'inner_split': False, 'inner_tex': 'D_5', 'inner_used': [2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 160], [2, 160]], 'label': '800.1168', '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.C2^3', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 14, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 320, 'number_divisions': 96, 'number_normal_subgroups': 370, 'number_subgroup_autclasses': 56, 'number_subgroup_classes': 606, 'number_subgroups': 1148, 'old_label': None, 'order': 800, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 15], [4, 80], [5, 24], [10, 360], [20, 320]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 168, 'outer_gen_orders': [4, 4, 4], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 3, 24, 121], [440, 47, 56, 645], [441, 446, 9, 245]], 'outer_group': None, 'outer_hash': 4889981620459245880, 'outer_nilpotent': False, 'outer_order': 172032, 'outer_permdeg': 22, 'outer_perms': [750751029560364497640, 423960296735593381831, 52487421356535384241], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^5.C_2^5.\\PSL(2,7)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 4, 5], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 8], [4, 32], [8, 40]], 'representations': {'PC': {'code': 2706664801236641630552115, 'gens': [1, 2, 4, 6], 'pres': [7, -2, -2, -2, -2, -5, -2, -5, 36, 80, 15132, 124, 15693]}, 'GLZN': {'d': 2, 'p': 44, 'gens': [2839969, 85193, 766665, 85207, 1788885, 86153, 2002815]}, 'Perm': {'d': 20, 'gens': [5169, 122002095511296016, 256096255903488000, 1307674368000, 6749568000, 16, 50520]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 20], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}^2.C_2^3', 'transitive_degree': 160, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '160.228', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 840, 'aut_gen_orders': [28, 12, 70], 'aut_gens': [[1, 2, 8, 80], [444, 442, 448, 161], [40, 2, 461, 244], [440, 727, 13, 481]], 'aut_group': None, 'aut_hash': 5428847875685733254, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1720320, 'aut_permdeg': 84, 'aut_perms': [2265954857620450689606847619055453162504586209952055671217166012886308362983585963676631047542063492902219920542037957038358848, 2278180922160465041896163300467684140310960245612017529321737711950167795175408483214599575318104511013699690323620251341353399, 1587521694437335063168396256607585063345659988675559092468325278533770535881677798087084748018091604132349671793193987152563145], 'aut_phi_ratio': 5376.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 14, 1], [4, 5, 16, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 8, 1], [10, 1, 4, 1], [10, 1, 56, 1], [10, 2, 2, 1], [10, 2, 8, 1], [10, 2, 28, 1], [10, 2, 112, 1], [20, 5, 64, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_2^5\\times D_5).C_2^5.\\PSL(2,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 168, 'autcent_group': '86016.bh', 'autcent_hash': 51025793962120521, 'autcent_nilpotent': False, 'autcent_order': 86016, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^2.C_2^7:\\GL(3,2)', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 1, 15], [4, 5, 16], [5, 1, 4], [5, 2, 10], [10, 1, 60], [10, 2, 150], [20, 5, 64]], 'center_label': '80.52', 'center_order': 80, 'central_product': True, 'central_quotient': '10.1', 'commutator_count': 1, 'commutator_label': '5.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1168, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['20.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15], [4, 5, 2, 8], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 4, 2], [10, 1, 4, 15], [10, 2, 2, 15], [10, 2, 4, 30], [20, 5, 8, 8]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72540, 'exponent': 20, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '400.219', 'hash': 1168, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [1, 2, 1, 5], 'inner_gens': [[1, 2, 8, 80], [1, 2, 8, 720], [1, 2, 8, 80], [1, 162, 8, 80]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 10, 'inner_split': False, 'inner_tex': 'D_5', 'inner_used': [2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 160], [2, 160]], 'label': '800.1168', '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.C2^3', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 14, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 320, 'number_divisions': 96, 'number_normal_subgroups': 370, 'number_subgroup_autclasses': 56, 'number_subgroup_classes': 606, 'number_subgroups': 1148, 'old_label': None, 'order': 800, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 15], [4, 80], [5, 24], [10, 360], [20, 320]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 168, 'outer_gen_orders': [4, 4, 4], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 3, 24, 121], [440, 47, 56, 645], [441, 446, 9, 245]], 'outer_group': None, 'outer_hash': 4889981620459245880, 'outer_nilpotent': False, 'outer_order': 172032, 'outer_permdeg': 22, 'outer_perms': [750751029560364497640, 423960296735593381831, 52487421356535384241], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^5.C_2^5.\\PSL(2,7)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 4, 5], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 8], [4, 32], [8, 40]], 'representations': {'PC': {'code': 2706664801236641630552115, 'gens': [1, 2, 4, 6], 'pres': [7, -2, -2, -2, -2, -5, -2, -5, 36, 80, 15132, 124, 15693]}, 'GLZN': {'d': 2, 'p': 44, 'gens': [2839969, 85193, 766665, 85207, 1788885, 86153, 2002815]}, 'Perm': {'d': 20, 'gens': [5169, 122002095511296016, 256096255903488000, 1307674368000, 6749568000, 16, 50520]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 20], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}^2.C_2^3', 'transitive_degree': 160, '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': '1.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': [], 'composition_length': 0, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1, 'exponent': 1, 'exponents_of_order': [], 'factors_of_aut_order': [], 'factors_of_order': [], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '1.1', 'hash': 1, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [], 'inner_gens': [], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 1]], 'label': '1.1', 'linC_count': 1, 'linC_degree': 0, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 0, 'linQ_degree_count': 1, 'linQ_dim': 0, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 0, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C1', 'ngens': 0, 'nilpotency_class': 0, 'nilpotent': True, 'normal_counts': [1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 1, 'number_characteristic_subgroups': 1, 'number_conjugacy_classes': 1, 'number_divisions': 1, 'number_normal_subgroups': 1, 'number_subgroup_autclasses': 1, 'number_subgroup_classes': 1, 'number_subgroups': 1, 'old_label': None, 'order': 1, 'order_factorization_type': 0, 'order_stats': [[1, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 0, 'perfect': True, 'permutation_degree': 1, 'pgroup': 1, 'primary_abelian_invariants': [], 'quasisimple': False, 'rank': 0, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 1]], 'representations': {'PC': {'code': 0, 'gens': [], 'pres': []}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_1', 'transitive_degree': 1, 'wreath_data': None, 'wreath_product': False}
