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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '648.297', 'ambient_counter': 297, 'ambient_order': 648, 'ambient_tex': 'C_2\\times C_3^2:D_{18}', 'central': False, 'central_factor': False, 'centralizer_order': 324, 'characteristic': True, 'core_order': 3, 'counter': 194, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '648.297.216.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '216.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '216.101', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 101, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 216, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'S_3\\times D_{18}', 'simple': True, 'solvable': True, 'special_labels': ['D2', 'C6'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '3.1', 'subgroup_hash': 1, 'subgroup_order': 3, 'subgroup_tex': 'C_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '648.297', 'aut_centralizer_order': 1944, 'aut_label': '216.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '2.1', 'aut_weyl_index': 1944, 'centralizer': '2.b1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['72.a1.a1', '72.b1.a1', '72.c1.a1', '108.a1.a1', '108.g1.a1', '108.g1.b1', '108.i1.a1', '108.i1.b1', '108.n1.a1', '108.n1.b1'], 'contains': ['648.a1.a1'], 'core': '216.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4316, 5895, 4337, 5662, 5358, 6017, 3269, 3803], 'generators': [216], 'label': '648.297.216.a1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '216.a1.a1', 'normal_contained_in': ['72.a1.a1', '72.b1.a1', '108.a1.a1'], 'normal_contains': ['648.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '216.a1.a1', 'projective_image': '648.297', 'quotient_action_image': '2.1', 'quotient_action_kernel': '108.27', 'quotient_action_kernel_order': 108, 'quotient_fusion': None, 'short_label': '216.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '3.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [2]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', '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], [3, 1, 2]], 'center_label': '3.1', 'center_order': 3, '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': ['3.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [3], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '3.1', 'hash': 1, 'hyperelementary': 3, '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 3]], 'label': '3.1', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3', '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': 3, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 3, 'order_factorization_type': 1, 'order_stats': [[1, 1], [3, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[2]], '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': 1, 'perfect': False, 'permutation_degree': 3, 'pgroup': 3, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -3]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [7]}, 'Lie': [{'d': 1, 'q': 3, 'gens': [3], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [7]}, 'Perm': {'d': 3, 'gens': [4]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [3], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3', 'transitive_degree': 3, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 18, 'aut_gen_orders': [6, 6, 6, 6, 18], 'aut_gens': [[1, 2, 12, 108], [433, 58, 60, 108], [221, 218, 492, 540], [333, 34, 240, 108], [441, 602, 276, 540], [109, 386, 48, 108]], 'aut_group': None, 'aut_hash': 4488834822908264931, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3888, 'aut_permdeg': 72, 'aut_perms': [42592552523603835658237493162212822377878690835023198355817560347027315588099878585274202070751185208957, 6973094132658639161081753543977034658186401790153717352811629913306509163456382336791595106651495795124, 37320079854059342553215069822202292412259847514241520405237458486931971500993695674806998862013846001118, 54789540716953076363800733498097168193683860921089265418986775376654868900588548243662715284193758303637, 11490110637790366514288010763094013808350260535275923862848385249087501694820401995476051208281735346386], 'aut_phi_ratio': 18.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 2, 1], [2, 27, 2, 2], [3, 2, 1, 2], [3, 4, 1, 1], [3, 6, 1, 1], [3, 12, 1, 1], [6, 2, 1, 2], [6, 4, 1, 1], [6, 6, 1, 1], [6, 12, 1, 1], [6, 18, 2, 1], [6, 54, 2, 2], [9, 6, 3, 1], [9, 12, 3, 1], [18, 6, 3, 1], [18, 12, 3, 1], [18, 18, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_3^4.C_3.C_2^3', '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': 18, 'autcentquo_group': '972.429', 'autcentquo_hash': 429, 'autcentquo_nilpotent': False, 'autcentquo_order': 972, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^3.S_3^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 9, 2], [2, 27, 4], [3, 2, 2], [3, 4, 1], [3, 6, 1], [3, 12, 1], [6, 2, 2], [6, 4, 1], [6, 6, 1], [6, 12, 1], [6, 18, 2], [6, 54, 4], [9, 6, 3], [9, 12, 3], [18, 6, 3], [18, 12, 3], [18, 18, 6]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '324.37', 'commutator_count': 1, 'commutator_label': '81.3', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 297, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['324.37', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [2, 27, 1, 4], [3, 2, 1, 2], [3, 4, 1, 1], [3, 6, 1, 1], [3, 12, 1, 1], [6, 2, 1, 2], [6, 4, 1, 1], [6, 6, 1, 1], [6, 12, 1, 1], [6, 18, 1, 2], [6, 54, 1, 4], [9, 6, 3, 1], [9, 12, 3, 1], [18, 6, 3, 1], [18, 12, 3, 1], [18, 