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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '162000.be', 'ambient_counter': 31, 'ambient_order': 162000, 'ambient_tex': 'C_3^3:D_5\\wr S_3', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 3, 'counter': 1227, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '162000.be.2250.q1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '2250.q1', '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': 2250, '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': '72.15', 'subgroup_hash': 15, 'subgroup_order': 72, 'subgroup_tex': 'C_2^2:D_9', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '162000.be', 'aut_centralizer_order': None, 'aut_label': '2250.q1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '162000.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['18.q1', '750.g1'], 'contains': ['4500.z1', '6750.n1', '9000.j1'], 'core': '54000.a1', 'coset_action_label': None, 'count': 750, 'diagramx': None, 'generators': [4369, 33480, 41992, 54024, 36], 'label': '162000.be.2250.q1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '2.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '750.g1', 'old_label': '2250.q1', 'projective_image': '162000.be', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '2250.q1', 'subgroup_fusion': None, 'weyl_group': '216.90'}
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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': 36, 'aut_gen_orders': [6, 9, 2, 2], 'aut_gens': [[1, 2, 18, 36], [1, 4, 54, 36], [15, 2, 36, 54], [37, 38, 18, 36], [37, 20, 18, 36]], 'aut_group': '216.90', 'aut_hash': 90, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 216, 'aut_permdeg': 13, 'aut_perms': [90887765, 1089856203, 7, 23], 'aut_phi_ratio': 9.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 18, 1, 1], [3, 2, 1, 1], [4, 18, 1, 1], [6, 6, 1, 1], [9, 8, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^2.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': 36, 'autcentquo_group': '216.90', 'autcentquo_hash': 90, 'autcentquo_nilpotent': False, 'autcentquo_order': 216, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^2.S_4', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 18, 1], [3, 2, 1], [4, 18, 1], [6, 6, 1], [9, 8, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '72.15', 'commutator_count': 1, 'commutator_label': '36.3', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 15, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 18, 1, 1], [3, 2, 1, 1], [4, 18, 1, 1], [6, 6, 1, 1], [9, 8, 3, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 9, 'exponent': 36, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 1, 1]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '24.12', 'hash': 15, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [2, 9, 2, 2], 'inner_gens': [[1, 16, 54, 36], [5, 2, 54, 18], [37, 38, 18, 36], [1, 56, 18, 36]], 'inner_hash': 15, 'inner_nilpotent': False, 'inner_order': 72, 'inner_split': True, 'inner_tex': 'C_2^2:D_9', 'inner_used': [1, 2, 3], 'irrC_degree': 6, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 6, 'irrep_stats': [[1, 2], [2, 4], [3, 2], [6, 1]], 'label': '72.15', 'linC_count': 6, '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': 6, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C2^2:D9', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, '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': 7, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 9, 'number_divisions': 7, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 18, 'number_subgroups': 80, 'old_label': None, 'order': 72, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 21], [3, 2], [4, 18], [6, 6], [9, 24]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 3, 'outer_gen_orders': [3], 'outer_gen_pows': [0], 'outer_gens': [[1, 8, 18, 36]], 'outer_group': '3.