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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '24200.ba', 'ambient_counter': 27, 'ambient_order': 24200, 'ambient_tex': 'C_2\\times C_{110}:F_{11}', 'central': False, 'central_factor': False, 'centralizer_order': 200, 'characteristic': False, 'core_order': 10, 'counter': 91, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '24200.ba.1210.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '1210.b1', 'outer_equivalence': True, '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': 1210, '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': '20.5', 'subgroup_hash': 5, 'subgroup_order': 20, 'subgroup_tex': 'C_2\\times C_{10}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '24200.ba', 'aut_centralizer_order': None, 'aut_label': '1210.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '121.a1', 'complements': None, 'conjugacy_class_count': 6, 'contained_in': ['110.d1', '110.e1', '110.f1', '242.a1', '605.a1'], 'contains': ['2420.a1', '2420.c1', '6050.b1'], 'core': '2420.a1', 'coset_action_label': None, 'count': 726, 'diagramx': [4251, -1, 5288, -1], 'generators': [1105, 4840, 12210], 'label': '24200.ba.1210.b1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '10.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '121.a1', 'old_label': '1210.b1', 'projective_image': '2420.z', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '1210.b1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 12], 'aut_gens': [[1, 2], [10, 13], [10, 15]], 'aut_group': '24.5', 'aut_hash': 5, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24, 'aut_permdeg': 7, 'aut_perms': [127, 857], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [5, 1, 4, 1], [10, 1, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.5', 'autcent_hash': 5, 'autcent_nilpotent': False, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4\\times S_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, 3], [5, 1, 4], [10, 1, 12]], 'center_label': '20.5', 'center_order': 20, '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', '2.1', '5.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [5, 1, 4, 1], [10, 1, 4, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 6, 'exponent': 10, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '20.5', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 20]], 'label': '20.5', 'linC_count': 72, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 6, 'linQ_dim': 5, 'linQ_dim_count': 6, 'linR_count': 12, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C10', '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': 20, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 20, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [5, 4], [10, 12]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 12], 'outer_gen_pows': [0, 0], 'outer_gens': [[10, 13], [10, 15]], 'outer_group': '24.5', 'outer_hash': 5, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 7, 'outer_perms': [127, 857], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4\\times S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [4, 4]], 'representations': {'PC': {'code': 2179, 'gens': [1, 2], 'pres': [3, -2, -2, -5, 16]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [141602720900, 706303489327]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [13320, 4001]}, 'Perm': {'d': 9, 'gens': [40320, 720, 96]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_{10}', 'transitive_degree': 20, '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': '200.52', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 660, 'aut_gen_orders': [20, 60, 30, 30, 10], 'aut_gens': [[1, 10, 220], [16011, 12190, 3740], [16091, 12300, 9570], [16431, 12240, 4730], [13371, 12200, 22330], [6481, 210, 1980]], 'aut_group': None, 'aut_hash': 2562126870514592781, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5808000, 'aut_permdeg': 539, 'aut_perms': 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'aut_phi_ratio': 660.