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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1188.39', 'ambient_counter': 39, 'ambient_order': 1188, 'ambient_tex': 'S_3\\times D_{99}', 'central': False, 'central_factor': False, 'centralizer_order': 594, 'characteristic': True, 'core_order': 11, 'counter': 50, 'cyclic': True, 'direct': False, 'hall': 11, 'label': '1188.39.108.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '108.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '108.16', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 16, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 108, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'S_3\\times D_9', 'simple': True, 'solvable': True, 'special_labels': ['C5'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '11.1', 'subgroup_hash': 1, 'subgroup_order': 11, 'subgroup_tex': 'C_{11}', 'supersolvable': True, 'sylow': 11}
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gps_subgroup_data • Show schema
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{'ambient': '1188.39', 'aut_centralizer_order': 3564, 'aut_label': '108.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '10.2', 'aut_weyl_index': 3564, 'centralizer': '2.a1.a1', 'complements': ['11.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['36.a1.a1', '36.b1.a1', '36.c1.a1', '54.a1.a1', '54.b1.a1', '54.c1.a1'], 'contains': ['1188.a1.a1'], 'core': '108.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [1725, 6226, 7624, 2294, 6809, 4993, 6803, 4993], 'generators': [108], 'label': '1188.39.108.a1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '108.a1.a1', 'normal_contained_in': ['36.b1.a1', '36.a1.a1'], 'normal_contains': ['1188.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '108.a1.a1', 'projective_image': '1188.39', 'quotient_action_image': '2.1', 'quotient_action_kernel': '54.4', 'quotient_action_kernel_order': 54, 'quotient_fusion': None, 'short_label': '108.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '11.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 10, 'aut_gen_orders': [10], 'aut_gens': [[1], [2]], 'aut_group': '10.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 10, 'aut_permdeg': 7, 'aut_perms': [753], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [11, 1, 10, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{10}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 10, 'autcent_group': '10.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 10, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{10}', '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], [11, 1, 10]], 'center_label': '11.1', 'center_order': 11, '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': ['11.1'], '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], [11, 1, 10, 1]], 'element_repr_type': 'PC', 'elementary': 11, 'eulerian_function': 1, 'exponent': 11, 'exponents_of_order': [1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [11], 'faithful_reps': [[1, 0, 10]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '11.1', 'hash': 1, 'hyperelementary': 11, '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': 10, 'irrQ_dim': 10, 'irrR_degree': 2, 'irrep_stats': [[1, 11]], 'label': '11.1', 'linC_count': 10, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 1, 'linQ_dim': 10, 'linQ_dim_count': 1, 'linR_count': 5, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C11', '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': 11, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 11, 'order_factorization_type': 1, 'order_stats': [[1, 1], [11, 10]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 10, 'outer_gen_orders': [10], 'outer_gen_pows': [0], 'outer_gens': [[2]], 'outer_group': '10.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 10, 'outer_permdeg': 7, 'outer_perms': [753], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{10}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 11, 'pgroup': 11, 'primary_abelian_invariants': [11], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [10, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -11]}, 'Lie': [{'d': 1, 'q': 11, 'gens': [4037913], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 11, 'gens': [1343]}, 'Perm': {'d': 11, 'gens': [36288000]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [11], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{11}', 'transitive_degree': 11, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, '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': 990, 'aut_gen_orders': [30, 10, 90, 30], 'aut_gens': [[1, 2, 12], [9, 478, 192], [9, 658, 636], [9, 646, 84], [5, 1082, 168]], 'aut_group': None, 'aut_hash': 7847050869488114835, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 35640, 'aut_permdeg': 102, 'aut_perms': [406418291063869129739674114442944799911296235623103579778248678581702537548745720033117870759545830397518179946640643286566565974221077945279791067453409664280112, 411329552169017794169720535254644763209185884550668858483245392123768078372456293879446458839938134966263430137141406855762779294818364512475827596015913087504824, 410579076539783953669954928088256751365537129377285892439900762385163418326903942963903836580207199421465523271412307751321811021764922120973951579126463817095319, 193618846551680029775380209837606738985088993792083246083037061844844759868577951900418379618496720483064250300922472428342334207766279455313741751552586268566172], 'aut_phi_ratio': 99.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 99, 1, 1], [2, 297, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [6, 6, 1, 1], [6, 198, 1, 1], [9, 2, 3, 1], [9, 4, 3, 1], [11, 2, 5, 1], [18, 6, 3, 1], [22, 6, 5, 1], [33, 2, 10, 1], [33, 4, 5, 1], [33, 4, 10, 1], [66, 6, 10, 1], [99, 2, 30, 1], [99, 4, 30, 1], [198, 6, 30, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_3\\times C_{99}).C_{30}.C_2^2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 990, 'autcentquo_group': None, 'autcentquo_hash': 7847050869488114835, 'autcentquo_nilpotent': False, 'autcentquo_order': 35640, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': '(C_3\\times C_{99}).C_{30}.C_2^2', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 99, 1], [2, 297, 1], [3, 2, 2], [3, 4, 1], [6, 6, 1], [6, 198, 1], [9, 2, 3], [9, 4, 3], [11, 2, 5], [18, 6, 3], [22, 6, 5], [33, 2, 10], [33, 4, 15], [66, 6, 10], [99, 2, 30], [99, 4, 30], [198, 6, 30]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '1188.39', 'commutator_count': 1, 'commutator_label': '297.