-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '81.11', 'ambient_counter': 11, 'ambient_order': 81, 'ambient_tex': 'C_3^2\\times C_9', 'central': True, 'central_factor': False, 'centralizer_order': 81, 'characteristic': False, 'core_order': 27, 'counter': 10, 'cyclic': False, 'direct': True, 'hall': 0, 'label': '81.11.3.a1.i1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '3.a1.i1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '3.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 3, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, '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': '81.11', 'aut_centralizer_order': 18, 'aut_label': '3.a1', 'aut_quo_index': 1, 'aut_stab_index': 12, 'aut_weyl_group': '108.28', 'aut_weyl_index': 216, 'centralizer': '1.a1.a1', 'complements': ['27.a1.b1', '27.a1.d1', '27.a1.g1', '27.a1.a1', '27.a1.c1', '27.a1.f1', '27.a1.h1', '27.a1.k1', '27.a1.l1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['9.a1.c1', '9.c1.c1', '9.c1.d1', '9.c1.i1'], 'core': '3.a1.i1', 'coset_action_label': None, 'count': 1, 'diagramx': [7579, 7579, 9843, 9843, 7378, 7378, 9713, 9713], 'generators': [10, 21], 'label': '81.11.3.a1.i1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '3.a1.i1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['9.a1.c1', '9.c1.c1', '9.c1.i1', '9.c1.d1'], 'normalizer': '1.a1.a1', 'old_label': '3.a1.i1', 'projective_image': '3.1', 'quotient_action_image': '1.1', 'quotient_action_kernel': '3.1', 'quotient_action_kernel_order': 3, 'quotient_fusion': None, 'short_label': '3.a1.i1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
-
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': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '81.11', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 6, 'aut_exponent': 72, 'aut_gen_orders': [3, 3, 3, 6, 3, 2, 3, 3], 'aut_gens': [[1, 3, 9], [1, 3, 12], [1, 3, 11], [1, 57, 9], [1, 3, 18], [8, 1, 9], [2, 3, 9], [28, 3, 9], [1, 3, 36]], 'aut_group': '23328.jy', 'aut_hash': 1553358039411983535, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 23328, 'aut_permdeg': 29, 'aut_perms': [2839173438533166817439515584, 47189591861074696224919104, 656687099319336000931957968984, 340130112265143164490742001695, 6981704913908209646597317926, 140255679591968766911575920, 3797805488087724401445267962904, 1222547042659181080356525190704], 'aut_phi_ratio': 432.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 1, 24, 1], [9, 1, 54, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6.C_3^4:\\GL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 72, 'autcent_group': '23328.jy', 'autcent_hash': 1553358039411983535, 'autcent_nilpotent': False, 'autcent_order': 23328, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_6.C_3^4:\\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], [3, 1, 26], [9, 1, 54]], 'center_label': '81.11', 'center_order': 81, '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', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 11, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 2], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 13], [9, 1, 6, 9]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 13, 'exponent': 9, 'exponents_of_order': [4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '27.5', 'hash': 11, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 3, 9], [1, 3, 9], [1, 3, 9]], '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, 81]], 'label': '81.11', 'linC_count': 50544, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 486, 'linQ_dim': 10, 'linQ_dim_count': 486, 'linR_count': 6318, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^2*C9', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 81, 'number_divisions': 23, 'number_normal_subgroups': 50, 'number_subgroup_autclasses': 9, 'number_subgroup_classes': 50, 'number_subgroups': 50, 'old_label': None, 'order': 81, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 26], [9, 54]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 72, 'outer_gen_orders': [3, 3, 3, 6, 3, 2, 3, 3], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[1, 3, 12], [1, 3, 11], [1, 57, 9], [1, 3, 18], [8, 1, 9], [2, 3, 9], [28, 3, 9], [1, 3, 36]], 'outer_group': '23328.jy', 'outer_hash': 1553358039411983535, 'outer_nilpotent': False, 'outer_order': 23328, 'outer_permdeg': 29, 'outer_perms': [2839173438533166817439515584, 47189591861074696224919104, 656687099319336000931957968984, 340130112265143164490742001695, 6981704913908209646597317926, 140255679591968766911575920, 3797805488087724401445267962904, 1222547042659181080356525190704], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_6.C_3^4:\\GL(2,3)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 15, 'pgroup': 3, 'primary_abelian_invariants': [3, 3, 9], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 13], [6, 9]], 'representations': {'PC': {'code': 516, 'gens': [1, 2, 3], 'pres': [4, -3, 3, 3, -3, 46]}, 'GLZN': {'d': 2, 'p': 18, 'gens': [10015, 40831, 5941, 96397]}, 'GLZq': {'d': 2, 'q': 27, 'gens': [140464, 196840, 203896, 19927]}, 'Perm': {'d': 15, 'gens': [357120, 174356582400, 79833600, 80884]}}, 'schur_multiplier': [3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 9], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2\\times C_9', 'transitive_degree': 81, 'wreath_data': None, 'wreath_product': False}
-
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}
