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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '560.167', 'ambient_counter': 167, 'ambient_order': 560, 'ambient_tex': 'D_4\\times C_{70}', 'central': False, 'central_factor': False, 'centralizer_order': 280, 'characteristic': False, 'core_order': 2, 'counter': 98, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '560.167.140.d1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '140.d1.b1', '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': 140, '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': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '560.167', 'aut_centralizer_order': 96, 'aut_label': '140.d1', 'aut_quo_index': None, 'aut_stab_index': 8, 'aut_weyl_group': '2.1', 'aut_weyl_index': 768, 'centralizer': '2.a1.b1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['20.d1.b1', '28.d1.b1', '70.a1.b1'], 'contains': ['280.a1.a1', '280.c1.b1', '280.c1.d1'], 'core': '280.a1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [7729, -1, 6317, -1, 2481, -1, 8434, -1], 'generators': [1, 422], 'label': '560.167.140.d1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '70.a1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.b1', 'old_label': '140.d1.b1', 'projective_image': '280.30', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '140.d1.b1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', '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, 3]], 'center_label': '4.2', 'center_order': 4, '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': ['2.1', '2.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 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': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, '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': '280.40', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [12, 12, 12, 12, 2, 4], 'aut_gens': [[1, 2, 4], [1, 143, 148], [281, 142, 12], [281, 2, 69], [1, 3, 69], [1, 2, 396], [281, 2, 445]], 'aut_group': '1536.6279920', 'aut_hash': 7617410279007845091, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 1536, 'aut_permdeg': 18, 'aut_perms': [5877552602844836, 3408583877362598, 863758220456524, 148459981946645, 1846535840714880, 2155153257738688], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 4, 1], [4, 2, 2, 1], [5, 1, 4, 1], [7, 1, 6, 1], [10, 1, 4, 1], [10, 1, 8, 1], [10, 2, 16, 1], [14, 1, 6, 1], [14, 1, 12, 1], [14, 2, 24, 1], [20, 2, 8, 1], [28, 2, 12, 1], [35, 1, 24, 1], [70, 1, 24, 1], [70, 1, 48, 1], [70, 2, 96, 1], [140, 2, 48, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3:(C_2^4.C_2^5)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '768.27015', 'autcent_hash': 27015, 'autcent_nilpotent': True, 'autcent_order': 768, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_6\\times C_2^3.C_2^4', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 4], [4, 2, 2], [5, 1, 4], [7, 1, 6], [10, 1, 12], [10, 2, 16], [14, 1, 18], [14, 2, 24], [20, 2, 8], [28, 2, 12], [35, 1, 24], [70, 1, 72], [70, 2, 96], [140, 2, 48]], 'center_label': '140.11', 'center_order': 140, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '5.1', '7.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 167, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['5.1', 1], ['7.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 4], [4, 2, 1, 2], [5, 1, 4, 1], [7, 1, 6, 1], [10, 1, 4, 3], [10, 2, 4, 4], [14, 1, 6, 3], [14, 2, 6, 4], [20, 2, 4, 2], [28, 2, 6, 2], [35, 1, 24, 1], [70, 1, 24, 3], [70, 2, 24, 4], [140, 2, 24, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 37107, 'exponent': 140, 'exponents_of_order': [4, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 5, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '280.40', 'hash': 167, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [1, 2, 2], 'inner_gens': [[1, 2, 4], [1, 2, 284], [1, 282, 4]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 280], [2, 70]], 'label': '560.167', 'linC_count': 9216, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 96, 'linQ_dim': 12, 'linQ_dim_count': 96, 'linR_count': 96, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D4*C70', 'ngens': 6, 'nilpotency_class': 2, 'nilpotent': True, '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': 20, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 350, 'number_divisions': 40, 'number_normal_subgroups': 76, 'number_subgroup_autclasses': 48, 'number_subgroup_classes': 108, 'number_subgroups': 140, 'old_label': None, 'order': 560, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 11], [4, 4], [5, 4], [7, 6], [10, 44], [14, 66], [20, 16], [28, 24], [35, 24], [70, 264], [140, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 4, 12], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 2, 5], [1, 2, 164], [281, 2, 4], [1, 2, 228], [1, 143, 325]], 'outer_group': '384.19882', 'outer_hash': 19882, 'outer_nilpotent': True, 'outer_order': 384, 'outer_permdeg': 192, 'outer_perms': [3746286772417177001825146178238837859810026046093105057964459253420143494280263931117121797999217102232271291057978007163695759302101865157502557487488533343583405162047531694437490828653149644243693460389520301609142589627714986511487576820117481296576526106360910340176777602870185971228918057476732615292569208875322342800056334053132960182010237034004, 68878262534689841024968566314929023114045129381114889634756270041089442358700609061526050570892721530267286767670431753822370732940503999951056395180430310823641841742043277877518521920651575614526504699316240937324671133639493567047172528987837490552020309278454887569280847146146662384604553832262258885919707835888187144086951241539368463944908800000000, 5605314434481021898663323989318679499246848887836150366127796514381053407142352838671691344630229929593880015664612984645561025485490170327649488974617271683029378415665765841775068516170082898923737754215321698747005015998442336372906730744673885947923721649339330762115943257339090733599982590237911366412114515058679662220888127376740601606544668164658, 7463990035797722054747059984034661653030981021146704554857739601685677980601141863068892871937579076952194246279069607802180707174700382598633881102047458189735471629302055356118505063531895864757623995315305839234854718778595907961030130385722392769804756731024890624711875495880086667896141060476839397015979859376401050479118822315040901308424807533536, 161849370140242824798548196985813283823062664470447617016588693437927680512326053847002761656046395318441253057547272780462518772624574821527101889492822250848907508890800717298534412102667235277987432107676065635414703045811128895000252963992163824493810107965166984301591810291895497731292581957502084950881133104938614036991572321544726249815468823858725], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_4\\times C_{12}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5, 7], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 8], [6, 8], [8, 2], [12, 2], [24, 8], [48, 2]], 'representations': {'PC': {'code': 446776109525239656215, 'gens': [1, 2, 3], 'pres': [6, -2, -2, -2, -2, -5, -7, 2564, 50, 69, 178]}, 'GLZN': {'d': 2, 'p': 77, 'gens': [29218176, 456576, 10127293, 457381, 19630962, 15522156]}, 'Perm': {'d': 18, 'gens': [356995102464000, 43159481164800, 753226662988800, 159667200, 4320, 753220435968000]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 70], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4\\times C_{70}', 'transitive_degree': 280, 'wreath_data': None, 'wreath_product': False}
