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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '1836.70', 'ambient_counter': 70, 'ambient_order': 1836, 'ambient_tex': 'C_3^2:D_{102}', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 17, 'counter': 26, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1836.70.18.e1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '18.e1.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': 18, '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': '102.2', 'subgroup_hash': 2, 'subgroup_order': 102, 'subgroup_tex': 'C_3\\times D_{17}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1836.70', 'aut_centralizer_order': 6, 'aut_label': '18.e1', 'aut_quo_index': None, 'aut_stab_index': 18, 'aut_weyl_group': '544.242', 'aut_weyl_index': 108, 'centralizer': '306.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['6.e1.a1', '9.a1.a1'], 'contains': ['36.b1.a1', '54.b1.a1', '306.e1.a1'], 'core': '108.a1.a1', 'coset_action_label': None, 'count': 9, 'diagramx': [7300, -1, 294, -1, 6448, -1, 256, -1], 'generators': [9, 1226, 108], 'label': '1836.70.18.e1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.a1.a1', 'old_label': '18.e1.a1', 'projective_image': '1836.70', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '18.e1.a1', 'subgroup_fusion': None, 'weyl_group': '34.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 272, 'aut_gen_orders': [16, 34], 'aut_gens': [[1, 2], [1, 74], [97, 70]], 'aut_group': '544.242', 'aut_hash': 242, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 544, 'aut_permdeg': 19, 'aut_perms': [3007368056298528, 12904176190351327], 'aut_phi_ratio': 17.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 17, 1, 1], [3, 1, 2, 1], [6, 17, 2, 1], [17, 2, 8, 1], [51, 2, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times F_{17}', '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': 272, 'autcentquo_group': '272.50', 'autcentquo_hash': 50, 'autcentquo_nilpotent': False, 'autcentquo_order': 272, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{17}', 'cc_stats': [[1, 1, 1], [2, 17, 1], [3, 1, 2], [6, 17, 2], [17, 2, 8], [51, 2, 16]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '34.1', 'commutator_count': 1, 'commutator_label': '17.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '17.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['34.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 17, 1, 1], [3, 1, 2, 1], [6, 17, 2, 1], [17, 2, 8, 1], [51, 2, 16, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 102, 'exponents_of_order': [1, 1, 1], 'factors_of_aut_order': [2, 17], 'factors_of_order': [2, 3, 17], 'faithful_reps': [[2, 0, 16]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '102.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 34, 'inner_gen_orders': [2, 17], 'inner_gens': [[1, 32], [73, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 34, 'inner_split': False, 'inner_tex': 'D_{17}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 4, 'irrep_stats': [[1, 6], [2, 24]], 'label': '102.2', 'linC_count': 16, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 2, 'linQ_dim': 18, 'linQ_dim_count': 2, 'linR_count': 24, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*D17', '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': 30, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 40, 'old_label': None, 'order': 102, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 17], [3, 2], [6, 34], [17, 16], [51, 32]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 8, 'outer_gen_orders': [2, 8], 'outer_gen_pows': [0, 1], 'outer_gens': [[1, 100], [1, 14]], 'outer_group': '16.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 10, 'outer_perms': [362880, 5913], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_8', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2], [16, 1], [32, 1]], 'representations': {'PC': {'code': 20522131487, 'gens': [1, 2], 'pres': [3, -2, -3, -17, 193, 22, 866]}, 'GLFp': {'d': 2, 'p': 67, 'gens': [19850359, 9689170]}, 'Perm': {'d': 20, 'gens': [6423384156578664, 3, 134491868721989304]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times D_{17}', 'transitive_degree': 51, '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': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 816, 'aut_gen_orders': [48, 24, 16, 48, 48], 'aut_gens': [[1, 6, 36], [1529, 6, 828], [31, 618, 900], [1381, 30, 720], [1529, 30, 792], [587, 30, 1440]], 'aut_group': None, 'aut_hash': 3102432471460601533, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 58752, 'aut_permdeg': 312, 