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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '740.10', 'ambient_counter': 10, 'ambient_order': 740, 'ambient_tex': 'C_{185}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 5, 'characteristic': True, 'core_order': 370, 'counter': 2, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '740.10.2.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '2.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '2.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 2, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2', 'simple': False, 'solvable': True, 'special_labels': ['C1'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '370.2', 'subgroup_hash': 2, 'subgroup_order': 370, 'subgroup_tex': 'C_5\\times D_{37}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '740.10', 'aut_centralizer_order': 5, 'aut_label': '2.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '5328.a', 'aut_weyl_index': 5, 'centralizer': '148.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1', '10.a1.a1', '74.a1.a1'], 'core': '2.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5478, 4244, 4572, 6428, 2039, 4851, 2039, 4851], 'generators': [2, 444, 20], 'label': '740.10.2.a1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['4.a1.a1', '10.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2.a1.a1', 'projective_image': '740.10', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '2.a1.a1', 'subgroup_fusion': None, 'weyl_group': '148.3'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '10.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 1332, 'aut_gen_orders': [36, 148], 'aut_gens': [[1, 2], [1, 112], [181, 76]], 'aut_group': '5328.a', 'aut_hash': 6895881831525668864, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5328, 'aut_permdeg': 41, 'aut_perms': [18787124304315855266773495975611497414441276695458, 10296104734705126977814633722553493510897345426233], 'aut_phi_ratio': 37.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 37, 1, 1], [5, 1, 4, 1], [10, 37, 4, 1], [37, 2, 18, 1], [185, 2, 72, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times F_{37}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 1332, 'autcentquo_group': '1332.31', 'autcentquo_hash': 31, 'autcentquo_nilpotent': False, 'autcentquo_order': 1332, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{37}', 'cc_stats': [[1, 1, 1], [2, 37, 1], [5, 1, 4], [10, 37, 4], [37, 2, 18], [185, 2, 72]], 'center_label': '5.1', 'center_order': 5, 'central_product': True, 'central_quotient': '74.1', 'commutator_count': 1, 'commutator_label': '37.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '5.1', '37.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['5.1', 1], ['74.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 37, 1, 1], [5, 1, 4, 1], [10, 37, 4, 1], [37, 2, 18, 1], [185, 2, 72, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 18, 'exponent': 370, 'exponents_of_order': [1, 1, 1], 'factors_of_aut_order': [2, 3, 37], 'factors_of_order': [2, 5, 37], 'faithful_reps': [[2, 0, 72]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '370.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 74, 'inner_gen_orders': [2, 37], 'inner_gens': [[1, 72], [301, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 74, 'inner_split': False, 'inner_tex': 'D_{37}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 144, 'irrQ_dim': 144, 'irrR_degree': 4, 'irrep_stats': [[1, 10], [2, 90]], 'label': '370.2', 'linC_count': 72, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 40, 'linQ_degree_count': 2, 'linQ_dim': 40, 'linQ_dim_count': 2, 'linR_count': 108, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5*D37', 'ngens': 3, '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': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 100, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 80, 'old_label': None, 'order': 370, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 37], [5, 4], [10, 148], [37, 36], [185, 144]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 36, 'outer_gen_orders': [2, 36], 'outer_gen_pows': [1, 0], 'outer_gens': [[1, 12], [1, 346]], 'outer_group': '72.9', 'outer_hash': 9, 'outer_nilpotent': True, 'outer_order': 72, 'outer_permdeg': 15, 'outer_perms': [88216128000, 522593433], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{36}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 42, 'pgroup': 0, 'primary_abelian_invariants': [2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 2], [36, 1], [144, 1]], 'representations': {'PC': {'code': 7157540163599, 'gens': [1, 2], 'pres': [3, -2, -5, -37, 433, 34, 3242]}, 'GLFq': {'d': 2, 'q': 961, 'gens': [1906147373, 621092126533, 3550014728]}, 'Perm': {'d': 42, 'gens': [816438678154187320793278649036190835909028258640, 33, 35085403969472181749756619351513133336970056516560]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [10], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times