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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1440.5010', 'ambient_counter': 5010, 'ambient_order': 1440, 'ambient_tex': 'C_{30}.(C_4\\times D_6)', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 9, 'counter': 171, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1440.5010.40.f1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '40.f1.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': 40, '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': '36.7', 'subgroup_hash': 7, 'subgroup_order': 36, 'subgroup_tex': 'C_3^2:C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1440.5010', 'aut_centralizer_order': 32, 'aut_label': '40.f1', 'aut_quo_index': None, 'aut_stab_index': 10, 'aut_weyl_group': '36.10', 'aut_weyl_index': 320, 'centralizer': '180.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['8.f1.a1', '20.e1.a1'], 'contains': ['80.b1.a1', '120.n1.a1', '120.o1.a1', '120.p1.a1'], 'core': '160.a1.a1', 'coset_action_label': None, 'count': 5, 'diagramx': [2981, -1, 8857, -1, 381, -1, 9469, -1], 'generators': [723, 732, 8, 960], 'label': '1440.5010.40.f1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '8.f1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '5.a1.a1', 'old_label': '40.f1.a1', 'projective_image': '1440.5010', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '40.f1.a1', 'subgroup_fusion': None, 'weyl_group': '36.10'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [2, 4, 2, 4, 3, 2, 6, 6], 'aut_gens': [[1, 4, 12], [3, 4, 12], [23, 28, 32], [25, 8, 24], [19, 16, 28], [9, 4, 12], [1, 32, 12], [19, 4, 12], [11, 20, 24]], 'aut_group': '864.4661', 'aut_hash': 4661, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 864, 'aut_permdeg': 11, 'aut_perms': [1, 10889689, 24054000, 20507935, 14605950, 2245249, 19496545, 16180705], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 4, 1], [4, 9, 2, 1], [6, 2, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_3^2:\\GL(2,3)', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '432.734', 'autcentquo_hash': 734, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 2, 4], [4, 9, 2], [6, 2, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '18.4', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 7, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 4], [4, 9, 2, 1], [6, 2, 1, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 28, 'exponent': 12, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '18.4', 'hash': 7, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3, 3], 'inner_gens': [[1, 8, 24], [9, 4, 12], [25, 4, 12]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 18, 'inner_split': False, 'inner_tex': 'C_3:S_3', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4], [2, 8]], 'label': '36.7', 'linC_count': 18, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 18, 'linQ_dim': 6, 'linQ_dim_count': 18, 'linR_count': 18, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^2:C4', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 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': 5, 'number_conjugacy_classes': 12, 'number_divisions': 11, 'number_normal_subgroups': 13, 'number_subgroup_autclasses': 9, 'number_subgroup_classes': 18, 'number_subgroups': 34, 'old_label': None, 'order': 36, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [3, 8], [4, 18], [6, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2], 'outer_gen_pows': [0, 0, 0, 1, 1], 'outer_gens': [[3, 24, 8], [3, 4, 12], [1, 24, 28], [1, 32, 20], [1, 20, 16]], 'outer_group': '48.48', 'outer_hash': 48, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 6, 'outer_perms': [143, 127, 576, 126, 121], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 9]], 'representations': {'PC': {'code': 11576083, 'gens': [1, 3, 4], 'pres': [4, -2, -2, -3, -3, 8, 98, 387]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125124616718753647, 58415899998881273, 125101736062734304]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [11689448, 17625105, 28816400, 37924170]}, 'Perm': {'d': 10, 'gens': [41185, 1680, 403203, 3]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2:C_4', 'transitive_degree': 36, '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': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 60, 'aut_gen_orders': [6, 20, 60, 60, 12, 30, 2], 'aut_gens': [[1, 2, 24], [737, 770, 1416], [1081, 218, 24], [1089, 986, 984], [1089, 1130, 984], [361, 34, 168], [1097, 1130, 744], [1, 442, 696]], 'aut_group': None, 'aut_hash': 3555297260956627131, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 11520, 'aut_permdeg': 72, 'aut_perms': [17934984785178264638378021674029392721292336189041559975144454882645738827723301681501127934961776587347, 49619403316558637895447253219175843564119549460439016115525801443547264333067592681295854018058317006358, 55513156026908319172435874163365180205628754534825569768007839164365081785545410669990897104247631449412, 55441284375207262871139882109632679118006847574943848515672247766228295892768661871888937606494012813612, 21678845095738879225948798728750813175025317004090499478341374500389015736085087728116709107483606969312, 3084158880278779900543543377706409636344369733744445813072779278364824452506477289449094807185582582211, 191982484347994560268850956851654597459912062108535923872402462698752415162616115120930922437145403242], 'aut_phi_ratio': 30.