'aut_perms': 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136825669846922187022911743703769195297770526590517542775571233030609806027470473199750090000923191890707078874096998533578987046564741293945149618590473954936259023566039258488622632792193053513757786407676843360457781132273540062873195295667739852505846678564227072552246818069689939377983562029766664550292736588312647810915263567024645283660837229488653881735832695598665313302571390339933194809209713977387835592911054777043193367913467343370540272162273142718976137207663473118032838023585421027461112300332106942625601470739021727789877158490211606386349798320519840433189004056403289704707205573929072273296756465607797600458355245354487], 'aut_phi_ratio': 102.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 153, 2, 1], [3, 2, 1, 1], [3, 3, 2, 1], [3, 6, 1, 1], [3, 6, 2, 1], [6, 2, 1, 1], [6, 3, 2, 1], [6, 6, 1, 1], [6, 6, 2, 1], [6, 153, 4, 1], [17, 2, 8, 1], [34, 2, 8, 1], [51, 2, 16, 1], [51, 6, 16, 2], [51, 6, 32, 1], [102, 2, 16, 1], [102, 6, 16, 2], [102, 6, 32, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_3\\times C_{51}).C_{48}.C_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': 816, 'autcentquo_group': None, 'autcentquo_hash': 6573069769348916178, 'autcentquo_nilpotent': False, 'autcentquo_order': 29376, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{51}.(C_{48}\\times S_3).C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 153, 2], [3, 2, 1], [3, 3, 2], [3, 6, 3], [6, 2, 1], [6, 3, 2], [6, 6, 3], [6, 153, 4], [17, 2, 8], [34, 2, 8], [51, 2, 16], [51, 6, 64], [102, 2, 16], [102, 6, 64]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '918.14', 'commutator_count': 1, 'commutator_label': '153.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '17.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 70, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['918.14', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 153, 1, 2], [3, 2, 1, 1], [3, 3, 2, 1], [3, 6, 1, 1], [3, 6, 2, 1], [6, 2, 1, 1], [6, 3, 2, 1], [6, 6, 1, 1], [6, 6, 2, 1], [6, 153, 2, 2], [17, 2, 8, 1], [34, 2, 8, 1], [51, 2, 16, 1], [51, 6, 16, 2], [51, 6, 32, 1], [102, 2, 16, 1], [102, 6, 16, 2], [102, 6, 32, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 102, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 3, 17], 'factors_of_order': [2, 3, 17], 'faithful_reps': [[6, 1, 16]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '612.33', 'hash': 70, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 102, 'inner_gen_orders': [6, 3, 51], 'inner_gens': [[1, 1254, 1800], [625, 6, 36], [73, 6, 36]], 'inner_hash': 14, 'inner_nilpotent': False, 'inner_order': 918, 'inner_split': True, 'inner_tex': 'C_3^2:D_{51}', 'inner_used': [1, 2, 3], 'irrC_degree': 6, 'irrQ_degree': 96, 'irrQ_dim': 96, 'irrR_degree': 6, 'irrep_stats': [[1, 12], [2, 150], [6, 34]], 'label': '1836.70', 'linC_count': 16, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 22, 'linQ_degree_count': 3, 'linQ_dim': 22, 'linQ_dim_count': 3, 'linR_count': 16, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^2:D102', '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, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 22, 'number_characteristic_subgroups': 22, 'number_conjugacy_classes': 196, 'number_divisions': 24, 'number_normal_subgroups': 26, 'number_subgroup_autclasses': 64, 'number_subgroup_classes': 78, 'number_subgroups': 1966, 'old_label': None, 'order': 1836, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 307], [3, 26], [6, 638], [17, 16], [34, 16], [51, 416], [102, 416]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 16, 'outer_gen_orders': [2, 2, 16], 'outer_gen_pows': [0, 4, 0], 'outer_gens': [[5, 6, 1800], [19, 1230, 576], [1, 6, 1332]], 'outer_group': '64.183', 'outer_hash': 183, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 20, 'outer_perms': [121645100408832000, 355687428096000, 2803205272625], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{16}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 28, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 2], [6, 2], [16, 2], [32, 4], [64, 2], [96, 2]], 'representations': {'PC': {'code': 88620801574483893233985777202572973637919, 'gens': [1, 3, 5], 'pres': [6, -2, -3, -2, -3, -3, -17, 12, 22574, 11078, 50, 15267, 7497, 54004, 118, 62213]}, 'Perm': {'d': 28, 'gens': [403913035968392044960591855, 3628800, 52676, 16371, 104197, 11696695522355136250210176000]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2:D_{102}', 'transitive_degree': 306, 'wreath_data': None, 'wreath_product': False}
