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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1728.47288', 'ambient_counter': 47288, 'ambient_order': 1728, 'ambient_tex': 'C_{12}.D_6^2', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 4, 'counter': 826, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1728.47288.72.eo1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '72.eo1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 72, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '24.5', 'subgroup_hash': 5, 'subgroup_order': 24, 'subgroup_tex': 'C_4\\times S_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1728.47288', 'aut_centralizer_order': None, 'aut_label': '72.eo1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '432.b1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['24.en1', '24.ep1', '24.eu1', '36.bi1'], 'contains': ['144.bc1', '144.ca1', '144.cc1', '216.bh1'], 'core': '432.b1', 'coset_action_label': None, 'count': 72, 'diagramx': None, 'generators': [1006, 432, 864, 632], 'label': '1728.47288.72.eo1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '8.bh1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '36.bi1', 'old_label': '72.eo1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '72.eo1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 2, 6], 'aut_gens': [[1, 2], [13, 2], [1, 10], [9, 14]], 'aut_group': '24.14', 'aut_hash': 14, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24, 'aut_permdeg': 7, 'aut_perms': [7, 136, 856], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [3, 2, 1, 1], [4, 1, 2, 1], [4, 3, 2, 1], [6, 2, 1, 1], [12, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_6', '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': '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, 1], [2, 3, 2], [3, 2, 1], [4, 1, 2], [4, 3, 2], [6, 2, 1], [12, 2, 2]], 'center_label': '4.1', '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': 5, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 2, 1, 1], [4, 1, 2, 1], [4, 3, 2, 1], [6, 2, 1, 1], [12, 2, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 12, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '12.4', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 10], [17, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 4]], 'label': '24.5', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 5, 'linQ_dim': 4, 'linQ_dim_count': 5, 'linR_count': 5, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4*S3', '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': 8, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 12, 'number_divisions': 9, 'number_normal_subgroups': 11, 'number_subgroup_autclasses': 14, 'number_subgroup_classes': 16, 'number_subgroups': 26, 'old_label': None, 'order': 24, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 8], [6, 2], [12, 4]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 14], [13, 2]], '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, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 1]], 'representations': {'PC': {'code': 2949342177, 'gens': [1, 2], 'pres': [4, -2, -2, -2, -3, 81, 21, 242, 34, 259]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [35931561, 26129103]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [539, 501]}, 'Perm': {'d': 7, 'gens': [127, 17, 7, 840]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4\\times S_3', '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': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': None, 'aut_cyclic': None, 'aut_derived_length': None, 'aut_exponent': None, 'aut_gen_orders': None, 'aut_gens': [[1, 2, 4, 24, 144], [865, 2, 20, 24, 720], [1, 866, 4, 24, 144], [1, 2, 868, 120, 144], [1, 2, 20, 888, 144], [1, 2, 4, 24, 1584], [3, 2, 588, 120, 1304], [505, 2, 524, 24, 144], [433, 2, 20, 120, 720], [1, 2, 20, 24, 144], [1, 2, 4, 120, 144], [1, 2, 4, 24, 720], [577, 578, 4, 24, 144], [1, 10, 4, 24, 144], [1, 2, 52, 24, 144]], 'aut_group': '3888.dy', 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 55296, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': None, 