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 3, 'outer_permdeg': 3, 'outer_perms': [3], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [3, 2], [6, 2]], 'representations': {'PC': {'code': 3057659244760652, 'gens': [1, 2, 4, 5], 'pres': [5, -2, -3, -3, -2, 2, 161, 36, 182, 1083, 548, 234]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [69485621029890892, 78750816549566953]}, 'GLZN': {'d': 2, 'p': 36, 'gens': [47089, 1342423, 1656919, 909811, 70633]}, 'Perm': {'d': 13, 'gens': [39929815, 43603258, 104227, 482630400, 1037836800]}}, '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': 'C_2^2:D_9', 'transitive_degree': 18, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 180, 'aut_gen_orders': [30, 30, 12, 12], 'aut_gens': [[1, 2, 12, 72, 2160, 10800], [136513, 31402, 75180, 110556, 2160, 57504], [21481, 24322, 492, 21300, 2160, 111048], [83101, 115942, 137868, 129276, 2160, 111480], [114929, 154234, 18360, 27948, 432, 136560]], 'aut_group': None, 'aut_hash': 7939387612149648616, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1944000, 'aut_permdeg': 675, 'aut_perms': 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'aut_phi_ratio': 45.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 75, 1, 1], [2, 270, 1, 1], [2, 405, 1, 1], [2, 450, 1, 1], [2, 3375, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 6, 3, 1], [3, 1800, 1, 1], [4, 450, 1, 1], [4, 6750, 1, 1], [5, 6, 2, 1], [5, 8, 2, 1], [5, 12, 2, 1], [5, 24, 1, 1], [5, 24, 2, 1], [6, 150, 1, 2], [6, 150, 3, 1], [6, 300, 1, 1], [6, 300, 3, 1], [6, 540, 1, 1], [6, 900, 1, 1], [6, 900, 3, 1], [6, 27000, 1, 1], [9, 3600, 1, 1], [10, 150, 2, 1], [10, 540, 2, 2], [10, 900, 2, 1], [10, 1080, 2, 2], [10, 1620, 2, 2], [10, 3240, 1, 1], [12, 900, 1, 1], [12, 900, 3, 1], [12, 13500, 1, 1], [15, 12, 2, 2], [15, 12, 6, 1], [15, 16, 2, 1], [15, 24, 2, 3], [15, 24, 4, 2], [15, 24, 6, 2], [15, 48, 1, 2], [15, 48, 2, 3], [15, 48, 3, 1], [15, 48, 4, 1], [15, 48, 6, 4], [15, 48, 12, 1], [15, 3600, 2, 1], [20, 900, 2, 1], [30, 300, 2, 2], [30, 300, 6, 1], [30, 540, 4, 1], [30, 600, 2, 1], [30, 600, 6, 1], [30, 1080, 2, 1], [30, 1080, 4, 2], [30, 1800, 2, 1], [30, 1800, 6, 1], [45, 3600, 4, 1], [60, 900, 4, 1], [60, 900, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_5^3.C_6^2.(C_{12}\\times S_3^2)', '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': 180, 'autcentquo_group': None, 'autcentquo_hash': 7939387612149648616, 'autcentquo_nilpotent': False, 'autcentquo_order': 1944000, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_5^3.C_6^2.(C_{12}\\times S_3^2)', 'cc_stats': [[1, 1, 1], [2, 75, 1], [2, 270, 1], [2, 405, 1], [2, 450, 1], [2, 3375, 1], [3, 2, 1], [3, 6, 4], [3, 1800, 1], [4, 450, 1], [4, 6750, 1], [5, 6, 2], [5, 8, 2], [5, 12, 2], [5, 24, 3], [6, 150, 5], [6, 300, 4], [6, 540, 1], [6, 900, 4], [6, 27000, 1], [9, 3600, 1], [10, 150, 2], [10, 540, 4], [10, 900, 2], [10, 1080, 4], [10, 1620, 4], [10, 3240, 1], [12, 900, 4], [12, 13500, 1], [15, 12, 10], [15, 16, 2], [15, 24, 26], [15, 48, 51], [15, 3600, 2], [20, 900, 2], [30, 300, 10], [30, 540, 4], [30, 600, 8], [30, 1080, 10], [30, 1800, 8], [45, 3600, 4], [60, 900, 16]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '162000.be', 'commutator_count': 1, 'commutator_label': '40500.i', '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', '3.1', '5.1', '5.1', '5.