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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '336.104', 'ambient_counter': 104, 'ambient_order': 336, 'ambient_tex': 'C_{21}:Q_{16}', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 28, 'counter': 11, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '336.104.6.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '6.c1.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': 6, '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': '56.1', 'subgroup_hash': 1, 'subgroup_order': 56, 'subgroup_tex': 'C_7:C_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '336.104', 'aut_centralizer_order': 8, 'aut_label': '6.c1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '168.47', 'aut_weyl_index': 24, 'centralizer': '84.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.c1.a1', '3.a1.a1'], 'contains': ['12.a1.a1', '42.c1.a1'], 'core': '12.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [8329, -1, 5450, -1, 8748, -1, 7836, -1], 'generators': [2, 8, 48, 4], 'label': '336.104.6.c1.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '6.c1.a1', 'projective_image': '168.38', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.c1.a1', 'subgroup_fusion': None, 'weyl_group': '28.3'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '8.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 42, 'aut_gen_orders': [2, 6, 14], 'aut_gens': [[1, 8], [5, 8], [1, 24], [51, 8]], 'aut_group': '168.47', 'aut_hash': 47, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 168, 'aut_permdeg': 11, 'aut_perms': [7, 414856, 18224776], 'aut_phi_ratio': 7.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [7, 2, 3, 1], [8, 7, 4, 1], [14, 2, 3, 1], [28, 2, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times F_7', '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': 42, 'autcentquo_group': '42.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 42, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 1, 2], [7, 2, 3], [8, 7, 4], [14, 2, 3], [28, 2, 6]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, 'central_quotient': '14.1', 'commutator_count': 1, 'commutator_label': '7.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '7.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [7, 2, 3, 1], [8, 7, 4, 1], [14, 2, 3, 1], [28, 2, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 56, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[2, 0, 6]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '14.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 7], 'inner_gens': [[1, 48], [17, 8]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 14, 'inner_split': True, 'inner_tex': 'D_7', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 12]], 'label': '56.1', 'linC_count': 6, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 2, 'linQ_dim': 10, 'linQ_dim_count': 1, 'linR_count': 9, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C7:C8', 'ngens': 4, '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': 7, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 20, 'number_divisions': 7, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 14, 'old_label': None, 'order': 56, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 1], [4, 2], [7, 6], [8, 28], [14, 6], [28, 12]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 8], [5, 40]], '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': 2, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 1], [6, 2], [12, 1]], 'representations': {'PC': {'code': 10673942837, 'gens': [1, 4], 'pres': [4, -2, -2, -2, -7, 8, 21, 771]}, 'GLFp': {'d': 2, 'p': 13, 'gens': [24763, 2950]}, 'Perm': {'d': 15, 'gens': [6267306537, 12568, 18498, 99712529280]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [8], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_7:C_8', 'transitive_degree': 56, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [6, 6, 6, 6, 2, 12], 'aut_gens': [[1, 2, 16], [13, 134, 304], [9, 130, 160], [1, 222, 256], [5, 30, 208], [9, 194, 320], [5, 242, 176]], 'aut_group': None, 'aut_hash': 866793471638122205, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 4032, 'aut_permdeg': 88, 'aut_perms': [135201295968350488538328436663429669036523106596020092920154517661310181171342832739045286485968941028713715913445654633686017060357385, 19964000498105589078491661650571585458834707083941432428831535436970354078599745118575679672331963447106950940784271749373361732267981, 1489727013132982946603354062295801232075646232327921820665532706623929642720073458279257692676772888323469920937434253314962074253374, 117375374617766346561687123119714664701646543727679578191298828796573919268694735319487171792262423118589994129612789475245135189791472, 20764194496033549846780517483487016746872378762156974863144740344962773900725437414698488134033649079557180936256700513372522026487317, 116133560187403006929062517733152896323753981201074208057966913232275761424348014188037994199159920974953777091882635708369616003411256], 'aut_phi_ratio': 42.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 1], [4, 2, 1, 1], [4, 4, 1, 1], [4, 84, 1, 1], [6, 2, 1, 1], [7, 2, 3, 1], [8, 42, 2, 1], [12, 4, 1, 1], [12, 4, 2, 1], [14, 2, 3, 1], [21, 2, 6, 1], [28, 4, 3, 1], [28, 4, 6, 1], [42, 2, 6, 1], [84, 4, 6, 1], [84, 4, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{42}.(C_2^4\\times C_6)', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '1008.906', 'autcentquo_hash': 906, 'autcentquo_nilpotent': False, 'autcentquo_order': 1008, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_6\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 2, 1], [4, 2, 1], [4, 4, 1], [4, 84, 1], [6, 2, 1], [7, 2, 3], [8, 42, 2], [12, 4, 3], [14, 2, 3], [21, 2, 6], [28, 4, 9], [42, 2, 6], [84, 4, 18]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '168.38', 'commutator_count': 1, 'commutator_label': '84.6', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 104, '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, 1], [4, 2, 1, 1], [4, 4, 1, 1], [4, 84, 1, 1], [6, 2, 1, 1], [7, 2, 3, 1], [8, 42, 2, 1], [12, 4, 1, 1], [12, 4, 2, 1], [14, 2, 3, 1], [21, 2, 6, 1], [28, 4, 3, 1], [28, 4, 6, 1], [42, 2, 6, 1], [84, 4, 6, 1], [84, 4, 12, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 168, 'exponents_of_order': [4, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[4, -1, 6]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '84.14', 'hash': 104, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 4, 21], 'inner_gens': [[1, 14, 16], [5, 2, 320], [1, 34, 16]], 'inner_hash': 38, 'inner_nilpotent': False, 'inner_order': 168, 'inner_split': True, 'inner_tex': 'C_{21}:D_4', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 24, 'irrQ_dim': 48, 'irrR_degree': 8, 'irrep_stats': [[1, 4], [2, 43], [4, 10]], 'label': '336.104', 'linC_count': 54, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 2, 'linQ_dim': 14, 'linQ_dim_count': 2, 'linR_count': 24, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C21:Q16', '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, 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': 57, 'number_divisions': 18, 'number_normal_subgroups': 19, 'number_subgroup_autclasses': 36, 'number_subgroup_classes': 36, 'number_subgroups': 184, 'old_label': None, 'order': 336, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 90], [6, 2], [7, 6], [8, 84], [12, 12], [14, 6], [21, 12], [28, 36], [42, 12], [84, 72]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 2, 128], [1, 10, 16], [1, 2, 160]], 'outer_group': '24.15', 'outer_hash': 15, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [744, 40320, 27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 3], [4, 3], [6, 2], [12, 4], [24, 2]], 'representations': {'PC': {'code': 119183521579554136784627611595, 'gens': [1, 2, 5], 'pres': [6, -2, -2, -2, -2, -3, -7, 48, 169, 31, 218, 50, 4810, 118, 5195]}, 'Perm': {'d': 26, 'gens': [621574835720323280888133, 2987351515484, 4404319792351, 5808408640626, 6758061133824000, 16754357281155028254720000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{21}:Q_{16}', 'transitive_degree': 336, 'wreath_data': None, 'wreath_product': False}
