-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1080.172', 'ambient_counter': 172, 'ambient_order': 1080, 'ambient_tex': 'C_{30}.C_6^2', 'central': False, 'central_factor': False, 'centralizer_order': 360, 'characteristic': False, 'core_order': 1, 'counter': 67, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '1080.172.360.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '360.b1', 'outer_equivalence': True, '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': 360, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': True, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '3.1', 'subgroup_hash': 1, 'subgroup_order': 3, 'subgroup_tex': 'C_3', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '1080.172', 'aut_centralizer_order': 1440, 'aut_label': '360.b1', 'aut_quo_index': None, 'aut_stab_index': 12, 'aut_weyl_group': '2.1', 'aut_weyl_index': 17280, 'centralizer': '3.a1', 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['72.b1', '120.a1', '180.b1', '180.d1'], 'contains': ['1080.a1'], 'core': '1080.a1', 'coset_action_label': None, 'count': 12, 'diagramx': [1439, -1, 6526, -1], 'generators': [720], 'label': '1080.172.360.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '120.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1', 'old_label': '360.b1', 'projective_image': '1080.172', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '360.b1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '3.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [2]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', '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': 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], [3, 1, 2]], 'center_label': '3.1', 'center_order': 3, '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': ['3.1'], '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], [3, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [3], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '3.1', 'hash': 1, 'hyperelementary': 3, '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 3]], 'label': '3.1', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3', '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': 3, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 3, 'order_factorization_type': 1, 'order_stats': [[1, 1], [3, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[2]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 3, 'pgroup': 3, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -3]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [7]}, 'Lie': [{'d': 1, 'q': 3, 'gens': [3], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [7]}, 'Perm': {'d': 3, 'gens': [4]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [3], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3', 'transitive_degree': 3, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '72.36', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 120, 'aut_gen_orders': [12, 8, 30, 6, 12], 'aut_gens': [[1, 6, 72], [603, 42, 196], [975, 30, 148], [713, 66, 74], [17, 42, 1012], [975, 66, 916]], 'aut_group': None, 'aut_hash': 8203041546889010050, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 34560, 'aut_permdeg': 36, 'aut_perms': [9958971235211374035182506467157966939858, 103712400469541577631153979940955899616011, 371869508589964100043584403391432018452544, 278557059630071156172164384767455477357350, 113159332264301409239668097968449449044042], 'aut_phi_ratio': 120.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 2, 1], [3, 1, 2, 1], [3, 3, 8, 1], [4, 1, 2, 1], [4, 5, 2, 1], [5, 2, 2, 1], [6, 1, 2, 1], [6, 3, 8, 1], [6, 5, 4, 1], [6, 15, 16, 1], [10, 2, 2, 1], [12, 1, 4, 1], [12, 3, 16, 1], [12, 5, 4, 1], [12, 15, 16, 1], [15, 2, 4, 1], [15, 6, 16, 1], [20, 2, 4, 1], [30, 2, 4, 1], [30, 6, 16, 1], [60, 2, 8, 1], [60, 6, 32, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2\\times \\AGL(2,3)\\times F_5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '36.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 36, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_6^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': '960.10901', 'autcentquo_hash': 10901, 'autcentquo_nilpotent': False, 'autcentquo_order': 960, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\times \\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 5, 2], [3, 1, 2], [3, 3, 8], [4, 1, 2], [4, 5, 2], [5, 2, 2], [6, 1, 2], [6, 3, 8], [6, 5, 4], [6, 15, 16], [10, 2, 2], [12, 1, 4], [12, 3, 16], [12, 5, 4], [12, 15, 16], [15, 2, 4], [15, 6, 16], [20, 2, 4], [30, 2, 4], [30, 6, 16], [60, 2, 8], [60, 6, 32]], 'center_label': '12.2', 'center_order': 12, 'central_product': True, 'central_quotient': '90.5', 'commutator_count': 1, 'commutator_label': '15.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 172, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['10.1', 1], ['27.3', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 1, 2], [3, 1, 2, 1], [3, 3, 2, 4], [4, 1, 2, 1], [4, 5, 2, 1], [5, 2, 2, 1], [6, 1, 2, 1], [6, 3, 2, 4], [6, 5, 2, 2], [6, 15, 2, 8], [10, 2, 2, 1], [12, 1, 4, 1], [12, 3, 4, 4], [12, 5, 4, 1], [12, 15, 4, 4], [15, 2, 4, 1], [15, 6, 4, 4], [20, 2, 4, 1], [30, 2, 4, 1], [30, 6, 4, 4], [60, 2, 8, 1], [60, 6, 8, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 60, 'exponents_of_order': [3, 3, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[6, 0, 8]], 'familial': False, 'frattini_label': '6.2', 'frattini_quotient': '180.32', 'hash': 172, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [6, 1, 15], 'inner_gens': [[1, 6, 336], [1, 6, 72], [889, 6, 72]], 'inner_hash': 5, 'inner_nilpotent': False, 'inner_order': 90, 'inner_split': False, 'inner_tex': 'C_3^2\\times D_5', 'inner_used': [1, 3], 'irrC_degree': 6, 'irrQ_degree': 48, 'irrQ_dim': 48, 'irrR_degree': 12, 'irrep_stats': [[1, 72], [2, 72], [3, 16], [6, 16]], 'label': '1080.172', 'linC_count': None, 'linC_degree': None, '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': False, 'monomial': True, 'name': 'C30.C6^2', 'ngens': 7, '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, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 24, 'number_characteristic_subgroups': 27, 'number_conjugacy_classes': 176, 'number_divisions': 54, 'number_normal_subgroups': 77, 'number_subgroup_autclasses': 70, 'number_subgroup_classes': 176, 'number_subgroups': 684, 'old_label': None, 'order': 1080, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 11], [3, 26], [4, 12], [5, 4], [6, 286], [10, 4], [12, 312], [15, 104], [20, 8], [30, 104], [60, 208]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [2, 2, 2, 6, 4, 4], 'outer_gen_pows': [0, 0, 722, 3, 0, 0], 'outer_gens': [[37, 6, 72], [1, 42, 72], [29, 30, 124], [759, 6, 194], [747, 6, 434], [389, 6, 74]], 'outer_group': '384.20055', 'outer_hash': 20055, 'outer_nilpotent': False, 'outer_order': 384, 'outer_permdeg': 14, 'outer_perms': [127, 288, 2566327680, 958729374, 31259294640, 14095025280], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^3\\times \\GL(2,3)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 18], [4, 10], [6, 4], [8, 9], [12, 2], [16, 4], [24, 2], [48, 1]], 'representations': {'PC': {'code': 52353389103677054596951436903447, 'gens': [1, 3, 6], 'pres': [7, -2, -3, -2, -2, -3, 3, -5, 14, 58, 80, 14117, 2028, 166, 42342]}, 'GLZN': {'d': 2, 'p': 45, 'gens': [1458001, 91801, 91144, 91531, 152806, 1731394, 3371662]}, 'Perm': {'d': 18, 'gens': [21010450809600, 1041465600, 11268, 57376, 482630400, 93313, 397620186163200]}}, 'schur_multiplier': [3, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6, 12], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{30}.C_6^2', 'transitive_degree': 180, 'wreath_data': None, 'wreath_product': False}
