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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1728.46346', 'ambient_counter': 46346, 'ambient_order': 1728, 'ambient_tex': 'C_2^3.S_3^3', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 2, 'counter': 717, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1728.46346.216.z1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '216.z1.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': 216, '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': '8.3', 'subgroup_hash': 3, 'subgroup_order': 8, 'subgroup_tex': 'D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1728.46346', 'aut_centralizer_order': 32, 'aut_label': '216.z1', 'aut_quo_index': None, 'aut_stab_index': 54, 'aut_weyl_group': '4.2', 'aut_weyl_index': 1728, 'centralizer': '216.f1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['72.cm1.a1', '72.cq1.a1', '108.m1.a1', '108.bk1.a1', '108.bl1.a1'], 'contains': ['432.e1.a1', '432.f1.a1', '432.k1.a1'], 'core': '864.a1.a1', 'coset_action_label': None, 'count': 54, 'diagramx': [8828, -1, 8790, -1, 8377, -1, 5130, -1], 'generators': [25, 170], 'label': '1728.46346.216.z1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.d1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '54.a1.a1', 'old_label': '216.z1.a1', 'projective_image': '864.4673', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '216.z1.a1', 'subgroup_fusion': None, 'weyl_group': '4.2'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 6], [3, 2]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [5, 9], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [4, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [4, 2, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [4, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 6], [5, 2]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 1]], 'label': '8.3', 'linC_count': 1, 'linC_degree': 2, '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': 'D4', 'ngens': 2, 'nilpotency_class': 2, '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': 5, 'number_divisions': 5, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 10, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 5], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [2], 'outer_gens': [[3, 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': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1]], 'representations': {'PC': {'code': 294, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 37, 16]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [29, 56, 24], 'family': 'COPlus'}, {'d': 1, 'q': 4, 'gens': [7, 16, 1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [12, 55, 56]}, 'Perm': {'d': 4, 'gens': [6, 16, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4', 'transitive_degree': 4, 'wreath_data': ['C_2', 'C_2', '2T1'], 'wreath_product': True}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 6, 6, 4, 4, 6, 2], 'aut_gens': [[1, 2, 8, 48, 288], [881, 2, 88, 960, 1296], [17, 578, 104, 960, 432], [1, 578, 1112, 48, 1440], [17, 582, 1144, 960, 720], [1045, 6, 908, 1104, 1440], [5, 578, 1100, 240, 432], [149, 1158, 152, 240, 1440]], 'aut_group': None, 'aut_hash': 5675430861318798311, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6912, 'aut_permdeg': 24, 'aut_perms': [561190631128529263149901, 437176593475581628913072, 1093277250884382676065, 436459252312795472661085, 588682875375884795637994, 162358223772396152309189, 188700361783939883161350], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 6, 1, 1], [2, 18, 1, 1], [2, 36, 1, 2], [3, 2, 1, 2], [3, 4, 1, 1], [3, 8, 1, 1], [3, 16, 1, 2], [3, 32, 1, 1], [4, 3, 2, 1], [4, 9, 2, 1], [4, 12, 1, 2], [4, 18, 1, 1], [4, 36, 1, 2], [4, 54, 1, 1], [4, 108, 1, 2], [6, 2, 1, 2], [6, 4, 1, 1], [6, 6, 1, 4], [6, 8, 1, 1], [6, 12, 1, 3], [6, 16, 