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 31, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 75, 1, 1], [2, 270, 1, 1], [2, 405, 1, 1], [2, 450, 1, 1], [2, 3375, 1, 1], [3, 2, 1, 1], [3, 6, 1, 4], [3, 1800, 1, 1], [4, 450, 1, 1], [4, 6750, 1, 1], [5, 6, 2, 1], [5, 8, 2, 1], [5, 12, 2, 1], [5, 24, 1, 1], [5, 24, 2, 1], [6, 150, 1, 5], [6, 300, 1, 4], [6, 540, 1, 1], [6, 900, 1, 4], [6, 27000, 1, 1], [9, 3600, 1, 1], [10, 150, 2, 1], [10, 540, 2, 2], [10, 900, 2, 1], [10, 1080, 2, 2], [10, 1620, 2, 2], [10, 3240, 1, 1], [12, 900, 1, 4], [12, 13500, 1, 1], [15, 12, 2, 5], [15, 16, 2, 1], [15, 24, 2, 9], [15, 24, 4, 2], [15, 48, 1, 5], [15, 48, 2, 15], [15, 48, 4, 4], [15, 3600, 2, 1], [20, 900, 2, 1], [30, 300, 2, 5], [30, 540, 4, 1], [30, 600, 2, 4], [30, 1080, 2, 1], [30, 1080, 4, 2], [30, 1800, 2, 4], [45, 3600, 4, 1], [60, 900, 4, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1674, 'exponent': 180, 'exponents_of_order': [4, 4, 3], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[12, 0, 12], [12, 1, 12], [24, 1, 24], [48, 1, 39]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '18000.o', 'hash': 5824175395591980579, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 180, 'inner_gen_orders': [30, 6, 6, 30, 5, 15], 'inner_gens': [[1, 51082, 12, 66948, 2160, 57072], [158753, 2, 115560, 65220, 864, 117528], [1, 124214, 12, 74808, 2160, 47520], [10093, 10526, 100236, 72, 8640, 43200], [1, 3458, 12, 4392, 2160, 10800], [128761, 68306, 136092, 129672, 2160, 10800]], 'inner_hash': 5824175395591980579, 'inner_nilpotent': False, 'inner_order': 162000, 'inner_split': True, 'inner_tex': 'C_3^3:D_5\\wr S_3', 'inner_used': [1, 2], 'irrC_degree': 12, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [2, 4], [3, 4], [4, 1], [6, 26], [8, 4], [12, 54], [16, 8], [24, 58], [48, 51]], 'label': '162000.be', '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': False, 'metacyclic': False, 'monomial': None, 'name': 'C3^3:D5wrS3', 'ngens': 11, '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, 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': 78, 'number_characteristic_subgroups': 21, 'number_conjugacy_classes': 214, 'number_divisions': 114, 'number_normal_subgroups': 21, 'number_subgroup_autclasses': 1744, 'number_subgroup_classes': 2672, 'number_subgroups': 1145664, 'old_label': None, 'order': 162000, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 4575], [3, 1826], [4, 7200], [5, 124], [6, 33090], [9, 3600], [10, 18300], [12, 17100], [15, 10424], [20, 1800], [30, 35160], [45, 14400], [60, 14400]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [12], 'outer_gen_pows': [0], 'outer_gens': [[60541, 131978, 12, 50232, 4320, 149040]], 'outer_group': '12.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [963], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{12}', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [3, 4], [4, 1], [6, 18], [12, 8], [16, 2], [24, 17], [32, 2], [48, 30], [64, 1], [96, 19], [192, 4]], 'representations': {'PC': {'code': '16190948630124733450280250699790963399418818846266578694008051753514879268290953694293394224160735354602253770746669386378523527989985892188854688876077402605071839780583094130338597101027144772614878131457634669047', 'gens': [1, 2, 4, 6, 9, 10], 'pres': [11, 2, 2, 3, 2, 3, 2, 3, 5, 5, 3, 5, 24288, 1123805, 56, 1276574, 2542334, 1632997, 124, 1485015, 743186, 4418573, 2152276, 382761, 411482, 192, 5714022, 2857025, 2273530, 3735, 303, 3231367, 1615698, 95069, 12712, 42787, 1817670, 11943, 6277929, 6464060, 2055931, 435642, 66064, 438, 836362, 287517, 1163084, 1328623, 217865]}, 'Perm': {'d': 24, 'gens': [55293319886856141537754, 28263638410332205312167]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3:D_5\\wr S_3', 'transitive_degree': 45, 'wreath_data': None, 'wreath_product': False}