D_{37}', 'transitive_degree': 185, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6660, 'aut_gen_orders': [36, 18, 12], 'aut_gens': [[1, 4], [465, 312], [97, 196], [581, 56]], 'aut_group': None, 'aut_hash': 1267107012035511543, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 26640, 'aut_permdeg': 185, 'aut_perms': [24452640428889607094275804779833913328370479263870281419260007735837755990078330595470606185924611085646458158818979575813768895425626874220988288787268488017798503223639740480178828631512279421056690667231731483531397287418394767091675841212173330758475514725249937151018588219010957102745192686834315942113973853472767793419592506631219501, 17010532108268305407185625416573042919814390055086272771285090274997398341335616742086801471207947082638565396844508528567820165914154000964625570271829969699563488040054463702224861899029521529052246768902954642109764521868404563944818015238721095089773241140500820388120127679059753837240050210364567849399879211535262992503234420744690354, 36953215998909788901291736630852957456575990354568574558280916441730561360580758510235636430278320636310682730012073636764297335867152249681587609758636538867555316757464940195675992262392133630359293930738794212972023286036419158954923228762390298384144340493584764350828189571987331905882783756921703642707559021910714679456465396076744676], 'aut_phi_ratio': 92.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 37, 1, 1], [4, 185, 1, 2], [5, 2, 2, 1], [10, 74, 2, 1], [37, 4, 9, 1], [185, 4, 36, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{185}.C_9.C_4^2', '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': 6660, 'autcentquo_group': None, 'autcentquo_hash': 1267107012035511543, 'autcentquo_nilpotent': False, 'autcentquo_order': 26640, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{185}.C_9.C_4^2', 'cc_stats': [[1, 1, 1], [2, 37, 1], [4, 185, 2], [5, 2, 2], [10, 74, 2], [37, 4, 9], [185, 4, 36]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '740.10', 'commutator_count': 1, 'commutator_label': '185.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1', '37.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 10, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 37, 1, 1], [4, 185, 2, 1], [5, 2, 2, 1], [10, 74, 2, 1], [37, 4, 9, 1], [185, 4, 36, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 740, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3, 5, 37], 'factors_of_order': [2, 5, 37], 'faithful_reps': [[4, 0, 36]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '740.10', 'hash': 10, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 740, 'inner_gen_orders': [4, 185], 'inner_gens': [[1, 716], [29, 4]], 'inner_hash': 10, 'inner_nilpotent': False, 'inner_order': 740, 'inner_split': True, 'inner_tex': 'C_{185}:C_4', 'inner_used': [1, 2], 'irrC_degree': 4, 'irrQ_degree': 144, 'irrQ_dim': 144, 'irrR_degree': 8, 'irrep_stats': [[1, 4], [2, 4], [4, 45]], 'label': '740.10', 'linC_count': 36, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 40, 'linQ_degree_count': 2, 'linQ_dim': 40, 'linQ_dim_count': 1, 'linR_count': 18, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C185:C4', 'ngens': 4, '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': 8, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 53, 'number_divisions': 7, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 12, 'number_subgroups': 308, 'old_label': None, 'order': 740, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 37], [4, 370], [5, 4], [10, 148], [37, 36], [185, 144]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 36, 'outer_gen_orders': [36], 'outer_gen_pows': [0], 'outer_gens': [[1, 208]], 'outer_group': '36.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 36, 'outer_permdeg': 13, 'outer_perms': [522593433], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{36}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 42, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 2], [36, 1], [144, 1]], 'representations': {'PC': {'code': 6680741259480771810366863, 'gens': [1, 3], 'pres': [4, -2, -2, -5, -37, 8, 8594, 870, 70, 9923, 5767]}, 'Perm': {'d': 42, 'gens': [837400554133458348960892098579121993814759464327, 1693079331819408831992984507949839947876676260320, 37, 35981934744669403826437909877171957673819734133120]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{185}:C_4', 'transitive_degree': 185, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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], [2, 1, 1]], 'center_label': '2.1', 'center_order': 2, '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': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', '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': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], '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': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, 'wreath_data': None, 'wreath_product': False}
