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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '26620.g', 'ambient_counter': 7, 'ambient_order': 26620, 'ambient_tex': 'C_{11}\\times C_{22}:F_{11}', 'central': True, 'central_factor': False, 'centralizer_order': 26620, 'characteristic': True, 'core_order': 11, 'counter': 53, 'cyclic': True, 'direct': True, 'hall': 0, 'label': '26620.g.2420.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '2420.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '2420.r', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 2157492669750725759, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 2420, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{22}:F_{11}', 'simple': True, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '11.1', 'subgroup_hash': 1, 'subgroup_order': 11, 'subgroup_tex': 'C_{11}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '26620.g', 'aut_centralizer_order': None, 'aut_label': '2420.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1.a1', 'complements': ['11.a1'], 'conjugacy_class_count': 1, 'contained_in': ['220.a1', '220.c1', '484.a1', '1210.a1', '1210.f1'], 'contains': ['26620.a1'], 'core': '2420.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5501, 1987, 5248, 1682], 'generators': [10], 'label': '26620.g.2420.a1', 'mobius_quo': -1, 'mobius_sub': -242, 'normal_closure': '2420.a1', 'normal_contained_in': ['220.a1', '1210.a1'], 'normal_contains': ['26620.a1'], 'normalizer': '1.a1', 'old_label': '2420.a1', 'projective_image': '2420.r', 'quotient_action_image': '1.1', 'quotient_action_kernel': '2420.r', 'quotient_action_kernel_order': 2420, 'quotient_fusion': None, 'short_label': '2420.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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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': '220.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 220, 'aut_gen_orders': [10, 20, 10, 10, 10], 'aut_gens': [[1, 110, 1210], [9401, 12540, 15730], [15329, 5610, 18480], [16079, 4950, 25630], [7589, 5060, 11330], [9501, 550, 6050]], 'aut_group': None, 'aut_hash': 7721490425071638488, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 484000, 'aut_permdeg': 494, 'aut_perms': 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'aut_phi_ratio': 50.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 121, 2, 1], [5, 121, 2, 2], [10, 121, 2, 2], [10, 121, 4, 2], [11, 1, 10, 1], [11, 10, 2, 1], [11, 10, 10, 1], [11, 10, 20, 1], [11, 10, 100, 1], [22, 1, 10, 1], [22, 10, 2, 1], [22, 10, 10, 1], [22, 10, 20, 1], [22, 10, 100, 1], [22, 121, 20, 1], [55, 121, 20, 2], [110, 121, 20, 2], [110, 121, 40, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_{11}^2.C_5.C_{10}^2.C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 10, 'autcent_group': '20.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 20, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{10}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 220, 'autcentquo_group': '24200.bg', 'autcentquo_hash': 3957277396302200749, 'autcentquo_nilpotent': False, 'autcentquo_order': 24200, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_{11}\\wr C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 121, 2], [5, 121, 4], [10, 121, 12], [11, 1, 10], [11, 10, 132], [22, 1, 10], [22, 10, 132], [22, 121, 20], [55, 121, 40], [110, 121, 120]], 'center_label': '22.2', 'center_order': 22, 'central_product': True, 'central_quotient': '1210.10', 'commutator_count': 1, 'commutator_label': '121.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1', '11.1', '11.1', '11.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 7, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['11.1', 1], ['1210.10', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 121, 1, 2], [5, 121, 4, 1], [10, 121, 4, 3], [11, 1, 10, 1], [11, 10, 1, 2], [11, 10, 5, 2], [11, 10, 10, 12], [22, 1, 10, 1], [22, 10, 1, 2], [22, 10, 5, 2], [22, 10, 10, 12], [22, 121, 10, 2], [55, 121, 40, 1], [110, 121, 40, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 432, 'exponent': 110, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [[10, 0, 100]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '26620.g', 'hash': 2427916003340982652, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 110, 'inner_gen_orders': [10, 11, 11], 'inner_gens': [[1, 2640, 20570], [25301, 110, 1210], [7261, 110, 1210]], 'inner_hash': 10, 'inner_nilpotent': False, 'inner_order': 1210, 'inner_split': False, 'inner_tex': 'C_{11}:F_{11}', 'inner_used': [1, 2], 'irrC_degree': 10, 'irrQ_degree': 100, 'irrQ_dim': 100, 'irrR_degree': 20, 'irrep_stats': [[1, 220], [10, 264]], 'label': '26620.g', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C11*C22:F11', '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': 26, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 484, 'number_divisions': 48, 'number_normal_subgroups': 32, 