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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '432.207', 'ambient_counter': 207, 'ambient_order': 432, 'ambient_tex': 'C_4:C_4\\times \\He_3', 'central': True, 'central_factor': False, 'centralizer_order': 432, 'characteristic': True, 'core_order': 2, 'counter': 49, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '432.207.216.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '216.c1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '216.77', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 77, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 216, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_4\\times \\He_3', 'simple': True, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '432.207', 'aut_centralizer_order': 13824, 'aut_label': '216.c1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '1.1', 'aut_weyl_index': 13824, 'centralizer': '1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['72.c1', '72.f1', '108.a1', '108.c1'], 'contains': ['432.a1'], 'core': '216.c1', 'coset_action_label': None, 'count': 1, 'diagramx': [5121, 5452, 5661, 6507], 'generators': [6], 'label': '432.207.216.c1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '216.c1', 'normal_contained_in': ['72.c1', '108.a1'], 'normal_contains': ['432.a1'], 'normalizer': '1.a1', 'old_label': '216.c1', 'projective_image': '216.77', 'quotient_action_image': '1.1', 'quotient_action_kernel': '216.77', 'quotient_action_kernel_order': 216, 'quotient_fusion': None, 'short_label': '216.c1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', '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': [[1]], '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], [2, 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], [2, 1, 1]], 'center_label': '2.1', 'center_order': 2, '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': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', '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': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 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': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, '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': '72.36', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [2, 2, 2, 3, 3, 2, 3, 2, 2], 'aut_gens': [[1, 12, 36], [223, 12, 36], [5, 24, 48], [1, 12, 42], [1, 12, 60], [317, 12, 328], [217, 12, 36], [13, 12, 36], [223, 12, 252], [325, 12, 252]], 'aut_group': '13824.kv', 'aut_hash': 541400511364950914, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 13824, 'aut_permdeg': 32, 'aut_perms': [33192584704479527649787313509528608, 8556278023028667447788357822825, 239043895232341744130973213260310, 1608920457056961515523156356828944, 6113643713923863788303492317373681, 99046290532650300930828129797272608, 2436012203746524888282336000, 33395945371231387200279078179992614, 115483136967881857404321973118104614], 'aut_phi_ratio': 96.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [3, 3, 8, 1], [4, 2, 2, 1], [4, 2, 4, 1], [6, 1, 2, 3], [6, 3, 8, 3], [12, 2, 4, 1], [12, 2, 8, 1], [12, 6, 16, 1], [12, 6, 32, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6^2:(D_4\\times \\GL(2,3))', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '144.197', 'autcent_hash': 197, 'autcent_nilpotent': True, 'autcent_order': 144, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_6^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '96.189', 'autcentquo_hash': 189, 'autcentquo_nilpotent': False, 'autcentquo_order': 96, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2\\times \\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 3], [3, 1, 2], [3, 3, 8], [4, 2, 6], [6, 1, 6], [6, 3, 24], [12, 2, 12], [12, 6, 48]], 'center_label': '12.5', 'center_order': 12, 'central_product': True, 'central_quotient': '36.14', 'commutator_count': 1, 'commutator_label': '6.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 207, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['16.4', 1], ['27.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [3, 3, 2, 4], [4, 2, 1, 2], [4, 2, 2, 2], [6, 1, 2, 3], [6, 3, 2, 12], [12, 2, 2, 2], [12, 2, 4, 2], [12, 6, 2, 8], [12, 6, 4, 8]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 12, 'exponents_of_order': [4, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '12.5', 'frattini_quotient': '36.14', 'hash': 207, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [6, 1, 6], 'inner_gens': [[1, 12, 264], [1, 12, 36], [241, 12, 36]], 'inner_hash': 14, 'inner_nilpotent': True, 'inner_order': 36, 'inner_split': True, 'inner_tex': 'C_6^2', 'inner_used': [1, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 72], [2, 18], [3, 16], [6, 4]], 'label': '432.207', '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': 'C4:C4*He3', 'ngens': 7, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 18, 'number_characteristic_subgroups': 21, 'number_conjugacy_classes': 110, 'number_divisions': 48, 'number_normal_subgroups': 77, 'number_subgroup_autclasses': 50, 'number_subgroup_classes': 143, 'number_subgroups': 