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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '480.1136', 'ambient_counter': 1136, 'ambient_order': 480, 'ambient_tex': 'C_{60}:C_2^3', 'central': False, 'central_factor': False, 'centralizer_order': 240, 'characteristic': True, 'core_order': 5, 'counter': 73, 'cyclic': True, 'direct': False, 'hall': 5, 'label': '480.1136.96.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '96.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '96.220', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 220, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 96, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^3\\times C_{12}', 'simple': True, 'solvable': True, 'special_labels': ['D', 'L1', 'D1', 'C6'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '5.1', 'subgroup_hash': 1, 'subgroup_order': 5, 'subgroup_tex': 'C_5', 'supersolvable': True, 'sylow': 5}
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gps_subgroup_data • Show schema
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{'ambient': '480.1136', 'aut_centralizer_order': 15360, 'aut_label': '96.a1', 'aut_quo_index': 14, 'aut_stab_index': 1, 'aut_weyl_group': '4.1', 'aut_weyl_index': 15360, 'centralizer': '2.c1', 'complements': ['5.a1'], 'conjugacy_class_count': 1, 'contained_in': ['32.a1', '48.a1', '48.b1', '48.c1'], 'contains': ['480.a1'], 'core': '96.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5691, 2341, 1832, 4534], 'generators': [96], 'label': '480.1136.96.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '96.a1', 'normal_contained_in': ['32.a1', '48.a1', '48.b1', '48.c1'], 'normal_contains': ['480.a1'], 'normalizer': '1.a1', 'old_label': '96.a1', 'projective_image': '480.1136', 'quotient_action_image': '2.1', 'quotient_action_kernel': '48.44', 'quotient_action_kernel_order': 48, 'quotient_fusion': None, 'short_label': '96.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '5.1', '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], [2]], '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], [5, 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], [5, 1, 4]], 'center_label': '5.1', 'center_order': 5, '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'], '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], [5, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 5, 'eulerian_function': 1, 'exponent': 5, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [5], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '1.1', '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': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 5]], 'label': '5.1', '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': 'C5', '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': 5, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 5, 'order_factorization_type': 1, 'order_stats': [[1, 1], [5, 4]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [0], 'outer_gens': [[2]], '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': 5, 'pgroup': 5, 'primary_abelian_invariants': [5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -5]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [26329372]}, 'Lie': [{'d': 1, 'q': 5, 'gens': [33], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [131]}, 'Perm': {'d': 5, 'gens': [96]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [5], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5', 'transitive_degree': 5, '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': '96.220', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 60, 'aut_gen_orders': [4, 4, 4, 20], 'aut_gens': [[1, 2, 4, 8], [193, 2, 6, 380], [295, 2, 244, 298], [193, 242, 6, 346], [293, 6, 244, 254]], 'aut_group': None, 'aut_hash': 5024676986004023821, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 61440, 'aut_permdeg': 50, 'aut_perms': [9228058849684918675933393026585537219546543422379422200519662203, 20925581272922222905374228616728027134980276930154418850106881636, 18349940936825503717609053085101413791745132207531087713577165362, 2463678523439877165525658857673548361318249213532255972012036236], 'aut_phi_ratio': 480.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [2, 5, 8, 1], [3, 1, 2, 1], [4, 1, 8, 1], [4, 5, 8, 1], [5, 2, 2, 1], [6, 1, 2, 1], [6, 1, 12, 1], [6, 5, 16, 1], [10, 2, 2, 1], [10, 2, 12, 1], [12, 1, 16, 1], [12, 5, 16, 1], [15, 2, 4, 1], [20, 2, 16, 1], [30, 2, 4, 1], [30, 2, 24, 1], [60, 2, 32, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4.C_2^4.C_5.C_6.C_2^3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': None, 'autcent_hash': 1930889827755753443, 'autcent_nilpotent': False, 'autcent_order': 3072, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^4.C_2^4.D_6', '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, 7], [2, 5, 8], [3, 1, 2], [4, 1, 8], [4, 5, 8], [5, 2, 2], [6, 1, 14], [6, 5, 16], [10, 2, 14], [12, 1, 16], [12, 5, 16], [15, 2, 4], [20, 2, 16], [30, 2, 28], [60, 2, 32]], 'center_label': '48.44', 'center_order': 48, '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', '3.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1136, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['10.1', 1], ['2.1', 2], ['3.