-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '486.109', 'ambient_counter': 109, 'ambient_order': 486, 'ambient_tex': 'C_9^2:C_6', 'central': False, 'central_factor': False, 'centralizer_order': 81, 'characteristic': True, 'core_order': 27, 'counter': 26, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '486.109.18.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '18.c1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '18.5', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 5, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 18, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3\\times C_6', 'simple': False, 'solvable': True, 'special_labels': ['D', 'L1', 'D1', 'C3'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '27.2', 'subgroup_hash': 2, 'subgroup_order': 27, 'subgroup_tex': 'C_3\\times C_9', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '486.109', 'aut_centralizer_order': 81, 'aut_label': '18.c1', 'aut_quo_index': 24, 'aut_stab_index': 1, 'aut_weyl_group': '12.5', 'aut_weyl_index': 81, 'centralizer': '6.c1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['6.b1.a1', '6.c1.a1', '6.d1.a1', '6.d1.b1', '9.a1.a1'], 'contains': ['54.a1.a1', '54.c1.a1', '54.h1.a1'], 'core': '18.c1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [1885, 6150, 2594, 6528, 2453, 7143, 3476, 7252], 'generators': [18, 378], 'label': '486.109.18.c1.a1', 'mobius_quo': 0, 'mobius_sub': -3, 'normal_closure': '18.c1.a1', 'normal_contained_in': ['6.b1.a1', '6.c1.a1', '6.d1.a1', '6.d1.b1', '9.a1.a1'], 'normal_contains': ['54.a1.a1', '54.c1.a1'], 'normalizer': '1.a1.a1', 'old_label': '18.c1.a1', 'projective_image': '162.36', 'quotient_action_image': '6.2', 'quotient_action_kernel': '3.1', 'quotient_action_kernel_order': 3, 'quotient_fusion': None, 'short_label': '18.c1.a1', 'subgroup_fusion': None, 'weyl_group': '6.2'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '27.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 6, 'aut_gen_orders': [3, 3, 2, 2, 6], 'aut_gens': [[1, 3], [1, 23], [1, 12], [2, 24], [2, 4], [11, 15]], 'aut_group': '108.28', 'aut_hash': 28, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 108, 'aut_permdeg': 11, 'aut_perms': [4556304, 11088384, 1, 169567, 26386807], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 1, 6, 1], [9, 1, 18, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^2:D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '108.28', 'autcent_hash': 28, 'autcent_nilpotent': False, 'autcent_order': 108, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^2:D_6', '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, 8], [9, 1, 18]], 'center_label': '27.2', 'center_order': 27, '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': ['3.1', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 4], [9, 1, 6, 3]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 4, 'exponent': 9, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '9.2', 'hash': 2, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 3], [1, 3]], '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, 27]], 'label': '27.2', 'linC_count': 216, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 9, 'linQ_dim': 8, 'linQ_dim_count': 9, 'linR_count': 54, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*C9', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 27, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 27, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 8], [9, 18]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [3, 3, 2, 2, 6], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 23], [1, 12], [2, 24], [2, 4], [11, 15]], 'outer_group': '108.28', 'outer_hash': 28, 'outer_nilpotent': False, 'outer_order': 108, 'outer_permdeg': 11, 'outer_perms': [4556304, 11088384, 1, 169567, 26386807], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3^2:D_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 12, 'pgroup': 3, 'primary_abelian_invariants': [3, 9], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4], [6, 3]], 'representations': {'PC': {'code': 34, 'gens': [1, 2], 'pres': [3, -3, 3, -3, 22]}, 'GLFp': {'d': 2, 'p': 19, 'gens': [34311, 75456]}, 'Perm': {'d': 12, 'gens': [357120, 79833600, 80884]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 9], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times C_9', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '18.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 18, 'aut_gen_orders': [6, 3, 6, 9], 'aut_gens': [[1, 6, 54], [1, 30, 54], [19, 6, 54], [1, 6, 270], [109, 6, 54]], 'aut_group': '972.783', 'aut_hash': 783, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 972, 'aut_permdeg': 24, 'aut_perms': [411328118282949344430, 408913963181210154390, 45013707368186996058096, 354706711202948572046988], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 9, 1, 2], [6, 9, 2, 1], [6, 27, 1, 2], [9, 3, 2, 1], [9, 6, 1, 1], [9, 6, 2, 2], [9, 6, 6, 1], [9, 9, 2, 2], [9, 18, 1, 2], [9, 18, 2, 2], [18, 27, 2, 3]], 'aut_supersolvable': True, 'aut_tex': 'C_3^3.S_3^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 3, 'autcent_group': '9.