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '11.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 39, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['198.3', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 99, 1, 1], [2, 297, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [6, 6, 1, 1], [6, 198, 1, 1], [9, 2, 3, 1], [9, 4, 3, 1], [11, 2, 5, 1], [18, 6, 3, 1], [22, 6, 5, 1], [33, 2, 10, 1], [33, 4, 5, 1], [33, 4, 10, 1], [66, 6, 10, 1], [99, 2, 30, 1], [99, 4, 30, 1], [198, 6, 30, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 198, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 11], 'faithful_reps': [[4, 1, 30]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '396.22', 'hash': 39, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 198, 'inner_gen_orders': [2, 6, 99], 'inner_gens': [[1, 10, 12], [5, 2, 1176], [1, 26, 12]], 'inner_hash': 39, 'inner_nilpotent': False, 'inner_order': 1188, 'inner_split': True, 'inner_tex': 'S_3\\times D_{99}', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 120, 'irrQ_dim': 120, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 100], [4, 49]], 'label': '1188.39', 'linC_count': 150, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 8, 'linQ_dim': 18, 'linQ_dim_count': 8, 'linR_count': 150, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'S3*D99', 'ngens': 6, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 21, 'number_characteristic_subgroups': 22, 'number_conjugacy_classes': 153, 'number_divisions': 21, 'number_normal_subgroups': 22, 'number_subgroup_autclasses': 68, 'number_subgroup_classes': 68, 'number_subgroups': 1892, 'old_label': None, 'order': 1188, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 399], [3, 8], [6, 204], [9, 18], [11, 10], [18, 18], [22, 30], [33, 80], [66, 60], [99, 180], [198, 180]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 30, 'outer_gen_orders': [30], 'outer_gen_pows': [0], 'outer_gens': [[5, 2, 168]], 'outer_group': '30.4', 'outer_hash': 4, 'outer_nilpotent': True, 'outer_order': 30, 'outer_permdeg': 10, 'outer_perms': [373024], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{30}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 23, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 1], [6, 2], [10, 2], [12, 1], [20, 3], [40, 1], [60, 2], [120, 1]], 'representations': {'PC': {'code': 31579982350277749202110920826001119, 'gens': [1, 2, 4], 'pres': [6, -2, -2, -3, -3, -3, -11, 121, 31, 146, 14121, 93, 17290, 118, 19451]}, 'Perm': {'d': 23, 'gens': [111926654118639728047, 479001600, 1284585462597685248000, 6706022400, 4364893, 2462597285793765120000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times D_{99}', 'transitive_degree': 198, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, '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': 18, 'aut_gen_orders': [6, 18, 3], 'aut_gens': [[1, 2, 12], [1, 2, 24], [1, 106, 12], [9, 2, 12]], 'aut_group': '324.118', 'aut_hash': 118, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 324, 'aut_permdeg': 12, 'aut_perms': [8442744, 259662265, 4], 'aut_phi_ratio': 9.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [6, 6, 1, 1], [6, 18, 1, 1], [9, 2, 3, 1], [9, 4, 3, 1], [18, 6, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^2.S_3^2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 18, 'autcentquo_group': '324.118', 'autcentquo_hash': 118, 'autcentquo_nilpotent': False, 'autcentquo_order': 324, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^2.S_3^2', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 9, 1], [2, 27, 1], [3, 2, 2], [3, 4, 1], [6, 6, 1], [6, 18, 1], [9, 2, 3], [9, 4, 3], [18, 6, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '108.16', 'commutator_count': 1, 'commutator_label': '27.2', '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': 16, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['18.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [6, 6, 1, 1], [6, 18, 1, 1], [9, 2, 3, 1], [9, 4, 3, 1], [18, 6, 3, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 18, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 1, 3]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '36.10', 'hash': 16, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [2, 6, 9], 'inner_gens': [[1, 10, 12], [5, 2, 96], [1, 26, 12]], 'inner_hash': 16, 'inner_nilpotent': False, 'inner_order': 108, 'inner_split': True, 'inner_tex': 'S_3\\times D_9', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 10], [4, 4]], 'label': '108.16', 'linC_count': 15, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 4, 'linQ_dim': 8, 'linQ_dim_count': 4, 'linR_count': 15, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'S3*D9', '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': 12, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 18, 'number_divisions': 12, 'number_normal_subgroups': 13, 'number_subgroup_autclasses': 34, 'number_subgroup_classes': 34, 'number_subgroups': 176, 'old_label': None, 'order': 108, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 39], [3, 8], [6, 24], [9, 18], [18, 18]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 3, 'outer_gen_orders': [3], 'outer_gen_pows': [6], 'outer_gens': [[1, 2, 24]], 'outer_group': '3.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 3, 'outer_permdeg': 3, 'outer_perms': [3], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 1], [6, 2], [12, 1]], 'representations': {'PC': {'code': 8066209008895815, 'gens': [1, 2, 4], 'pres': [5, -2, -2, -3, -3, -3, 101, 26, 122, 968, 78, 909]}, 'GLZN': {'d': 2, 'p': 18, 'gens': [49742, 9805, 6137, 61399, 5941]}, 'Perm': {'d': 12, 'gens': [1, 8719944, 51902880, 3, 95973864]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times D_9', 'transitive_degree': 18, 'wreath_data': None, 'wreath_product': False}
