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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '25920.a', 'ambient_counter': 1, 'ambient_order': 25920, 'ambient_tex': '\\SU(4,2)', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 1, 'counter': 59, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '25920.a.1080.i1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '1080.i1.a1', 'outer_equivalence': False, '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': 1080, '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': '24.12', 'subgroup_hash': 12, 'subgroup_order': 24, 'subgroup_tex': 'S_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '25920.a', 'aut_centralizer_order': None, 'aut_label': '1080.i1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '12960.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['40.b1.a1', '216.a1.a1', '540.e1.a1'], 'contains': ['2160.f1.a1', '3240.f1.a1', '4320.d1.a1'], 'core': '25920.a1.a1', 'coset_action_label': None, 'count': 540, 'diagramx': [5393, -1, 4164, -1, 4756, -1, 4781, -1], 'generators': [1426395393, 2454495048, 685184840, 1430327621], 'label': '25920.a.1080.i1.a1', 'mobius_quo': None, 'mobius_sub': 2, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '540.e1.a1', 'old_label': '1080.i1.a1', 'projective_image': '25920.a', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '1080.i1.a1', 'subgroup_fusion': None, 'weyl_group': '24.12'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 3, 2, 2], 'aut_gens': [[2, 4, 16, 7], [5, 3, 23, 7], [5, 4, 7, 23], [2, 8, 16, 7], [21, 19, 16, 7]], 'aut_group': '24.12', 'aut_hash': 12, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24, 'aut_permdeg': 4, 'aut_perms': [2, 4, 16, 7], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 6, 1, 1], [3, 8, 1, 1], [4, 6, 1, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_4', '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': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 6, 1], [3, 8, 1], [4, 6, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '24.12', 'commutator_count': 1, 'commutator_label': '12.3', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 12, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 6, 1, 1], [3, 8, 1, 1], [4, 6, 1, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 9, 'exponent': 12, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[3, 1, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '24.12', 'hash': 12, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 3, 2, 2], 'inner_gens': [[2, 3, 7, 16], [5, 4, 7, 23], [21, 19, 16, 7], [21, 15, 16, 7]], 'inner_hash': 12, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': True, 'inner_tex': 'S_4', 'inner_used': [1, 2, 3], 'irrC_degree': 3, 'irrQ_degree': 3, 'irrQ_dim': 3, 'irrR_degree': 3, 'irrep_stats': [[1, 2], [2, 1], [3, 2]], 'label': '24.12', 'linC_count': 2, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 2, 'linQ_dim': 3, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'S4', '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': 4, 'number_conjugacy_classes': 5, 'number_divisions': 5, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 11, 'number_subgroup_classes': 11, 'number_subgroups': 30, 'old_label': None, 'order': 24, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 9], [3, 8], [4, 6]], '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': 4, 'perfect': False, 'permutation_degree': 4, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [3, 2]], 'representations': {'PC': {'code': 8281755524, 'gens': [1, 2, 3, 4], 'pres': [4, -2, -3, -2, 2, 33, 146, 114, 99, 55]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [10644, 10320]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [29, 23], 'family': 'PGL'}, {'d': 3, 'q': 3, 'gens': [1699, 13205], 'family': 'SO'}, {'d': 3, 'q': 3, 'gens': [1699, 13205], 'family': 'PSO'}, {'d': 3, 'q': 3, 'gens': [1557, 1699, 13205], 'family': 'PGO'}, {'d': 2, 'q': 3, 'gens': [362, 4377], 'family': 'PGU'}, {'d': 3, 'q': 3, 'gens': [1699, 13205], 'family': 'CSO'}, {'d': 2, 'q': 3, 'gens': [29, 23], 'family': 'PGammaL'}, {'d': 2, 'q': 3, 'gens': [362, 4377], 'family': 'PGammaU'}, {'d': 2, 'q': 2, 'gens': [266, 337, 275], 'family': 'AGL'}, {'d': 2, 'q': 2, 'gens': [266, 337, 275], 'family': 'ASL'}, {'d': 1, 'q': 4, 'gens': [3, 7, 1], 'family': 'AGammaL'}, {'d': 2, 'q': 2, 'gens': [7, 2, 5], 'family': 'AGammaL'}, {'d': 2, 'q': 2, 'gens': [7, 2, 5], 'family': 'ASigmaL'}], 'GLFp': {'d': 3, 'p': 2, 'gens': [465, 458, 314, 124]}, 'Perm': {'d': 4, 'gens': [2, 4, 16, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'S_4', 'transitive_degree': 4, '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': '1.1', 'all_subgroups_known': True, 'almost_simple': True, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 360, 'aut_gen_orders': [3, 