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 9, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 18, 'autcentquo_group': '108.26', 'autcentquo_hash': 26, 'autcentquo_nilpotent': False, 'autcentquo_order': 108, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{18}:C_6', 'cc_stats': [[1, 1, 1], [2, 9, 1], [3, 1, 2], [3, 2, 3], [3, 9, 2], [6, 9, 2], [6, 27, 2], [9, 3, 2], [9, 6, 11], [9, 9, 4], [9, 18, 6], [18, 27, 6]], 'center_label': '3.1', 'center_order': 3, 'central_product': False, 'central_quotient': '162.36', 'commutator_count': 1, 'commutator_label': '27.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 109, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 9, 2, 1], [6, 9, 2, 1], [6, 27, 2, 1], [9, 3, 2, 1], [9, 6, 1, 1], [9, 6, 2, 2], [9, 6, 6, 1], [9, 9, 2, 2], [9, 18, 2, 3], [18, 27, 2, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72, 'exponent': 18, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 0, 6]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '54.12', 'hash': 109, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [6, 3, 9], 'inner_gens': [[1, 24, 108], [37, 6, 54], [433, 6, 54]], 'inner_hash': 36, 'inner_nilpotent': False, 'inner_order': 162, 'inner_split': True, 'inner_tex': 'D_9:C_3^2', 'inner_used': [1, 2, 3], 'irrC_degree': 6, 'irrQ_degree': 36, 'irrQ_dim': 36, 'irrR_degree': 12, 'irrep_stats': [[1, 18], [2, 9], [3, 4], [6, 11]], 'label': '486.109', 'linC_count': 6, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 2, 'linQ_dim': 12, 'linQ_dim_count': 2, 'linR_count': 5, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C9^2:C6', 'ngens': 6, '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': 24, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 42, 'number_divisions': 21, 'number_normal_subgroups': 24, 'number_subgroup_autclasses': 57, 'number_subgroup_classes': 66, 'number_subgroups': 238, 'old_label': None, 'order': 486, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 9], [3, 26], [6, 72], [9, 216], [18, 162]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[1, 30, 54]], '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': 3, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 9], [4, 4], [6, 3], [12, 2], [36, 1]], 'representations': {'PC': {'code': 445686852996498839801550175, 'gens': [1, 3, 5], 'pres': [6, -2, -3, -3, -3, -3, -3, 12, 434, 386, 68, 3244, 3250, 118, 11669]}, 'Perm': {'d': 18, 'gens': [47089213920000, 64276829176068, 57401, 93313, 443308396608000, 779748506208000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_9^2:C_6', 'transitive_degree': 54, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '18.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 3], [1, 17], [7, 3]], 'aut_group': '48.29', 'aut_hash': 29, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 48, 'aut_permdeg': 8, 'aut_perms': [475, 23888], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 8, 1], [6, 1, 8, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '48.29', 'autcent_hash': 29, 'autcent_nilpotent': False, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(2,3)', '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, 1], [3, 1, 8], [6, 1, 8]], 'center_label': '18.5', 'center_order': 18, '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', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 4], [6, 1, 2, 4]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '18.5', 'hash': 5, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 3], [1, 3]], '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, 18]], 'label': '18.5', 'linC_count': 72, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 18, 'linQ_dim': 4, 'linQ_dim_count': 18, 'linR_count': 18, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*C6', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 18, 'number_divisions': 10, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 12, 'number_subgroups': 12, 'old_label': None, 'order': 18, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [3, 8], [6, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 17], [7, 3]], 'outer_group': '48.29', 'outer_hash': 29, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 8, 'outer_perms': [475, 23888], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,3)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 8, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 8]], 'representations': {'PC': {'code': 277, 'gens': [1, 2], 'pres': [3, -3, -2, -3, 16]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [35931151, 16858245]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [1030, 1374]}, 'Perm': {'d': 8, 'gens': [5040, 240, 4]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 6], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times C_6', 'transitive_degree': 18, 'wreath_data': None, 'wreath_product': False}