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 121, 4, 1], [5, 1, 4, 1], [5, 11, 5, 4], [10, 1, 12, 1], [10, 11, 15, 4], [10, 121, 16, 1], [10, 121, 20, 4], [11, 2, 5, 1], [11, 10, 1, 1], [11, 10, 10, 1], [22, 2, 15, 1], [22, 10, 3, 1], [22, 10, 30, 1], [55, 2, 20, 1], [55, 10, 4, 1], [55, 10, 40, 1], [55, 22, 25, 4], [110, 2, 60, 1], [110, 10, 12, 1], [110, 10, 120, 1], [110, 22, 75, 4]], 'aut_supersolvable': False, 'aut_tex': 'C_{22}^2.C_{15}.C_5.C_{20}.C_2^3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 60, 'autcent_group': '480.1189', 'autcent_hash': 1189, 'autcent_nilpotent': False, 'autcent_order': 480, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'F_5\\times S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 110, 'autcentquo_group': '12100.s', 'autcentquo_hash': 6992133204288697182, 'autcentquo_nilpotent': False, 'autcentquo_order': 12100, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{11}^2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 121, 4], [5, 1, 4], [5, 11, 20], [10, 1, 12], [10, 11, 60], [10, 121, 96], [11, 2, 5], [11, 10, 11], [22, 2, 15], [22, 10, 33], [55, 2, 20], [55, 10, 44], [55, 22, 100], [110, 2, 60], [110, 10, 132], [110, 22, 300]], 'center_label': '20.5', 'center_order': 20, 'central_product': True, 'central_quotient': '1210.16', 'commutator_count': 1, 'commutator_label': '121.2', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '5.1', '5.1', '11.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 27, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['1210.16', 1], ['2.1', 2], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 121, 1, 4], [5, 1, 4, 1], [5, 11, 4, 5], [10, 1, 4, 3], [10, 11, 4, 15], [10, 121, 4, 24], [11, 2, 5, 1], [11, 10, 1, 1], [11, 10, 5, 2], [22, 2, 5, 3], [22, 10, 1, 3], [22, 10, 5, 6], [55, 2, 20, 1], [55, 10, 4, 1], [55, 10, 20, 2], [55, 22, 20, 5], [110, 2, 20, 3], [110, 10, 4, 3], [110, 10, 20, 6], [110, 22, 20, 15]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 749952, 'exponent': 110, 'exponents_of_order': [3, 2, 2], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '24200.ba', 'hash': 7198318507082171453, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 110, 'inner_gen_orders': [10, 11, 11], 'inner_gens': [[1, 190, 4620], [41, 10, 220], [19801, 10, 220]], 'inner_hash': 16, 'inner_nilpotent': False, 'inner_order': 1210, 'inner_split': True, 'inner_tex': 'C_{11}:F_{11}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': None, 'irrep_stats': [[1, 200], [2, 500], [10, 220]], 'label': '24200.ba', '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': 'C2*C110:F11', 'ngens': 7, '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, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 38, 'number_characteristic_subgroups': 23, 'number_conjugacy_classes': 920, 'number_divisions': 108, 'number_normal_subgroups': 188, 'number_subgroup_autclasses': 108, 'number_subgroup_classes': 576, 'number_subgroups': 15952, 'old_label': None, 'order': 24200, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 487], [5, 224], [10, 12288], [11, 120], [22, 360], [55, 2680], [110, 8040]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 60, 'outer_gen_orders': [60, 30, 20, 20], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[12101, 12150, 8470], [2531, 12190, 20790], [2531, 210, 18370], [2421, 190, 2860]], 'outer_group': None, 'outer_hash': 4397560633071774595, 'outer_nilpotent': False, 'outer_order': 4800, 'outer_permdeg': 16, 'outer_perms': [4023146056297, 4103618324977, 5405544673608, 5405541442567], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '(C_5\\times A_4).C_{10}.C_2^3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 31, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [4, 48], [10, 8], [40, 28], [50, 8], [200, 8]], 'representations': {'PC': {'code': '81368289751303699397152971342210966262806071641479740690309039', 'gens': [1, 3, 5], 'pres': [7, -2, -5, -2, -11, -2, -5, -11, 14, 3992, 954, 58, 4483, 2530, 161704, 102, 388085, 250, 1078006]}, 'GLFp': {'d': 4, 'p': 11, 'gens': [36687300164533435, 1977210383605740, 5387309583705790, 12531822321003912, 33418192856010432, 13581699985061301, 31619393949140358]}, 'Perm': {'d': 31, 'gens': [379225566720000, 122024325975552001, 265895349721003134987525162919015, 379226702788200, 549147929537406778327408360107768, 776847279061728, 122421946373839566]}}, 'schur_multiplier': [2, 2, 10], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_{110}:F_{11}', 'transitive_degree': 1100, 'wreath_data': None, 'wreath_product': False}