6, 2], 'aut_gens': [[2150632450, 71565585], [3231746111, 3771445358], [3290499075, 814879308], [3433170959, 246877160]], 'aut_group': '51840.b', 'aut_hash': 7911298588026999171, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 51840, 'aut_permdeg': 45, 'aut_perms': [27624667074147629476400292389168271178259146281872107783, 118944471835762390221799774680973018192366437004614902604, 26704987568569967532501857924165238007975094822157225938], 'aut_phi_ratio': 7.5, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 45, 1, 1], [2, 270, 1, 1], [3, 40, 2, 1], [3, 240, 1, 1], [3, 480, 1, 1], [4, 540, 1, 1], [4, 3240, 1, 1], [5, 5184, 1, 1], [6, 360, 2, 1], [6, 720, 2, 1], [6, 1440, 1, 1], [6, 2160, 1, 1], [9, 2880, 2, 1], [12, 2160, 2, 1]], 'aut_supersolvable': False, 'aut_tex': '\\SO(5,3)', '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': 360, 'autcentquo_group': '51840.b', 'autcentquo_hash': 7911298588026999171, 'autcentquo_nilpotent': False, 'autcentquo_order': 51840, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\SO(5,3)', 'cc_stats': [[1, 1, 1], [2, 45, 1], [2, 270, 1], [3, 40, 2], [3, 240, 1], [3, 480, 1], [4, 540, 1], [4, 3240, 1], [5, 5184, 1], [6, 360, 2], [6, 720, 2], [6, 1440, 1], [6, 2160, 1], [9, 2880, 2], [12, 2160, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '25920.a', 'commutator_count': 1, 'commutator_label': '25920.a', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['25920.a'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 45, 1, 1], [2, 270, 1, 1], [3, 40, 2, 1], [3, 240, 1, 1], [3, 480, 1, 1], [4, 540, 1, 1], [4, 3240, 1, 1], [5, 5184, 1, 1], [6, 360, 2, 1], [6, 720, 2, 1], [6, 1440, 1, 1], [6, 2160, 1, 1], [9, 2880, 2, 1], [12, 2160, 2, 1]], 'element_repr_type': 'Lie', 'elementary': 1, 'eulerian_function': 11505, 'exponent': 180, 'exponents_of_order': [6, 4, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[5, 0, 2], [6, 1, 1], [10, 0, 2], [15, 1, 2], [20, 1, 1], [24, 1, 1], [30, 0, 2], [30, 1, 1], [40, 0, 2], [45, 0, 2], [60, 1, 1], [64, 1, 1], [81, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '25920.a', 'hash': 1186890006331080433, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 180, 'inner_gen_orders': [3, 6], 'inner_gens': [[2150632450, 147063585], [3290499075, 71565585]], 'inner_hash': 1186890006331080433, 'inner_nilpotent': False, 'inner_order': 25920, 'inner_split': True, 'inner_tex': '\\SU(4,2)', 'inner_used': [1, 2], 'irrC_degree': 5, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 6, 'irrep_stats': [[1, 1], [5, 2], [6, 1], [10, 2], [15, 2], [20, 1], [24, 1], [30, 3], [40, 2], [45, 2], [60, 1], [64, 1], [81, 1]], 'label': '25920.a', 'linC_count': 2, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'SU(4,2)', 'ngens': 2, '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, 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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 15, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 20, 'number_divisions': 15, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 116, 'number_subgroup_classes': 116, 'number_subgroups': 45649, 'old_label': None, 'order': 25920, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 315], [3, 800], [4, 3780], [5, 5184], [6, 5760], [9, 5760], [12, 4320]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [2222001158], 'outer_gens': [[2297629706, 3904659290]], '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': None, 'perfect': True, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [], 'quasisimple': True, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [6, 1], [10, 1], [15, 2], [20, 2], [24, 1], [30, 1], [60, 2], [64, 1], [80, 1], [81, 1], [90, 1]], 'representations': {'Lie': [{'d': 4, 'q': 2, 'gens': [2150632450, 71565585], 'family': 'SU'}, {'d': 4, 'q': 3, 'gens': [3385567, 28757108], 'family': 'PSp'}, {'d': 4, 'q': 2, 'family': 'PSU'}, {'d': 5, 'q': 3, 'gens': [282820695283, 94158610302], 'family': 'Omega'}, {'d': 6, 'q': 2, 'gens': [34629271681, 17200073026], 'family': 'OmegaMinus'}, {'d': 4, 'q': 2, 'gens': [71565585, 2148533250], 'family': 'PGU'}, {'d': 5, 'q': 3, 'family': 'POmega'}, {'d': 6, 'q': 2, 'family': 'POmegaMinus'}, {'d': 6, 'q': 2, 'gens': [86400144245811134927144401160993243137, 85070997379664633660922008431921201153, 148873860048321150973274748855897227265, 85070916254977979211381833725333733377, 85074810472670157953893574004900626445, 106338564182894358030527074258573918209, 85072214324240871650613377003277254661], 'family': 'SpinMinus'}, {'d': 4, 'q': 2, 'family': 'CSU'}, {'d': 4, 'q': 3, 'family': 'PSigmaSp'}], 'Perm': {'d': 27, 'gens': [691971538873295846472731478, 6184119457851162821995261590]}}, 'schur_multiplier': [2], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\SU(4,2)', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}
