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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1728.34389', 'ambient_counter': 34389, 'ambient_order': 1728, 'ambient_tex': '(C_2\\times C_4).S_3^3', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 144, 'counter': 97, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1728.34389.6.bq1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '6.bq1', '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': 6, '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': '288.543', 'subgroup_hash': 543, 'subgroup_order': 288, 'subgroup_tex': 'C_6^2.C_2^3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1728.34389', 'aut_centralizer_order': None, 'aut_label': '6.bq1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '432.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.l1', '3.c1'], 'contains': ['12.t1', '12.dc1', '12.df1', '12.dg1', '12.ej1', '12.ep1', '12.es1', '18.bm1', '18.bo1'], 'core': '12.t1', 'coset_action_label': None, 'count': 3, 'diagramx': None, 'generators': [1, 76, 864, 438, 576, 36, 72], 'label': '1728.34389.6.bq1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '2.l1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.c1', 'old_label': '6.bq1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.bq1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', '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, 2, 2, 2, 6, 2, 2], 'aut_gens': [[1, 2, 24], [165, 166, 24], [165, 166, 132], [21, 214, 180], [165, 10, 180], [13, 22, 36], [13, 146, 132], [1, 98, 168], [153, 22, 180], [13, 262, 264]], 'aut_group': None, 'aut_hash': 6573854338112168884, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2304, 'aut_permdeg': 28, 'aut_perms': [83738577597162494128866296965, 80495933232837991512095709737, 112132595089834682264741804491, 76641552649068317904190381024, 158719794160873233045241915344, 162171450806465726251897513110, 5760583749142459862804858964, 235361317376224888600372372148, 160072994428992395427459506623], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 18, 2, 1], [3, 2, 1, 2], [3, 4, 1, 1], [4, 4, 1, 1], [4, 6, 2, 2], [4, 12, 1, 2], [4, 36, 1, 1], [6, 2, 1, 6], [6, 4, 1, 3], [12, 4, 2, 2], [12, 4, 4, 1], [12, 12, 2, 4]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times D_6^2.C_2^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '36.10', 'autcentquo_hash': 10, 'autcentquo_nilpotent': False, 'autcentquo_order': 36, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3^2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 18, 2], [3, 2, 2], [3, 4, 1], [4, 4, 1], [4, 6, 4], [4, 12, 2], [4, 36, 1], [6, 2, 6], [6, 4, 3], [12, 4, 8], [12, 12, 8]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '72.46', 'commutator_count': 1, 'commutator_label': '36.14', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 543, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 18, 1, 2], [3, 2, 1, 2], [3, 4, 1, 1], [4, 4, 1, 1], [4, 6, 1, 2], [4, 6, 2, 1], [4, 12, 1, 2], [4, 36, 1, 1], [6, 2, 1, 6], [6, 4, 1, 3], [12, 4, 2, 2], [12, 4, 4, 1], [12, 12, 1, 2], [12, 12, 2, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2688, 'exponent': 12, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '72.46', 'hash': 543, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6, 6], 'inner_gens': [[1, 10, 36], [17, 2, 264], [13, 50, 24]], 'inner_hash': 46, 'inner_nilpotent': False, 'inner_order': 72, 'inner_split': True, 'inner_tex': 'S_3\\times D_6', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 22], [4, 12]], 'label': '288.543', 'linC_count': 16, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 133, 'linQ_dim': 10, 'linQ_dim_count': 148, 'linR_count': 8, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6^2.