-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '324.9', 'ambient_counter': 9, 'ambient_order': 324, 'ambient_tex': 'C_9:C_{36}', 'central': False, 'central_factor': False, 'centralizer_order': 18, 'characteristic': False, 'core_order': 54, 'counter': 4, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '324.9.3.b1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '3.b1.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': 3, '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': '108.7', 'subgroup_hash': 7, 'subgroup_order': 108, 'subgroup_tex': 'C_3:C_{36}', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '324.9', 'aut_centralizer_order': 3, 'aut_label': '3.b1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '36.12', 'aut_weyl_index': 9, 'centralizer': '18.c1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['6.a1.a1', '9.b1.a1', '9.c1.a1'], 'core': '6.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [3215, -1, 3198, -1, 3459, -1, 5151, -1], 'generators': [9, 108, 4, 18, 12], 'label': '324.9.3.b1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.b1.a1', 'old_label': '3.b1.a1', 'projective_image': '54.6', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '3.b1.a1', 'subgroup_fusion': None, 'weyl_group': '6.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '36.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 6, 6], 'aut_gens': [[1, 36], [19, 36], [13, 72], [53, 36]], 'aut_group': '72.48', 'aut_hash': 48, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 72, 'aut_permdeg': 10, 'aut_perms': [720, 40350, 403201], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 3, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [9, 1, 6, 1], [9, 2, 6, 1], [12, 3, 4, 1], [18, 1, 6, 1], [18, 2, 6, 1], [36, 3, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_6\\times D_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '12.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 12, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 1, 2], [3, 2, 3], [4, 3, 2], [6, 1, 2], [6, 2, 3], [9, 1, 6], [9, 2, 6], [12, 3, 4], [18, 1, 6], [18, 2, 6], [36, 3, 12]], 'center_label': '18.2', 'center_order': 18, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 7, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.1', 1], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 3, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [9, 1, 6, 1], [9, 2, 6, 1], [12, 3, 4, 1], [18, 1, 6, 1], [18, 2, 6, 1], [36, 3, 12, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72, 'exponent': 36, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 6]], 'familial': False, 'frattini_label': '6.2', 'frattini_quotient': '18.3', 'hash': 7, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 72], [73, 36]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 4, 'irrep_stats': [[1, 36], [2, 18]], 'label': '108.7', 'linC_count': 6, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 2, 'linQ_dim': 10, 'linQ_dim_count': 6, 'linR_count': 9, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3:C36', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 15, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 54, 'number_divisions': 15, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 22, 'number_subgroup_classes': 22, 'number_subgroups': 32, 'old_label': None, 'order': 108, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 1], [3, 8], [4, 6], [6, 8], [9, 18], [12, 12], [18, 18], [36, 36]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[17, 36], [7, 36]], 'outer_group': '12.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [24, 724], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [4, 9], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 5], [4, 3], [6, 2], [12, 3]], 'representations': {'PC': {'code': 679321538791, 'gens': [1, 5], 'pres': [5, -2, -2, -3, -3, -3, 10, 26, 57, 1804]}, 'GLFp': {'d': 2, 'p': 19, 'gens': [399, 13720, 82316]}, 'Perm': {'d': 16, 'gens': [87700838400, 52023, 1037836800, 98356, 1394852659200]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [36], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3:C_{36}', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '36.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 18, 'aut_gen_orders': [3, 6, 18], 'aut_gens': [[1, 36], [25, 36], [1, 72], [199, 36]], 'aut_group': '324.140', 'aut_hash': 140, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 324, 'aut_permdeg': 27, 'aut_perms': [854858857083005990406392, 9276351242488576249402884121, 7983782056443550898433528540], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 1, 2], [3, 2, 1, 3], [4, 9, 2, 1], [6, 1, 1, 2], [6, 2, 1, 3], [9, 3, 3, 2], [9, 6, 1, 3], [9, 6, 3, 2], [12, 9, 2, 2], [18, 3, 3, 2], [18, 6, 1, 3], [18, 6, 3, 2], [36, 9, 6, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_9:C_6^2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 6, 'autcent_group': '6.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 18, 'autcentquo_group': '54.6', 'autcentquo_hash': 6, 'autcentquo_nilpotent': False, 'autcentquo_order': 54, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_9:C_6', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 1, 2], [3, 2, 3], [4, 9, 2], [6, 1, 2], [6, 2, 3], [9, 3, 6], [9, 6, 9], [12, 9, 4], [18, 3, 6], [18, 6, 9], [36, 9, 12]], 'center_label': '6.2', 'center_order': 6, 'central_product': False, 'central_quotient': '54.6', 'commutator_count': 1, 'commutator_label': '9.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 9, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 9, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [9, 3, 6, 1], [9, 6, 1, 1], [9, 6, 2, 1], [9, 6, 6, 1], [12, 9, 4, 1], [18, 3, 6, 1], [18, 6, 1, 1], [18, 6, 2, 1], [18, 6, 6, 1], [36, 9, 12, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 144, 'exponent': 36, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 0, 2]], 'familial': False, 'frattini_label': '18.5', 'frattini_quotient': '18.3', 'hash': 9, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [6, 9], 'inner_gens': [[1, 72], [289, 36]], 'inner_hash': 6, 'inner_nilpotent': False, 'inner_order': 54, 'inner_split': True, 'inner_tex': 'C_9:C_6', 'inner_used': [1, 2], 'irrC_degree': 6, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 36], [2, 18], [6, 6]], 'label': '324.9', 'linC_count': 2, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 3, 'linQ_dim': 12, 'linQ_dim_count': 1, 'linR_count': 6, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C9:C36', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 31, 'number_characteristic_subgroups': 19, 'number_conjugacy_classes': 60, 'number_divisions': 19, 'number_normal_subgroups': 19, 'number_subgroup_autclasses': 35, 'number_subgroup_classes': 35, 'number_subgroups': 85, 'old_label': None, 'order': 324, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 1], [3, 8], [4, 18], [6, 8], [9, 72], [12, 36], [18, 72], [36, 108]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[7, 36]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [4, 9], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 5], [4, 3], [6, 4], [12, 5]], 'representations': {'PC': {'code': 50384474091699982234338767, 'gens': [1, 5], 'pres': [6, -2, -2, -3, -3, -3, -3, 12, 31, 68, 2164, 2170, 1906, 118, 7781]}, 'GLZN': {'d': 2, 'p': 108, 'gens': [103621652, 1895401, 91959013, 1263601, 1261009, 31492849]}, 'Perm': {'d': 22, 'gens': [5353116802812979209, 7434644093525830440, 16, 1089851040, 61187864731209216000, 109994244946877491200]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [36], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_9:C_{36}', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