18, 3, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 18144, 'exponent': 18, 'exponents_of_order': [4, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 1]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '72.46', 'hash': 297, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [2, 6, 9, 3], 'inner_gens': [[1, 10, 12, 540], [5, 2, 528, 540], [1, 242, 12, 108], [217, 218, 12, 108]], 'inner_hash': 37, 'inner_nilpotent': False, 'inner_order': 324, 'inner_split': True, 'inner_tex': 'C_3^2:D_{18}', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 8], [2, 20], [4, 8], [6, 4], [12, 2]], 'label': '648.297', 'linC_count': 36, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 13, 'linQ_dim': 12, 'linQ_dim_count': 13, 'linR_count': 36, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C2*C3^2:D18', '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, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 23, 'number_characteristic_subgroups': 23, 'number_conjugacy_classes': 42, 'number_divisions': 30, 'number_normal_subgroups': 39, 'number_subgroup_autclasses': 134, 'number_subgroup_classes': 206, 'number_subgroups': 2160, 'old_label': None, 'order': 648, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 127], [3, 26], [6, 278], [9, 54], [18, 162]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[325, 334, 12, 540], [1, 326, 60, 540]], 'outer_group': '12.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [24, 724], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [4, 2], [6, 8], [12, 4]], 'representations': {'PC': {'code': 290878033624618925006194142990401159984413, 'gens': [1, 2, 4, 6], 'pres': [7, -2, -2, -3, -3, -3, -2, -3, 141, 36, 170, 7402, 3125, 108, 1271, 22685, 11352, 124, 21174, 10597]}, 'Perm': {'d': 20, 'gens': [1446375508300801, 14231681682707160, 1, 135522408568781208, 19649121759590400, 8888760, 270407619022464000]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_3^2:D_{18}', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, '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': 18, 'aut_gen_orders': [2, 6, 6, 18], 'aut_gens': [[1, 2, 12], [1, 10, 12], [1, 2, 132], [9, 110, 12], [109, 62, 12]], 'aut_group': '1296.3034', 'aut_hash': 3034, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1296, 'aut_permdeg': 16, 'aut_perms': [127, 194119067536, 1463, 1495055076487], 'aut_phi_ratio': 18.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [2, 9, 2, 1], [2, 27, 2, 1], [3, 2, 1, 2], [3, 4, 1, 1], [6, 2, 1, 2], [6, 4, 1, 1], [6, 6, 2, 1], [6, 18, 2, 1], [9, 2, 3, 1], [9, 4, 3, 1], [18, 2, 3, 1], [18, 4, 3, 1], [18, 6, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_6^2.S_3^2', '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': 18, 'autcentquo_group': '324.118', 'autcentquo_hash': 118, 'autcentquo_nilpotent': False, 'autcentquo_order': 324, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^2.S_3^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [2, 9, 2], [2, 27, 2], [3, 2, 2], [3, 4, 1], [6, 2, 2], [6, 4, 1], [6, 6, 2], [6, 18, 2], [9, 2, 3], [9, 4, 3], [18, 2, 3], [18, 4, 3], [18, 6, 6]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '108.16', 'commutator_count': 1, 'commutator_label': '27.2', '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': 101, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['18.1', 1], ['2.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 9, 1, 2], [2, 27, 1, 2], [3, 2, 1, 2], [3, 4, 1, 1], [6, 2, 1, 2], [6, 4, 1, 1], [6, 6, 1, 2], [6, 18, 1, 2], [9, 2, 3, 1], [9, 4, 3, 1], [18, 2, 3, 1], [18, 4, 3, 1], [18, 6, 3, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2016, 'exponent': 18, 'exponents_of_order': [3, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 1, 3]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '72.46', 'hash': 101, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [2, 6, 9], 'inner_gens': [[1, 10, 12], [5, 2, 204], [1, 26, 12]], 'inner_hash': 16, 'inner_nilpotent': False, 'inner_order': 108, 'inner_split': True, 'inner_tex': 'S_3\\times D_9', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 20], [4, 8]], 'label': '216.101', 'linC_count': 39, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 12, 'linQ_dim': 8, 'linQ_dim_count': 12, 'linR_count': 39, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'S3*D18', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, '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': 18, 'number_characteristic_subgroups': 19, 'number_conjugacy_classes': 36, 'number_divisions': 24, 'number_normal_subgroups': 35, 'number_subgroup_autclasses': 68, 'number_subgroup_classes': 106, 'number_subgroups': 622, 'old_label': None, 'order': 216, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 79], [3, 8], [6, 56], [9, 18], [18, 54]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[109, 118, 12], [1, 110, 60]], 'outer_group': '12.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [720, 28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [4, 2], [6, 4], [12, 2]], 'representations': {'PC': {'code': 1506496786155930995531403, 'gens': [1, 2, 4], 'pres': [6, -2, -2, -3, -2, -3, -3, 121, 31, 146, 2457, 69, 2890, 118, 2603]}, 'GLZN': {'d': 2, 'p': 18, 'gens': [99161, 5869, 61399, 5849, 5941, 5995]}, 'Perm': {'d': 14, 'gens': [6, 1118437200, 1, 7784552160, 30, 14545077840]}}, 'schur_multiplier': [2, 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': 'S_3\\times D_{18}', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