'number_subgroup_autclasses': 64, 'number_subgroup_classes': 148, 'number_subgroups': 3500, 'old_label': None, 'order': 26620, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 243], [5, 484], [10, 1452], [11, 1330], [22, 3750], [55, 4840], [110, 14520]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 10, 'outer_gen_orders': [2, 2, 10, 10], 'outer_gen_pows': [0, 5, 0, 0], 'outer_gens': [[21, 15620, 1210], [89, 25080, 4180], [13411, 22880, 15730], [1, 19580, 1210]], 'outer_group': '400.221', 'outer_hash': 221, 'outer_nilpotent': True, 'outer_order': 400, 'outer_permdeg': 18, 'outer_perms': [6227020800, 39916800, 355687428504960, 1307674368037], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{10}^2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 35, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5, 11], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [4, 4], [10, 8], [40, 4], [50, 4], [100, 24]], 'representations': {'PC': {'code': '528813456519975220209744021034204504538003039909', 'gens': [1, 4, 5], 'pres': [6, -2, -5, -11, -11, -2, -11, 12, 67, 63363, 237609, 617104, 54460, 88, 522725, 130691]}, 'GLFp': {'d': 4, 'p': 11, 'gens': [25960483174700722, 18721992341619836, 20595883671717277, 1754382938723032, 32182222080390795, 41772741070013040]}, 'Perm': {'d': 35, 'gens': [3660370560, 1, 1528272406707723354336080590416745728000, 1182569150567880324329024375814087936000, 70824919749089501245204251625763635200, 296599418055876144539989709234376192001]}}, 'schur_multiplier': [22], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 110], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{11}\\times C_{22}:F_{11}', 'transitive_degree': 220, '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': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 220, 'aut_gen_orders': [20, 10, 22, 22], 'aut_gens': [[1, 10, 110], [9, 1100, 1300], [61, 90, 1870], [711, 100, 110], [1261, 10, 110]], 'aut_group': '48400.l', 'aut_hash': 3341594860267235989, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 48400, 'aut_permdeg': 484, 'aut_perms': 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2, 'central_product': True, 'central_quotient': '1210.10', 'commutator_count': 1, 'commutator_label': '121.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1', '11.1', '11.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 18, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['1210.10', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 121, 1, 2], [5, 121, 4, 1], [10, 121, 4, 3], [11, 10, 1, 2], [11, 10, 5, 2], [22, 10, 1, 2], [22, 10, 5, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 36, 'exponent': 110, 'exponents_of_order': [2, 2, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [[10, 1, 10]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '2420.r', 'hash': 2157492669750725759, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 110, 'inner_gen_orders': [10, 11, 11], 'inner_gens': [[1, 70, 2090], [51, 10, 110], [441, 10, 110]], 'inner_hash': 10, 'inner_nilpotent': False, 'inner_order': 1210, 'inner_split': False, 'inner_tex': 'C_{11}:F_{11}', 'inner_used': [1, 2, 3], 'irrC_degree': 10, 'irrQ_degree': 50, 'irrQ_dim': 50, 'irrR_degree': 10, 'irrep_stats': [[1, 20], [10, 24]], 'label': '2420.r', 'linC_count': 10, 'linC_degree': 10, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 20, 'linQ_degree_count': 3, 'linQ_dim': 20, 'linQ_dim_count': 3, 'linR_count': 10, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C22:F11', 'ngens': 5, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 13, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 44, 'number_divisions': 16, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 28, 'number_subgroup_classes': 50, 'number_subgroups': 1510, 'old_label': None, 'order': 2420, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 243], [5, 484], [10, 1452], [11, 120], [22, 120]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 10, 'outer_gen_orders': [2, 2, 10], 'outer_gen_pows': [0, 6, 0], 'outer_gens': [[9, 220, 1270], [1211, 20, 1870], [1, 20, 110]], 'outer_group': '40.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 40, 'outer_permdeg': 11, 'outer_perms': [3629520, 41040, 766], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{10}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [4, 4], [10, 4], [50, 4]], 'representations': {'PC': {'code': '6372473303491881090677902743303909', 'gens': [1, 3, 4], 'pres': [5, -2, -5, -11, -2, -11, 10, 1052, 382, 41803, 9908, 58, 44004, 24759]}, 'GLFp': {'d': 4, 'p': 11, 'gens': [13240838059471336, 13669369333096837, 20595883671717277, 32182222080390795, 41772741070013040]}, 'Perm': {'d': 24, 'gens': [479001600, 10540080640000325105282, 38777365, 262160289644134033152000, 1129232178557701308856]}}, 'schur_multiplier': [22], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{22}:F_{11}', 'transitive_degree': 110, 'wreath_data': None, 'wreath_product': False}
