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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '480.767', 'ambient_counter': 767, 'ambient_order': 480, 'ambient_tex': 'C_{60}:Q_8', 'central': True, 'central_factor': False, 'centralizer_order': 480, 'characteristic': True, 'core_order': 20, 'counter': 73, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '480.767.24.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '24.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '24.14', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 14, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 24, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times D_6', 'simple': False, 'solvable': True, 'special_labels': ['Z', 'U1'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '20.5', 'subgroup_hash': 5, 'subgroup_order': 20, 'subgroup_tex': 'C_2\\times C_{10}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '480.767', 'aut_centralizer_order': 768, 'aut_label': '24.a1', 'aut_quo_index': 12, 'aut_stab_index': 1, 'aut_weyl_group': '4.1', 'aut_weyl_index': 768, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['8.a1.a1', '12.a1.a1', '12.b1.a1', '12.b1.b1', '12.c1.a1', '12.d1.a1', '12.e1.a1', '12.e1.b1'], 'contains': ['48.a1.a1', '48.b1.a1', '48.c1.a1', '120.a1.a1'], 'core': '24.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4918, 3698, 6416, 2464, 4969, 6130, 6415, 6679], 'generators': [12, 96, 240], 'label': '480.767.24.a1.a1', 'mobius_quo': -2, 'mobius_sub': 24, 'normal_closure': '24.a1.a1', 'normal_contained_in': ['8.a1.a1', '12.a1.a1', '12.b1.b1', '12.b1.a1'], 'normal_contains': ['48.a1.a1', '48.b1.a1', '48.c1.a1', '120.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '24.a1.a1', 'projective_image': '24.14', 'quotient_action_image': '1.1', 'quotient_action_kernel': '24.14', 'quotient_action_kernel_order': 24, 'quotient_fusion': None, 'short_label': '24.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 12], 'aut_gens': [[1, 2], [10, 13], [10, 15]], 'aut_group': '24.5', 'aut_hash': 5, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24, 'aut_permdeg': 7, 'aut_perms': [127, 857], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [5, 1, 4, 1], [10, 1, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.5', 'autcent_hash': 5, 'autcent_nilpotent': False, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4\\times S_3', '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, 3], [5, 1, 4], [10, 1, 12]], 'center_label': '20.5', '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': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [5, 1, 4, 1], [10, 1, 4, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 6, 'exponent': 10, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '20.5', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 20]], 'label': '20.5', 'linC_count': 72, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 6, 'linQ_dim': 5, 'linQ_dim_count': 6, 'linR_count': 12, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C10', '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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 20, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 20, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [5, 4], [10, 12]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 12], 'outer_gen_pows': [0, 0], 'outer_gens': [[10, 13], [10, 15]], 'outer_group': '24.5', 'outer_hash': 5, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 7, 'outer_perms': [127, 857], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4\\times S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [4, 4]], 'representations': {'PC': {'code': 2179, 'gens': [1, 2], 'pres': [3, -2, -2, -5, 16]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [141602720900, 706303489327]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [13320, 4001]}, 'Perm': {'d': 9, 'gens': [40320, 720, 96]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_{10}', 'transitive_degree': 20, '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': '40.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [6, 4, 4, 4, 4, 6, 2, 4], 'aut_gens': [[1, 2, 24], [249, 362, 264], [241, 130, 24], [253, 134, 216], [253, 370, 180], [261, 22, 168], [257, 2, 228], [5, 142, 264], [13, 10, 84]], 'aut_group': None, 'aut_hash': 2284238130320887519, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3072, 'aut_permdeg': 24, 'aut_perms': [472409903639566669439594, 556294550498731827994376, 271043657979142337572822, 482304577644600278440639, 353661702568810240632036, 4326279247330568306146, 283133213706249923238416, 12275681751636533566196], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 1], [4, 2, 2, 1], [4, 4, 2, 1], [4, 6, 2, 2], [4, 12, 2, 1], [5, 1, 4, 1], [6, 2, 1, 3], [10, 1, 4, 3], [12, 4, 2, 1], [12, 4, 4, 1], [15, 2, 4, 1], [20, 2, 8, 1], [20, 4, 8, 1], [20, 6, 8, 2], [20, 12, 8, 1], [30, 2, 4, 3], [60, 4, 8, 1], [60, 4, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3:(C_2^5.C_2^5)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '256.56082', 'autcent_hash': 56082, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '12.4', 'autcentquo_hash': 4, 'autcentquo_nilpotent': False, 'autcentquo_order': 12, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6', 'cc_stats': [[1, 1, 1], [2, 1, 3], [3, 2, 1], [4, 2, 2], [4, 4, 2], [4, 6, 4], [4, 12, 2], [5, 1, 4], [6, 2, 3], [10, 1, 12], [12, 4, 6], [15, 2, 4], [20, 2, 8], [20, 4, 8], [20, 6, 16], [20, 12, 8], [30, 2, 12], [60, 4, 24]], 'center_label': '20.5', 'center_order': 20, 'central_product': True, 'central_quotient': '24.14', 'commutator_count': 1, 'commutator_label': '12.