'aut_solvable': None, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 6, 2, 1], [2, 9, 2, 1], [2, 18, 1, 1], [2, 18, 4, 1], [2, 54, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 4, 1, 1], [3, 4, 2, 1], [3, 8, 1, 1], [4, 2, 1, 2], [4, 6, 2, 1], [4, 6, 8, 1], [4, 18, 1, 2], [4, 18, 4, 1], [4, 54, 2, 1], [6, 2, 1, 1], [6, 2, 2, 2], [6, 4, 1, 1], [6, 4, 2, 2], [6, 4, 4, 1], [6, 8, 1, 2], [6, 8, 2, 1], [6, 12, 4, 1], [6, 18, 2, 2], [6, 24, 2, 1], [6, 36, 4, 1], [12, 4, 1, 2], [12, 4, 2, 3], [12, 8, 1, 1], [12, 8, 2, 4], [12, 12, 4, 1], [12, 12, 8, 2], [12, 24, 2, 1], [12, 24, 8, 1], [12, 36, 1, 2], [12, 36, 4, 1]], 'aut_supersolvable': None, 'aut_tex': 'C_3\\wr C_4:D_6', 'autcent_abelian': None, 'autcent_cyclic': None, 'autcent_exponent': None, 'autcent_group': None, 'autcent_hash': None, 'autcent_nilpotent': None, 'autcent_order': None, 'autcent_solvable': None, 'autcent_split': None, 'autcent_supersolvable': None, 'autcent_tex': None, 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': None, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 6, 2], [2, 9, 2], [2, 18, 5], [2, 54, 2], [3, 2, 3], [3, 4, 3], [3, 8, 1], [4, 2, 2], [4, 6, 10], [4, 18, 6], [4, 54, 2], [6, 2, 5], [6, 4, 9], [6, 8, 4], [6, 12, 4], [6, 18, 4], [6, 24, 2], [6, 36, 4], [12, 4, 8], [12, 8, 9], [12, 12, 20], [12, 24, 10], [12, 36, 6]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '864.4704', 'commutator_count': 1, 'commutator_label': '54.15', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 47288, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 6, 1, 2], [2, 9, 1, 2], [2, 18, 1, 5], [2, 54, 1, 2], [3, 2, 1, 3], [3, 4, 1, 3], [3, 8, 1, 1], [4, 2, 1, 2], [4, 6, 1, 10], [4, 18, 1, 6], [4, 54, 1, 2], [6, 2, 1, 3], [6, 2, 2, 1], [6, 4, 1, 5], [6, 4, 2, 2], [6, 8, 1, 2], [6, 8, 2, 1], [6, 12, 1, 4], [6, 18, 1, 2], [6, 18, 2, 1], [6, 24, 1, 2], [6, 36, 1, 4], [12, 4, 1, 6], [12, 4, 2, 1], [12, 8, 1, 5], [12, 8, 2, 2], [12, 12, 1, 20], [12, 24, 1, 10], [12, 36, 1, 6]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 79994880000, 'exponent': 12, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '864.4704', 'hash': 47288, 'hyperelementary': 1, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': None, 'inner_gens': None, 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': 864, 'inner_split': None, 'inner_tex': 'S_3\\times D_6^2', 'inner_used': None, 'irrC_degree': 8, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 16, 'irrep_stats': [[1, 32], [2, 48], [4, 30], [8, 16]], 'label': '1728.47288', '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': 'C12.D6^2', 'ngens': 9, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 54, 'number_characteristic_subgroups': 66, 'number_conjugacy_classes': 126, 'number_divisions': 118, 'number_normal_subgroups': 640, 'number_subgroup_autclasses': 1114, 'number_subgroup_classes': 3672, 'number_subgroups': 25236, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 231], [3, 26], [4, 280], [6, 390], [12, 800]], 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': True, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': '64.267', 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 64, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': 'C_2^6', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 25, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 48], [4, 26], [8, 8], [16, 3], [32, 1]], 'representations': {'PC': {'code': 229070623739173035642210258119991960129674048351237527102925353, 'gens': [1, 2, 3, 5, 7], 'pres': [9, -2, -2, -2, -3, -2, -3, -2, -2, -3, 7776, 281, 74, 300, 11092, 130, 1319, 99798, 22695, 186, 103687, 51856, 214, 93320, 46673]}, 'Perm': {'d': 25, 'gens': [626337036168515972334720, 1401651785355361106515343, 2042528898251588046048136, 2698125482978524017619200, 3346573550380886304057600, 3984710308819541882380800, 325, 45360, 435]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 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_{12}.D_6^2', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