285, 'old_label': None, 'order': 432, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 3], [3, 26], [4, 12], [6, 78], [12, 312]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [4, 3, 2, 2, 2, 4, 4, 2], 'outer_gen_pows': [0, 0, 0, 108, 0, 0, 0, 108], 'outer_gens': [[315, 12, 328], [305, 12, 328], [7, 12, 36], [397, 24, 180], [23, 12, 192], [253, 12, 190], [109, 12, 42], [131, 12, 192]], 'outer_group': '384.17972', 'outer_hash': 17972, 'outer_nilpotent': False, 'outer_order': 384, 'outer_permdeg': 12, 'outer_perms': [127019656, 128995200, 16, 806534, 40284856, 175800378, 18, 14], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'D_4\\times \\GL(2,3)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 20], [4, 16], [6, 4], [12, 4]], 'representations': {'PC': {'code': 8750127277325805554813208684502522299619737, 'gens': [1, 4, 5], 'pres': [7, 2, 2, 3, 3, 2, 2, 3, 14, 36, 9244, 1061, 438, 102, 4037, 1776, 1027, 124, 7650, 4129, 1931]}, 'GLZN': {'d': 2, 'p': 72, 'gens': [13998097, 18538585, 374977, 26503901, 373273, 375841, 13810213]}, 'Perm': {'d': 17, 'gens': [2703049574400, 22318680384000, 138838, 276480, 44460928512000, 1037836800, 80884]}}, 'schur_multiplier': [3, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6, 12], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4:C_4\\times \\He_3', 'transitive_degree': 144, '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': '36.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [12, 6, 8], 'aut_gens': [[1, 6, 36], [186, 1, 180], [181, 70, 180], [118, 181, 180]], 'aut_group': '3456.in', 'aut_hash': 5403615404265510450, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3456, 'aut_permdeg': 13, 'aut_perms': [610167613, 526266840, 47946362], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [3, 1, 2, 1], [3, 3, 8, 1], [4, 2, 1, 1], [6, 1, 2, 1], [6, 2, 4, 1], [6, 3, 8, 1], [6, 6, 16, 1], [12, 2, 2, 1], [12, 6, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4.\\SL(3,3)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '36.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 36, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_6^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '96.189', 'autcentquo_hash': 189, 'autcentquo_nilpotent': False, 'autcentquo_order': 96, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2\\times \\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [3, 1, 2], [3, 3, 8], [4, 2, 1], [6, 1, 2], [6, 2, 4], [6, 3, 8], [6, 6, 16], [12, 2, 2], [12, 6, 8]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '36.14', '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', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 77, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['27.3', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [3, 1, 2, 1], [3, 3, 2, 4], [4, 2, 1, 1], [6, 1, 2, 1], [6, 2, 2, 2], [6, 3, 2, 4], [6, 6, 2, 8], [12, 2, 2, 1], [12, 6, 2, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 12, 'exponents_of_order': [3, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 0, 2]], 'familial': False, 'frattini_label': '6.2', 'frattini_quotient': '36.14', 'hash': 77, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [6, 6, 1], 'inner_gens': [[1, 42, 36], [181, 6, 36], [1, 6, 36]], 'inner_hash': 14, 'inner_nilpotent': True, 'inner_order': 36, 'inner_split': True, 'inner_tex': 'C_6^2', 'inner_used': [1, 2], 'irrC_degree': 6, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 36], [2, 9], [3, 8], [6, 2]], 'label': '216.77', 'linC_count': 72, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 4, 'linQ_dim': 8, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D4*He3', 'ngens': 6, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 12, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 55, 'number_divisions': 30, 'number_normal_subgroups': 42, 'number_subgroup_autclasses': 30, 'number_subgroup_classes': 88, 'number_subgroups': 190, 'old_label': None, 'order': 216, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 5], [3, 26], [4, 2], [6, 130], [12, 52]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [3, 2, 2, 2, 4, 4], 'outer_gen_pows': [0, 98, 0, 0, 0, 0], 'outer_gens': [[15, 34, 36], [15, 164, 180], [22, 27, 36], [92, 87, 36], [34, 161, 36], [89, 80, 36]], 'outer_group': '96.189', 'outer_hash': 189, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 10, 'outer_perms': [23040, 363168, 368047, 1, 1641001, 2442654], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times \\GL(2,3)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 17], [4, 4], [6, 4], [12, 1]], 'representations': {'PC': {'code': 13662848333516353254741, 'gens': [1, 3, 5], 'pres': [6, -2, -3, -2, -3, -2, -3, 12, 758, 710, 50, 2019, 1881, 88]}, 'GLZN': {'d': 2, 'p': 36, 'gens': [47089, 1166401, 46675, 47305, 638209, 793493]}, 'Perm': {'d': 13, 'gens': [1037836800, 479001600, 138838, 276480, 482630400, 80884]}}, 'schur_multiplier': [3, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6, 6], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4\\times \\He_3', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
