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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '20160.l', 'ambient_counter': 12, 'ambient_order': 20160, 'ambient_tex': 'C_2^2\\times S_7', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': True, 'core_order': 2520, 'counter': 7, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '20160.l.8.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': True, 'nilpotent': False, 'normal': True, 'old_label': '8.a1', 'outer_equivalence': True, 'perfect': True, 'proper': True, 'quotient': '8.5', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 5, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^3', 'simple': True, 'solvable': False, 'special_labels': ['D', 'PC', 'L1', 'D1', 'D', 'PC', 'L1', 'D1'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '2520.a', 'subgroup_hash': 1290983011404564636, 'subgroup_order': 2520, 'subgroup_tex': 'A_7', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '20160.l', 'aut_centralizer_order': None, 'aut_label': '8.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '5040.a1', 'complements': ['2520.l1', '2520.n1', '2520.b1', '2520.d1', '2520.k1', '2520.z1', '2520.s1'], 'conjugacy_class_count': 1, 'contained_in': ['4.a1', '4.b1'], 'contains': ['56.a1', '120.b1', '168.g1', '280.p1'], 'core': '8.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [896, 4370, 757, 4169], 'generators': [21819000, 7988520], 'label': '20160.l.8.a1', 'mobius_quo': -1, 'mobius_sub': -8, 'normal_closure': '8.a1', 'normal_contained_in': ['4.a1', '4.b1'], 'normal_contains': ['20160.a1'], 'normalizer': '1.a1', 'old_label': '8.a1', 'projective_image': '20160.l', 'quotient_action_image': '8.5', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '8.a1', 'subgroup_fusion': None, 'weyl_group': '5040.w'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '1.1', 'all_subgroups_known': True, 'almost_simple': True, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 420, 'aut_gen_orders': [7, 2], 'aut_gens': [[33, 840], [4145, 45], [42, 840]], 'aut_group': '5040.w', 'aut_hash': 6283920599749723413, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5040, 'aut_permdeg': 70, 'aut_perms': [5245992651530022266563763351825463834307819705336033065908348996722012518310301576120941430907541232, 219380546888633531027591269463931186019039330636630289354406296882091772835168706932550141952632], 'aut_phi_ratio': 8.75, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 105, 1, 1], [3, 70, 1, 1], [3, 280, 1, 1], [4, 630, 1, 1], [5, 504, 1, 1], [6, 210, 1, 1], [7, 360, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_7', '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': 420, 'autcentquo_group': '5040.w', 'autcentquo_hash': 6283920599749723413, 'autcentquo_nilpotent': False, 'autcentquo_order': 5040, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_7', 'cc_stats': [[1, 1, 1], [2, 105, 1], [3, 70, 1], [3, 280, 1], [4, 630, 1], [5, 504, 1], [6, 210, 1], [7, 360, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '2520.a', 'commutator_count': 1, 'commutator_label': '2520.a', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2520.a'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 105, 1, 1], [3, 70, 1, 1], [3, 280, 1, 1], [4, 630, 1, 1], [5, 504, 1, 1], [6, 210, 1, 1], [7, 360, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 916, 'exponent': 420, 'exponents_of_order': [3, 2, 1, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 3, 5, 7], 'faithful_reps': [[6, 1, 1], [10, 0, 2], [14, 1, 2], [15, 1, 1], [21, 1, 1], [35, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2520.a', 'hash': 1290983011404564636, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 420, 'inner_gen_orders': [5, 3], 'inner_gens': [[33, 984], [2313, 840]], 'inner_hash': 1290983011404564636, 'inner_nilpotent': False, 'inner_order': 2520, 'inner_split': True, 'inner_tex': 'A_7', 'inner_used': [1, 2], 'irrC_degree': 6, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 6, 'irrep_stats': [[1, 1], [6, 1], [10, 2], [14, 2], [15, 1], [21, 1], [35, 1]], 'label': '2520.a', 'linC_count': 1, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'A7', 'ngens': 2, '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, 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': 8, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 9, 'number_divisions': 8, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 37, 'number_subgroup_classes': 40, 'number_subgroups': 3786, 'old_label': None, 'order': 2520, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 105], [3, 350], [4, 630], [5, 504], [6, 210], [7, 720]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [1917], 'outer_gens': [[966, 4472]], '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': None, 'perfect': True, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [], 'quasisimple': True, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [6, 1], [14, 2], [15, 1], [20, 1], [21, 1], [35, 1]], 'representations': {'Perm': {'d': 7, 'gens': [33, 840]}}, 'schur_multiplier': [6], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'A_7', 'transitive_degree': 7, 'wreath_data': None, 'wreath_product': False}