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 767, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['5.1', 1], ['96.95', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 1], [4, 2, 1, 2], [4, 4, 1, 2], [4, 6, 1, 4], [4, 12, 1, 2], [5, 1, 4, 1], [6, 2, 1, 3], [10, 1, 4, 3], [12, 4, 1, 2], [12, 4, 2, 2], [15, 2, 4, 1], [20, 2, 4, 2], [20, 4, 4, 2], [20, 6, 4, 4], [20, 12, 4, 2], [30, 2, 4, 3], [60, 4, 4, 2], [60, 4, 8, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 10416, 'exponent': 60, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '120.45', 'hash': 767, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6, 2], 'inner_gens': [[1, 22, 24], [5, 2, 264], [1, 242, 24]], 'inner_hash': 14, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': True, 'inner_tex': 'C_2\\times D_6', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 40], [2, 70], [4, 10]], 'label': '480.767', 'linC_count': 384, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 512, 'linQ_dim': 12, 'linQ_dim_count': 324, 'linR_count': 16, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C60:Q8', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 30, 'number_characteristic_subgroups': 38, 'number_conjugacy_classes': 120, 'number_divisions': 44, 'number_normal_subgroups': 74, 'number_subgroup_autclasses': 88, 'number_subgroup_classes': 136, 'number_subgroups': 276, 'old_label': None, 'order': 480, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 60], [5, 4], [6, 6], [10, 12], [12, 24], [15, 8], [20, 240], [30, 24], [60, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4, 4], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[13, 2, 24], [13, 2, 36], [241, 2, 24], [13, 14, 408], [13, 134, 36]], 'outer_group': '128.2154', 'outer_hash': 2154, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 12, 'outer_perms': [5167, 87091327, 40284846, 408, 87459246], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4^2:C_2^3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [4, 12], [8, 10], [16, 4]], 'representations': {'PC': {'code': 43636101896632587599116402450874435, 'gens': [1, 2, 5], 'pres': [7, -2, -2, -2, -3, -2, -2, -5, 84, 309, 36, 422, 58, 451, 4631, 102, 124]}, 'GLZN': {'d': 2, 'p': 33, 'gens': [898450, 826574, 359380, 428131, 48302, 1010002, 265522]}, 'Perm': {'d': 20, 'gens': [135936679844005200, 87454080, 263295183970022400, 33, 392327162440012800, 40279680, 5760]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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_{60}:Q_8', 'transitive_degree': 480, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 3, 3, 2, 2], 'aut_gens': [[1, 2, 4], [1, 2, 20], [1, 2, 5], [12, 15, 5], [1, 10, 4], [1, 14, 4], [1, 15, 4]], 'aut_group': '144.183', 'aut_hash': 183, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 144, 'aut_permdeg': 7, 'aut_perms': [1, 120, 144, 3, 744, 1680], 'aut_phi_ratio': 18.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 3, 4, 1], [3, 2, 1, 1], [6, 2, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.12', 'autcent_hash': 12, 'autcent_nilpotent': False, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 3, 4], [3, 2, 1], [6, 2, 3]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 14, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 1, 4], [3, 2, 1, 1], [6, 2, 1, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 28, 'exponent': 6, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '24.14', 'hash': 14, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [1, 2, 3], 'inner_gens': [[1, 2, 4], [1, 2, 20], [1, 10, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 4]], 'label': '24.14', 'linC_count': 12, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 12, 'linQ_dim': 3, 'linQ_dim_count': 12, 'linR_count': 12, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*D6', 'ngens': 4, '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': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 12, 'number_divisions': 12, 'number_normal_subgroups': 21, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 32, 'number_subgroups': 54, 'old_label': None, 'order': 24, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 15], [3, 2], [6, 6]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 3, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 2, 5], [13, 14, 17], [1, 15, 4], [1, 14, 4]], 'outer_group': '24.12', 'outer_hash': 12, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 4, 'outer_perms': [2, 4, 16, 7], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4]], 'representations': {'PC': {'code': 5123137, 'gens': [1, 2, 3], 'pres': [4, -2, -2, -2, -3, 126, 34, 135]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [16325, 16295, 3362]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [7426, 8156, 16849, 13286]}, 'Perm': {'d': 7, 'gens': [127, 7, 16, 840]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times D_6', 'transitive_degree': 12, 'wreath_data': None, 'wreath_product': False}
