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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '2023.a', 'ambient_counter': 1, 'ambient_order': 2023, 'ambient_tex': 'C_{17}\\times C_{119}', 'central': True, 'central_factor': False, 'centralizer_order': 2023, 'characteristic': False, 'core_order': 119, 'counter': 14, 'cyclic': True, 'direct': True, 'hall': 0, 'label': '2023.a.17.a1.l1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '17.a1.l1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '17.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 17, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{17}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '119.1', 'subgroup_hash': 1, 'subgroup_order': 119, 'subgroup_tex': 'C_{119}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '2023.a', 'aut_centralizer_order': 272, 'aut_label': '17.a1', 'aut_quo_index': 1, 'aut_stab_index': 18, 'aut_weyl_group': '96.59', 'aut_weyl_index': 4896, 'centralizer': '1.a1.a1', 'complements': ['119.a1.a1', '119.a1.n1', '119.a1.c1', '119.a1.o1', '119.a1.j1', '119.a1.k1', '119.a1.m1', '119.a1.p1', '119.a1.h1', '119.a1.g1', '119.a1.f1', '119.a1.e1', '119.a1.d1', '119.a1.b1', '119.a1.q1', '119.a1.i1', '119.a1.r1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['119.a1.l1', '289.a1.a1'], 'core': '17.a1.l1', 'coset_action_label': None, 'count': 1, 'diagramx': [7105, 7105, 8558, 8558, 7899, 7899, 5221, 5221], 'generators': [1445, 120], 'label': '2023.a.17.a1.l1', 'mobius_quo': 1, 'mobius_sub': -1, 'normal_closure': '17.a1.l1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['119.a1.l1', '289.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '17.a1.l1', 'projective_image': '17.1', 'quotient_action_image': '1.1', 'quotient_action_kernel': '17.1', 'quotient_action_kernel_order': 17, 'quotient_fusion': None, 'short_label': '17.a1.l1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '119.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 48, 'aut_gen_orders': [2, 48], 'aut_gens': [[1], [69], [3]], 'aut_group': '96.59', 'aut_hash': 59, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 96, 'aut_permdeg': 21, 'aut_perms': [2439325392333218664, 19984155209275203], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [7, 1, 6, 1], [17, 1, 16, 1], [119, 1, 96, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{48}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 48, 'autcent_group': '96.59', 'autcent_hash': 59, 'autcent_nilpotent': True, 'autcent_order': 96, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{48}', '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], [7, 1, 6], [17, 1, 16], [119, 1, 96]], 'center_label': '119.1', 'center_order': 119, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['7.1', '17.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['17.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [7, 1, 6, 1], [17, 1, 16, 1], [119, 1, 96, 1]], 'element_repr_type': 'PC', 'elementary': 119, 'eulerian_function': 1, 'exponent': 119, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [7, 17], 'faithful_reps': [[1, 0, 96]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '119.1', 'hash': 1, 'hyperelementary': 119, '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': 96, 'irrQ_dim': 96, 'irrR_degree': 2, 'irrep_stats': [[1, 119]], 'label': '119.1', 'linC_count': 96, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 22, 'linQ_degree_count': 1, 'linQ_dim': 22, 'linQ_dim_count': 1, 'linR_count': 48, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C119', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 119, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 119, 'order_factorization_type': 11, 'order_stats': [[1, 1], [7, 6], [17, 16], [119, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 48, 'outer_gen_orders': [2, 48], 'outer_gen_pows': [0, 0], 'outer_gens': [[69], [3]], 'outer_group': '96.59', 'outer_hash': 59, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 21, 'outer_perms': [2439325392333218664, 19984155209275203], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{48}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [7, 17], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [6, 1], [16, 1], [96, 1]], 'representations': {'PC': {'code': 1375, 'gens': [1], 'pres': [2, -7, -17, 14]}, 'GLFp': {'d': 2, 'p': 239, 'gens': [1092153667]}, 'Perm': {'d': 24, 'gens': [155112100433309859840000, 334764638208000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [119], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{119}', 'transitive_degree': 119, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '2023.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 4896, 'aut_gen_orders': [6, 16, 3], 'aut_gens': [[1, 17], [1, 884], [3, 17], [135, 1457]], 'aut_group': None, 'aut_hash': 5815074320858863402, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 470016, 'aut_permdeg': 294, 'aut_perms': [1506820833680603076600359765831554812745916387157488184769567864405495649254387282683750679728117781517013398887704471279851168743313188228494112347978620816369929075340826201303111963044948222771280730337516736395630506223482289198552531286100414749225310636310122169633139157385896929654165426676183268293400399953174819981624885245516983429393228622461491456439792638819662620718494535750462090323765144125375114939944525227866279595933964730478849977161845079403764520752826860506959312222765004202251701161203973400811274240000000000000000000000000000000000000000000000000000000000000000000000, 2516383469581658609195894121611764146938255641577776258251658408208518577056974783422784536586685964437622016461931288553139915082143065934550386935890676280450554906801268120533344820875891690581056614477723979354115547969591959640671914231969146010882293383155995979633757103135771260549166694511694760936476945262040265229921501439655593466711616792284728453381560861446577515241741910290006788662958782462589939947037405948598338105772468843062092079578441189847628379141141181982610352572730854557184315941251097270801237821072783414670379330181534487432890001660509263986792034, 5023998838109634358808824654221457332027365022194734537942585538968674740607738337401620554335474432545520770518111482958179231443969943418817303225476232934075179090487684561583405288490865534479785364769135064454753003134725178700031753261223659845860208037310408293026651706300874825062282522958342072837923820511999570750706263611796295923040749361086666992242151455273966649769301756150925009823921606916997145403270867154273298820904888508130766438829089531433015999864412077398754724509013027539790353051796130717735353605997536459486861516945893756990736520752635011730144704], 'aut_phi_ratio': 288.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [7, 1, 6, 1], [17, 1, 288, 1], [119, 1, 1728, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6\\times C_{16}.