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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': 420, 'aut_gen_orders': [12, 6], 'aut_gens': [[4405687, 8386576, 7390223], [14924880, 1623736, 19248487], [4077360, 7741463, 11023207]], 'aut_group': '120960.e', 'aut_hash': 8389248707221534269, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 120960, 'aut_permdeg': 84, 'aut_perms': [672722999719274767744501941729726188400089427181724023220727738610240287956584555999137162466904149977921655048589147911960258, 61121167021398419473087545622799035140002012130573973758839065685480913758333283290029238733533933298693673240538792908121719], 'aut_phi_ratio': 26.25, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 21, 4, 1], [2, 105, 1, 1], [2, 105, 3, 1], [2, 105, 4, 1], [3, 70, 1, 1], [3, 280, 1, 1], [4, 210, 4, 1], [4, 630, 1, 1], [4, 630, 3, 1], [5, 504, 1, 1], [6, 70, 3, 1], [6, 210, 1, 1], [6, 210, 3, 1], [6, 280, 3, 1], [6, 420, 4, 1], [6, 840, 4, 1], [7, 720, 1, 1], [10, 504, 3, 1], [10, 504, 4, 1], [12, 420, 4, 1], [14, 720, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_4\\times S_7', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.12', 'autcent_hash': 12, 'autcent_nilpotent': False, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 420, 'autcentquo_group': '5040.w', 'autcentquo_hash': 6283920599749723413, 'autcentquo_nilpotent': False, 'autcentquo_order': 5040, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 21, 4], [2, 105, 8], [3, 70, 1], [3, 280, 1], [4, 210, 4], [4, 630, 4], [5, 504, 1], [6, 70, 3], [6, 210, 4], [6, 280, 3], [6, 420, 4], [6, 840, 4], [7, 720, 1], [10, 504, 7], [12, 420, 4], [14, 720, 3]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '5040.w', 'commutator_count': 1, 'commutator_label': '2520.a', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2520.a'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 12, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['5040.w', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 21, 1, 4], [2, 105, 1, 8], [3, 70, 1, 1], [3, 280, 1, 1], [4, 210, 1, 4], [4, 630, 1, 4], [5, 504, 1, 1], [6, 70, 1, 3], [6, 210, 1, 4], [6, 280, 1, 3], [6, 420, 1, 4], [6, 840, 1, 4], [7, 720, 1, 1], [10, 504, 1, 7], [12, 420, 1, 4], [14, 720, 1, 3]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 21720132, 'exponent': 420, 'exponents_of_order': [6, 2, 1, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 3, 5, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '20160.l', 'hash': 8804660164701733268, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 420, 'inner_gen_orders': [4, 5, 7], 'inner_gens': [[4405687, 408976, 18871343], [18637927, 8386576, 12394103], [14878087, 22307776, 7390223]], 'inner_hash': 6283920599749723413, 'inner_nilpotent': False, 'inner_order': 5040, 'inner_split': True, 'inner_tex': 'S_7', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [6, 8], [14, 16], [15, 8], [20, 4], [21, 8], [35, 8]], 'label': '20160.l', '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': False, 'metacyclic': False, 'monomial': False, 'name': 'C2^2*S7', 'ngens': 3, '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, 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, 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': 23, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 60, 'number_divisions': 60, 'number_normal_subgroups': 21, 'number_subgroup_autclasses': 426, 'number_subgroup_classes': 1692, 'number_subgroups': 264796, 'old_label': None, 'order': 20160, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 927], [3, 350], [4, 3360], [5, 504], [6, 6930], [7, 720], [10, 3528], [12, 1680], [14, 2160]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 3, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[4405687, 8386583, 7390216], [4405687, 8386567, 7390216], [4405696, 8386576, 7390223], [4405703, 8386576, 7390223]], 'outer_group': '24.12', 'outer_hash': 12, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 4, 'outer_perms': [2, 4, 16, 7], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'S_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [6, 8], [14, 16], [15, 8], [20, 4], [21, 8], [35, 8]], 'representations': {'Perm': {'d': 11, 'gens': [4405687, 8386576, 7390223]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times S_7', 'transitive_degree': 28, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 84, 'aut_gen_orders': [3, 3], 'aut_gens': [[1, 2, 4], [4, 5, 3], [2, 4, 1]], 'aut_group': '168.42', 'aut_hash': 42, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 168, 'aut_permdeg': 7, 'aut_perms': [4361, 244], 'aut_phi_ratio': 42.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1]], 'aut_supersolvable': False, 'aut_tex': '\\PSL(2,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '168.42', 'autcent_hash': 42, 'autcent_nilpotent': False, 'autcent_order': 168, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\PSL(2,7)', '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, 7]], 'center_label': '8.5', 'center_order': 8, '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', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 3]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '8.5', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], '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, 8]], 'label': '8.5', 'linC_count': 28, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 28, 'linQ_dim': 3, 'linQ_dim_count': 28, 'linR_count': 28, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3', 'ngens': 3, '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': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 8, 'number_divisions': 8, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 16, 'number_subgroups': 16, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 84, 'outer_gen_orders': [3, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[4, 5, 3], [2, 4, 1]], 'outer_group': '168.42', 'outer_hash': 42, 'outer_nilpotent': False, 'outer_order': 168, 'outer_permdeg': 7, 'outer_perms': [4361, 244], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\PSL(2,7)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3], 'pres': [3, -2, 2, 2]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [16482, 16322, 3362]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [8156, 13286, 13933]}, 'Perm': {'d': 6, 'gens': [120, 6, 1]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}