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 1, 2], [3, 2, 1, 2], [3, 4, 1, 1], [4, 2, 1, 1], [4, 6, 2, 1], [4, 10, 1, 1], [4, 30, 2, 3], [4, 45, 2, 2], [4, 90, 1, 2], [5, 4, 1, 1], [6, 2, 1, 2], [6, 4, 1, 1], [6, 10, 1, 4], [6, 20, 1, 2], [10, 4, 1, 1], [12, 4, 1, 2], [12, 4, 2, 1], [12, 12, 2, 1], [12, 20, 1, 2], [12, 20, 2, 1], [12, 60, 2, 3], [15, 4, 2, 1], [15, 8, 1, 1], [15, 8, 2, 1], [20, 8, 1, 1], [20, 24, 2, 1], [30, 4, 2, 1], [30, 8, 1, 1], [30, 8, 2, 1], [60, 8, 2, 2], [60, 8, 4, 1], [60, 24, 4, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_6\\times S_3\\times D_5).C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '1440.5889', 'autcentquo_hash': 5889, 'autcentquo_nilpotent': False, 'autcentquo_order': 1440, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_5.D_6^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 5, 2], [3, 2, 2], [3, 4, 1], [4, 2, 1], [4, 6, 2], [4, 10, 1], [4, 30, 6], [4, 45, 4], [4, 90, 2], [5, 4, 1], [6, 2, 2], [6, 4, 1], [6, 10, 4], [6, 20, 2], [10, 4, 1], [12, 4, 4], [12, 12, 2], [12, 20, 4], [12, 60, 6], [15, 4, 2], [15, 8, 3], [20, 8, 1], [20, 24, 2], [30, 4, 2], [30, 8, 3], [60, 8, 8], [60, 24, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '720.804', 'commutator_count': 1, 'commutator_label': '90.10', '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', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 5010, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 1, 2], [3, 2, 1, 2], [3, 4, 1, 1], [4, 2, 1, 1], [4, 6, 1, 2], [4, 10, 1, 1], [4, 30, 1, 2], [4, 30, 2, 2], [4, 45, 2, 2], [4, 90, 2, 1], [5, 4, 1, 1], [6, 2, 1, 2], [6, 4, 1, 1], [6, 10, 1, 4], [6, 20, 1, 2], [10, 4, 1, 1], [12, 4, 1, 2], [12, 4, 2, 1], [12, 12, 1, 2], [12, 20, 1, 2], [12, 20, 2, 1], [12, 60, 1, 2], [12, 60, 2, 2], [15, 4, 2, 1], [15, 8, 1, 1], [15, 8, 2, 1], [20, 8, 1, 1], [20, 24, 1, 2], [30, 4, 2, 1], [30, 8, 1, 1], [30, 8, 2, 1], [60, 8, 2, 2], [60, 8, 4, 1], [60, 24, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 64512, 'exponent': 60, 'exponents_of_order': [5, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[8, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '720.804', 'hash': 5010, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 12, 30], 'inner_gens': [[1, 10, 744], [17, 2, 1128], [721, 338, 24]], 'inner_hash': 804, 'inner_nilpotent': False, 'inner_order': 720, 'inner_split': True, 'inner_tex': 'D_{10}.S_3^2', 'inner_used': [1, 2, 3], 'irrC_degree': 8, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 16, 'irrep_stats': [[1, 16], [2, 20], [4, 24], [8, 15]], 'label': '1440.5010', 'linC_count': 324, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 384, 'linQ_dim': 12, 'linQ_dim_count': 224, 'linR_count': 16, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C30.(C4*D6)', 'ngens': 8, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 50, 'number_characteristic_subgroups': 58, 'number_conjugacy_classes': 75, 'number_divisions': 55, 'number_normal_subgroups': 88, 'number_subgroup_autclasses': 234, 'number_subgroup_classes': 310, 'number_subgroups': 2388, 'old_label': None, 'order': 1440, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 11], [3, 8], [4, 564], [5, 4], [6, 88], [10, 4], [12, 480], [15, 32], [20, 56], [30, 32], [60, 160]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 360], 'outer_gens': [[1, 10, 24], [1, 2, 984], [721, 10, 24], [361, 362, 24]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 18], [4, 15], [8, 9], [16, 4], [32, 1]], 'representations': {'PC': {'code': 3325937840427595664100828815536067748258052365523813900062952001044743, 'gens': [1, 2, 5], 'pres': [8, -2, -2, -2, -3, -2, -2, -3, -5, 5760, 161, 41, 482, 66, 515, 29764, 22572, 11780, 116, 19597, 10965, 141, 5390, 5398, 222, 18447, 18455]}, 'Perm': {'d': 19, 'gens': [377918376086184, 7348561, 12516744, 753220451269704, 15301704, 3, 6706022400, 6799906713600000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{30}.(C_4\\times D_6)', 'transitive_degree': 120, 'wreath_data': None, 'wreath_product': False}