1, 2], [6, 24, 2, 1], [6, 32, 1, 1], [6, 36, 1, 1], [6, 36, 2, 1], [6, 48, 2, 1], [6, 72, 1, 1], [12, 6, 2, 1], [12, 12, 2, 2], [12, 18, 2, 1], [12, 24, 1, 2], [12, 24, 2, 3], [12, 36, 2, 1], [12, 48, 2, 1], [12, 72, 1, 1], [12, 72, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_2\\times S_3\\times S_4\\times D_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '864.4673', 'autcentquo_hash': 4673, 'autcentquo_nilpotent': False, 'autcentquo_order': 864, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4\\times S_3^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [2, 6, 1], [2, 18, 1], [2, 36, 2], [3, 2, 2], [3, 4, 1], [3, 8, 1], [3, 16, 2], [3, 32, 1], [4, 3, 2], [4, 9, 2], [4, 12, 2], [4, 18, 1], [4, 36, 2], [4, 54, 1], [4, 108, 2], [6, 2, 2], [6, 4, 1], [6, 6, 4], [6, 8, 1], [6, 12, 3], [6, 16, 2], [6, 24, 2], [6, 32, 1], [6, 36, 3], [6, 48, 2], [6, 72, 1], [12, 6, 2], [12, 12, 4], [12, 18, 2], [12, 24, 8], [12, 36, 2], [12, 48, 2], [12, 72, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '864.4673', 'commutator_count': 1, 'commutator_label': '216.173', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 46346, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 6, 1, 1], [2, 18, 1, 1], [2, 36, 1, 2], [3, 2, 1, 2], [3, 4, 1, 1], [3, 8, 1, 1], [3, 16, 1, 2], [3, 32, 1, 1], [4, 3, 2, 1], [4, 9, 2, 1], [4, 12, 1, 2], [4, 18, 1, 1], [4, 36, 1, 2], [4, 54, 1, 1], [4, 108, 1, 2], [6, 2, 1, 2], [6, 4, 1, 1], [6, 6, 1, 4], [6, 8, 1, 1], [6, 12, 1, 3], [6, 16, 1, 2], [6, 24, 2, 1], [6, 32, 1, 1], [6, 36, 1, 1], [6, 36, 2, 1], [6, 48, 2, 1], [6, 72, 1, 1], [12, 6, 2, 1], [12, 12, 2, 2], [12, 18, 2, 1], [12, 24, 1, 2], [12, 24, 2, 3], [12, 36, 2, 1], [12, 48, 2, 1], [12, 72, 1, 1], [12, 72, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 161280, 'exponent': 12, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, -1, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '864.4673', 'hash': 46346, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 6, 6, 6], 'inner_gens': [[1, 2, 44, 48, 432], [1, 2, 8, 48, 1440], [21, 2, 8, 960, 432], [1, 2, 1112, 48, 288], [145, 578, 152, 48, 288]], 'inner_hash': 4673, 'inner_nilpotent': False, 'inner_order': 864, 'inner_split': True, 'inner_tex': 'S_4\\times S_3^2', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': 12, 'irrQ_degree': 24, 'irrQ_dim': 48, 'irrR_degree': 24, 'irrep_stats': [[1, 8], [2, 22], [3, 8], [4, 13], [6, 14], [8, 2], [12, 5]], 'label': '1728.46346', 'linC_count': 144, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 9, 'linQ_degree_count': 32, 'linQ_dim': 11, 'linQ_dim_count': 144, 'linR_count': 64, 'linR_degree': 9, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C2^3.S3^3', 'ngens': 9, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 57, 'number_characteristic_subgroups': 60, 'number_conjugacy_classes': 72, 'number_divisions': 57, 'number_normal_subgroups': 60, 'number_subgroup_autclasses': 786, 'number_subgroup_classes': 802, 'number_subgroups': 8924, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 103], [3, 80], [4, 408], [6, 464], [12, 672]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 6, 8, 240, 288], [1, 2, 8, 240, 288], [1, 2, 40, 48, 432]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [3, 8], [4, 8], [6, 8], [8, 7], [12, 4], [24, 2]], 'representations': {'PC': {'code': 533821624648748000704164390858422329744756437401015611794478613, 'gens': [1, 2, 4, 6, 8], 'pres': [9, 2, 2, 2, 2, 3, 2, 3, 2, 3, 36, 46, 1587, 102, 1444, 6512, 3119, 158, 1545, 31111, 51856, 3922, 5875, 214, 46673]}, 'Perm': {'d': 18, 'gens': [482762184, 8847625, 43153265201784, 15810744, 22230464256000, 6706022400, 3, 356995102464000, 753220435968000]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3.S_3^3', 'transitive_degree': 144, 'wreath_data': None, 'wreath_product': False}