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 5, 1, 8], [3, 1, 2, 1], [4, 1, 2, 4], [4, 5, 2, 4], [5, 2, 2, 1], [6, 1, 2, 7], [6, 5, 2, 8], [10, 2, 2, 7], [12, 1, 4, 4], [12, 5, 4, 4], [15, 2, 4, 1], [20, 2, 4, 4], [30, 2, 4, 7], [60, 2, 8, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 260400, 'exponent': 60, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '240.205', 'hash': 1136, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 1, 1, 5], 'inner_gens': [[1, 2, 4, 392], [1, 2, 4, 8], [1, 2, 4, 8], [97, 2, 4, 8]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 10, 'inner_split': False, 'inner_tex': 'D_5', 'inner_used': [1, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 96], [2, 96]], 'label': '480.1136', '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': 'C60: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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 20, 'number_characteristic_subgroups': 22, 'number_conjugacy_classes': 192, 'number_divisions': 72, 'number_normal_subgroups': 290, 'number_subgroup_autclasses': 84, 'number_subgroup_classes': 472, 'number_subgroups': 1200, 'old_label': None, 'order': 480, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 47], [3, 2], [4, 48], [5, 4], [6, 94], [10, 28], [12, 96], [15, 8], [20, 32], [30, 56], [60, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [4, 4, 4, 6], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 2, 6, 332], [5, 2, 244, 344], [7, 2, 244, 88], [7, 246, 2, 12]], 'outer_group': None, 'outer_hash': 549362732464081010, 'outer_nilpotent': False, 'outer_order': 6144, 'outer_permdeg': 20, 'outer_perms': [621808496415577831, 891398637895449544, 621096947115696940, 122423253836110608], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^4.C_2^4.D_6.C_2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 4, 3], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 24], [4, 16], [8, 12], [16, 4]], 'representations': {'PC': {'code': 80742441736725980972518866951, 'gens': [1, 2, 3, 4], 'pres': [7, -2, -2, -2, -2, -2, -3, -5, 10979, 80, 10644, 102, 5381, 166, 18822]}, 'GLZN': {'d': 2, 'p': 20, 'gens': [8009, 136017, 72009, 76219, 8081, 24003, 130311]}, 'Perm': {'d': 16, 'gens': [3669246, 479001601, 482670720, 487025281, 1394852659200, 3669120, 966]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 12], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{60}:C_2^3', 'transitive_degree': 240, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '96.220', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 168, 'aut_gen_orders': [12, 6], 'aut_gens': [[1, 2, 4, 8], [7, 50, 3, 92], [2, 7, 4, 58]], 'aut_group': '43008.cc', 'aut_hash': 6154471930669946344, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 43008, 'aut_permdeg': 18, 'aut_perms': [1728407203183825, 4278499466741424], 'aut_phi_ratio': 1344.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 14, 1], [3, 1, 2, 1], [4, 1, 16, 1], [6, 1, 2, 1], [6, 1, 28, 1], [12, 1, 32, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2.C_2^7:\\GL(3,2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 168, 'autcent_group': '43008.cc', 'autcent_hash': 6154471930669946344, 'autcent_nilpotent': False, 'autcent_order': 43008, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2.C_2^7:\\GL(3,2)', '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, 15], [3, 1, 2], [4, 1, 16], [6, 1, 30], [12, 1, 32]], 'center_label': '96.220', 'center_order': 96, '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', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 220, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['3.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15], [3, 1, 2, 1], [4, 1, 2, 8], [6, 1, 2, 15], [12, 1, 4, 8]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 600, 'exponent': 12, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '48.52', 'hash': 220, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1, 1], 'inner_gens': [[1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 8]], '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, 96]], 'label': '96.220', 'linC_count': None, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, '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': 'C2^3*C12', 'ngens': 6, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 96, 'number_divisions': 48, 'number_normal_subgroups': 236, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 236, 'number_subgroups': 236, 'old_label': None, 'order': 96, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 15], [3, 2], [4, 16], [6, 30], [12, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 168, 'outer_gen_orders': [12, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[7, 50, 3, 92], [2, 7, 4, 58]], 'outer_group': '43008.cc', 'outer_hash': 6154471930669946344, 'outer_nilpotent': False, 'outer_order': 43008, 'outer_permdeg': 18, 'outer_perms': [1728407203183825, 4278499466741424], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2.C_2^7:\\GL(3,2)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 4, 3], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 24], [4, 8]], 'representations': {'PC': {'code': 6643779073, 'gens': [1, 2, 3, 4], 'pres': [6, -2, -2, -2, -2, -2, -3, 69, 88]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101743012391910, 25038659807093174, 24992906222183252, 125101750005091861]}, 'GLFq': {'d': 4, 'q': 4, 'gens': [1083295489, 1083311105, 1087522049, 2149582850, 1087570945, 40503424]}, 'Perm': {'d': 13, 'gens': [2400, 479001600, 3628800, 40320, 4, 744]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 12], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3\\times C_{12}', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
