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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1320.30', 'ambient_counter': 30, 'ambient_order': 1320, 'ambient_tex': 'C_{132}:C_{10}', 'central': False, 'central_factor': True, 'centralizer_order': 110, 'characteristic': True, 'core_order': 24, 'counter': 37, 'cyclic': False, 'direct': True, 'hall': 6, 'label': '1320.30.55.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '55.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '55.1', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 55, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{11}:C_5', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '24.6', 'subgroup_hash': 6, 'subgroup_order': 24, 'subgroup_tex': 'D_{12}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1320.30', 'aut_centralizer_order': 110, 'aut_label': '55.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '48.38', 'aut_weyl_index': 110, 'centralizer': '12.a1.a1', 'complements': ['24.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['5.a1.a1', '11.a1.a1'], 'contains': ['110.a1.a1', '110.a1.b1', '110.b1.a1', '165.a1.a1'], 'core': '55.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [9265, 1129, 9454, 1244, 8635, 9506, 2500, 9231], 'generators': [5, 330, 660, 880], 'label': '1320.30.55.a1.a1', 'mobius_quo': 0, 'mobius_sub': 11, 'normal_closure': '55.a1.a1', 'normal_contained_in': ['5.a1.a1'], 'normal_contains': ['110.a1.b1', '110.a1.a1', '110.b1.a1'], 'normalizer': '1.a1.a1', 'old_label': '55.a1.a1', 'projective_image': '660.18', 'quotient_action_image': '1.1', 'quotient_action_kernel': '55.1', 'quotient_action_kernel_order': 55, 'quotient_fusion': None, 'short_label': '55.a1.a1', 'subgroup_fusion': None, 'weyl_group': '12.4'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 12], 'aut_gens': [[1, 2], [1, 10], [1, 14], [15, 2]], 'aut_group': '48.38', 'aut_hash': 38, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 48, 'aut_permdeg': 7, 'aut_perms': [745, 24, 1707], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 2, 1], [3, 2, 1, 1], [4, 2, 1, 1], [6, 2, 1, 1], [12, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3\\times D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', '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, 1], [2, 6, 2], [3, 2, 1], [4, 2, 1], [6, 2, 1], [12, 2, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '12.4', 'commutator_count': 1, 'commutator_label': '6.2', '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': 6, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 2], [3, 2, 1, 1], [4, 2, 1, 1], [6, 2, 1, 1], [12, 2, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 12, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 1, 2]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '12.4', 'hash': 6, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6], 'inner_gens': [[1, 22], [5, 2]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'D_6', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 5]], 'label': '24.6', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 3, 'linQ_dim': 4, 'linQ_dim_count': 3, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D12', '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': 7, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 9, 'number_divisions': 8, 'number_normal_subgroups': 9, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 16, 'number_subgroups': 34, 'old_label': None, 'order': 24, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 13], [3, 2], [4, 2], [6, 2], [12, 4]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [6, 0], 'outer_gens': [[7, 2], [1, 14]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 3], [4, 1]], 'representations': {'PC': {'code': 2968216545, 'gens': [1, 2], 'pres': [4, -2, -2, -2, -3, 177, 21, 242, 34, 259]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [1381251, 40593880]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [11822, 13311]}, 'Perm': {'d': 7, 'gens': [745, 1464, 2424, 3]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{12}', 'transitive_degree': 12, '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': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 660, 'aut_gen_orders': [10, 30, 5, 20, 60], 'aut_gens': [[1, 10], [1201, 1150], [701, 730], [1, 970], [1051, 290], [771, 370]], 'aut_group': None, 'aut_hash': 3017774860523347021, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5280, 'aut_permdeg': 23, 'aut_perms': [14158905024148560604551, 16574054407752916349384, 249018008154058433, 6876250192192193352350, 17000906405241668263498], 'aut_phi_ratio': 16.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 2, 1], [3, 2, 1, 1], [4, 2, 1, 1], [5, 11, 1, 4], [6, 2, 1, 1], [10, 11, 1, 4], [10, 66, 2, 4], [11, 5, 2, 1], [12, 2, 2, 1], [15, 22, 1, 4], [20, 22, 1, 4], [22, 5, 2, 1], [22, 30, 4, 1], [30, 22, 1, 4], [33, 10, 2, 1], [44, 10, 2, 1], [60, 22, 2, 4], [66, 10, 2, 1], [132, 10, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{66}.C_{10}.C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 330, 'autcentquo_group': '1320.142', 'autcentquo_hash': 142, 'autcentquo_nilpotent': False, 'autcentquo_order': 1320, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 6, 2], [3, 2, 1], [4, 2, 1], [5, 11, 4], [6, 2, 1], [10, 11, 4], [10, 66, 8], [11, 5, 2], [12, 2, 2], [15, 22, 4], [20, 22, 4], [22, 5, 2], [22, 30, 4], [30, 22, 4], [33, 10, 2], [44, 10, 2], [60, 22, 8], [66, 10, 2], [132, 10, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '660.18', 'commutator_count': 1, 'commutator_label': '66.