\\PSL(2,17).C_2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4896, 'autcent_group': None, 'autcent_hash': 5815074320858863402, 'autcent_nilpotent': False, 'autcent_order': 470016, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_6\\times C_{16}.\\PSL(2,17).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], [7, 1, 6], [17, 1, 288], [119, 1, 1728]], 'center_label': '2023.a', 'center_order': 2023, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['7.1', '17.1', '17.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['17.1', 2], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [7, 1, 6, 1], [17, 1, 16, 18], [119, 1, 96, 18]], 'element_repr_type': 'PC', 'elementary': 17, 'eulerian_function': 8, 'exponent': 119, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3, 17], 'factors_of_order': [7, 17], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '2023.a', 'hash': 2, 'hyperelementary': 17, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 17], [1, 17]], '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': None, 'irrep_stats': [[1, 2023]], 'label': '2023.a', 'linC_count': None, 'linC_degree': 2, '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': True, 'monomial': True, 'name': 'C17*C119', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 2023, 'number_divisions': 38, 'number_normal_subgroups': 40, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 40, 'number_subgroups': 40, 'old_label': None, 'order': 2023, 'order_factorization_type': 22, 'order_stats': [[1, 1], [7, 6], [17, 288], [119, 1728]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4896, 'outer_gen_orders': [6, 16, 3], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 884], [3, 17], [135, 1457]], 'outer_group': None, 'outer_hash': 5815074320858863402, 'outer_nilpotent': False, 'outer_order': 470016, 'outer_permdeg': 294, 'outer_perms': [1506820833680603076600359765831554812745916387157488184769567864405495649254387282683750679728117781517013398887704471279851168743313188228494112347978620816369929075340826201303111963044948222771280730337516736395630506223482289198552531286100414749225310636310122169633139157385896929654165426676183268293400399953174819981624885245516983429393228622461491456439792638819662620718494535750462090323765144125375114939944525227866279595933964730478849977161845079403764520752826860506959312222765004202251701161203973400811274240000000000000000000000000000000000000000000000000000000000000000000000, 2516383469581658609195894121611764146938255641577776258251658408208518577056974783422784536586685964437622016461931288553139915082143065934550386935890676280450554906801268120533344820875891690581056614477723979354115547969591959640671914231969146010882293383155995979633757103135771260549166694511694760936476945262040265229921501439655593466711616792284728453381560861446577515241741910290006788662958782462589939947037405948598338105772468843062092079578441189847628379141141181982610352572730854557184315941251097270801237821072783414670379330181534487432890001660509263986792034, 5023998838109634358808824654221457332027365022194734537942585538968674740607738337401620554335474432545520770518111482958179231443969943418817303225476232934075179090487684561583405288490865534479785364769135064454753003134725178700031753261223659845860208037310408293026651706300874825062282522958342072837923820511999570750706263611796295923040749361086666992242151455273966649769301756150925009823921606916997145403270867154273298820904888508130766438829089531433015999864412077398754724509013027539790353051796130717735353605997536459486861516945893756990736520752635011730144704], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_6\\times C_{16}.\\PSL(2,17).C_2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 41, 'pgroup': 0, 'primary_abelian_invariants': [7, 17, 17], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 1], [6, 1], [16, 18], [96, 18]], 'representations': {'PC': {'code': '143199', 'gens': [1, 2], 'pres': [3, -17, -7, -17, 46]}, 'GLFp': {'d': 2, 'p': 239, 'gens': [1187717014, 1747445843]}, 'Perm': {'d': 41, 'gens': [4895491699487386406073667617576695365632000000000, 138933081900990183928291110420480000000, 334764638208000]}}, 'schur_multiplier': [17], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [17, 119], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{17}\\times C_{119}', 'transitive_degree': 2023, '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': '17.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 16, 'aut_gen_orders': [16], 'aut_gens': [[1], [3]], 'aut_group': '16.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 16, 'aut_permdeg': 16, 'aut_perms': [1401602636313], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [17, 1, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{16}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 16, 'autcent_group': '16.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{16}', '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], [17, 1, 16]], 'center_label': '17.1', 'center_order': 17, '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': ['17.1'], '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], [17, 1, 16, 1]], 'element_repr_type': 'PC', 'elementary': 17, 'eulerian_function': 1, 'exponent': 17, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [17], 'faithful_reps': [[1, 0, 16]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '17.1', 'hash': 1, 'hyperelementary': 17, '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': 16, 'irrQ_dim': 16, 'irrR_degree': 2, 'irrep_stats': [[1, 17]], 'label': '17.1', 'linC_count': 16, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 1, 'linQ_dim': 16, 'linQ_dim_count': 1, 'linR_count': 8, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C17', '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': 17, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 17, 'order_factorization_type': 1, 'order_stats': [[1, 1], [17, 16]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 16, 'outer_gen_orders': [16], 'outer_gen_pows': [0], 'outer_gens': [[3]], 'outer_group': '16.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 16, 'outer_perms': [1401602636313], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{16}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 17, 'pgroup': 17, 'primary_abelian_invariants': [17], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [16, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -17]}, 'Lie': [{'d': 1, 'q': 17, 'gens': [22324392524313], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 17, 'gens': [4931]}, 'Perm': {'d': 17, 'gens': [334764638208000]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [17], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{17}', 'transitive_degree': 17, 'wreath_data': None, 'wreath_product': False}