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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 30, 'number_characteristic_subgroups': 44, 'number_conjugacy_classes': 42, 'number_divisions': 33, 'number_normal_subgroups': 50, 'number_subgroup_autclasses': 143, 'number_subgroup_classes': 165, 'number_subgroups': 698, 'old_label': None, 'order': 288, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 39], [3, 8], [4, 88], [6, 24], [12, 128]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[13, 10, 120], [145, 10, 24], [1, 22, 24], [1, 146, 24], [1, 2, 36]], 'outer_group': '32.51', 'outer_hash': 51, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 10, 'outer_perms': [362880, 5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [4, 9], [8, 4]], 'representations': {'PC': {'code': 19560033491580647236981757884051356553, 'gens': [1, 2, 5], 'pres': [7, 2, 2, 2, 3, 2, 2, 3, 84, 141, 36, 422, 58, 451, 1264, 4631, 102, 5052, 124, 4717]}, 'GLZN': {'d': 2, 'p': 66, 'gens': [17335236, 6612431, 433456, 8063983, 4773214, 4949389, 12362371]}, 'Perm': {'d': 18, 'gens': [358477173701119, 359791121901439, 127020593, 40298299, 18619, 776764801612800, 1129924059264000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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_6^2.C_2^3', 'transitive_degree': 48, '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': '16.14', '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, 12, 144], [73, 2, 60, 720], [865, 2, 60, 720], [1, 74, 12, 144], [1, 874, 12, 720], [1, 10, 132, 720], [1, 10, 876, 144], [1, 2, 60, 216], [1, 10, 60, 1008], [1, 10, 12, 144], [1, 2, 60, 144], [1, 2, 12, 720], [77, 2, 12, 144], [1, 50, 12, 144], [1, 2, 588, 144]], 'aut_group': None, '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, 3], [2, 36, 1, 1], [2, 54, 2, 1], [2, 108, 1, 1], [3, 2, 1, 3], [3, 4, 1, 3], [3, 8, 1, 1], [4, 4, 1, 1], [4, 6, 2, 1], [4, 12, 1, 5], [4, 18, 2, 1], [4, 36, 1, 4], [6, 2, 1, 9], [6, 4, 1, 9], [6, 8, 1, 3], [6, 72, 1, 1], [12, 4, 2, 2], [12, 4, 4, 1], [12, 8, 1, 1], [12, 8, 2, 2], [12, 8, 4, 1], [12, 12, 2, 6], [12, 12, 4, 2], [12, 24, 1, 6], [12, 24, 2, 4], [12, 36, 2, 1], [12, 72, 1, 4]], 'aut_supersolvable': None, 'aut_tex': None, '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, 3], [2, 36, 1], [2, 54, 2], [2, 108, 1], [3, 2, 3], [3, 4, 3], [3, 8, 1], [4, 4, 1], [4, 6, 2], [4, 12, 5], [4, 18, 2], [4, 36, 4], [6, 2, 9], [6, 4, 9], [6, 8, 3], [6, 72, 1], [12, 4, 8], [12, 8, 9], [12, 12, 20], [12, 24, 14], [12, 36, 2], [12, 72, 4]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '432.759', 'commutator_count': 1, 'commutator_label': '108.45', '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': 34389, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 36, 1, 1], [2, 54, 1, 2], [2, 108, 1, 1], [3, 2, 1, 3], [3, 4, 1, 3], [3, 8, 1, 1], [4, 4, 1, 1], [4, 6, 2, 1], [4, 12, 1, 5], [4, 18, 2, 1], [4, 36, 1, 4], [6, 2, 1, 9], [6, 4, 1, 9], [6, 8, 1, 3], [6, 72, 1, 1], [12, 4, 2, 4], [12, 8, 1, 1], [12, 8, 2, 4], [12, 12, 2, 8], [12, 12, 4, 1], [12, 24, 1, 8], [12, 24, 2, 3], [12, 36, 2, 1], [12, 72, 1, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 44291520, 'exponent': 12, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '432.759', 'hash': 34389, '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': 432, 'inner_split': None, 'inner_tex': 'C_2\\times S_3^3', 'inner_used': None, 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 28], [4, 52], [8, 12]], 'label': '1728.34389', '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': '(C2*C4).S3^3', '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': 78, 'number_characteristic_subgroups': 164, 'number_conjugacy_classes': 108, 'number_divisions': 83, 'number_normal_subgroups': 166, 'number_subgroup_autclasses': 1092, 'number_subgroup_classes': 1192, 'number_subgroups': 12292, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 255], [3, 26], [4, 256], [6, 150], [12, 1040]], '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': '128.2328', 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 128, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': 'C_2^7', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 25, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 24], [4, 16], [8, 23], [16, 4]], 'representations': {'PC': {'code': 98477231995722719070643908296781553060419596718591453751789115493301833, 'gens': [1, 2, 4, 7], 'pres': [9, 2, 2, 3, 2, 2, 3, 2, 2, 3, 7776, 15733, 46, 2162, 506, 31539, 16644, 102, 2713, 130, 2606, 68046, 6819, 3813, 186, 8674, 214, 7811]}, 'Perm': {'d': 25, 'gens': [703682966903089024051336, 1292603276293179862636943, 626229105017526010430640, 1965987003255273782784000, 21010450809600, 2615754178952280317952000, 325, 435, 45360]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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_2\\times C_4).S_3^3', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