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '5.1', '11.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 30, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['24.6', 1], ['55.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 2], [3, 2, 1, 1], [4, 2, 1, 1], [5, 11, 4, 1], [6, 2, 1, 1], [10, 11, 4, 1], [10, 66, 4, 2], [11, 5, 2, 1], [12, 2, 2, 1], [15, 22, 4, 1], [20, 22, 4, 1], [22, 5, 2, 1], [22, 30, 2, 2], [30, 22, 4, 1], [33, 10, 2, 1], [44, 10, 2, 1], [60, 22, 8, 1], [66, 10, 2, 1], [132, 10, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72, 'exponent': 660, 'exponents_of_order': [3, 1, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 5, 11], 'faithful_reps': [[10, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '660.18', 'hash': 30, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 330, 'inner_gen_orders': [10, 66], 'inner_gens': [[1, 710], [621, 10]], 'inner_hash': 18, 'inner_nilpotent': False, 'inner_order': 660, 'inner_split': True, 'inner_tex': 'C_{66}:C_{10}', 'inner_used': [1, 2], 'irrC_degree': 10, 'irrQ_degree': 40, 'irrQ_dim': 40, 'irrR_degree': 20, 'irrep_stats': [[1, 20], [2, 25], [5, 8], [10, 10]], 'label': '1320.30', 'linC_count': 80, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 12, 'linQ_dim': 14, 'linQ_dim_count': 12, 'linR_count': 8, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C132:C10', 'ngens': 6, '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, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 42, 'number_characteristic_subgroups': 21, 'number_conjugacy_classes': 63, 'number_divisions': 24, 'number_normal_subgroups': 27, 'number_subgroup_autclasses': 48, 'number_subgroup_classes': 64, 'number_subgroups': 476, 'old_label': None, 'order': 1320, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 13], [3, 2], [4, 2], [5, 44], [6, 2], [10, 572], [11, 10], [12, 4], [15, 88], [20, 88], [22, 130], [30, 88], [33, 20], [44, 20], [60, 176], [66, 20], [132, 40]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [330, 0, 220], 'outer_gens': [[331, 1090], [1, 650], [881, 1090]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 3], [4, 5], [8, 3], [10, 4], [16, 1], [20, 3], [40, 1]], 'representations': {'PC': {'code': 44592077204886695310986078158248311013341030019, 'gens': [1, 3], 'pres': [6, -2, -5, -2, -2, -3, -11, 12, 12782, 2258, 50, 2403, 6009, 69, 6004, 15010, 118, 21605, 6491]}, 'GLZN': {'d': 2, 'p': 66, 'gens': [291149, 287521, 6612431, 433456, 6806977, 287893]}, 'Perm': {'d': 18, 'gens': [745, 1464, 23819883050880, 2424, 3, 401650985393280]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{132}:C_{10}', 'transitive_degree': 132, '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': '5.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 110, 'aut_gen_orders': [10, 11], 'aut_gens': [[1, 5], [21, 35], [6, 5]], 'aut_group': '110.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 110, 'aut_permdeg': 11, 'aut_perms': [955523, 8529212], 'aut_phi_ratio': 2.75, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [5, 11, 1, 4], [11, 5, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'F_{11}', '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': 110, 'autcentquo_group': '110.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 110, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{11}', 'cc_stats': [[1, 1, 1], [5, 11, 4], [11, 5, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '55.1', 'commutator_count': 1, 'commutator_label': '11.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['5.1', '11.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [5, 11, 4, 1], [11, 5, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 55, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [5, 11], 'faithful_reps': [[5, 0, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '55.1', 'hash': 1, 'hyperelementary': 5, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 55, 'inner_gen_orders': [5, 11], 'inner_gens': [[1, 20], [41, 5]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 55, 'inner_split': True, 'inner_tex': 'C_{11}:C_5', 'inner_used': [1, 2], 'irrC_degree': 5, 'irrQ_degree': 10, 'irrQ_dim': 10, 'irrR_degree': 10, 'irrep_stats': [[1, 5], [5, 2]], 'label': '55.1', 'linC_count': 2, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 1, 'linQ_dim': 10, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C11:C5', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 7, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 14, 'old_label': None, 'order': 55, 'order_factorization_type': 11, 'order_stats': [[1, 1], [5, 44], [11, 10]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [4], 'outer_gens': [[1, 30]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 1], [10, 1]], 'representations': {'PC': {'code': 1439, 'gens': [1, 2], 'pres': [2, -5, -11, 81]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [1343, 1335]}, 'Perm': {'d': 11, 'gens': [1332490, 36288000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [5], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{11}:C_5', 'transitive_degree': 11, 'wreath_data': None, 'wreath_product': False}
