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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '3960.o', 'ambient_counter': 15, 'ambient_order': 3960, 'ambient_tex': 'S_3\\times \\PSL(2,11)', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 1, 'counter': 64, 'cyclic': False, 'direct': None, 'hall': 2, 'label': '3960.o.495.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '495.a1.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': 495, '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': '8.5', 'subgroup_hash': 5, 'subgroup_order': 8, 'subgroup_tex': 'C_2^3', 'supersolvable': True, 'sylow': 2}
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gps_subgroup_data • Show schema
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{'ambient': '3960.o', 'aut_centralizer_order': 8, 'aut_label': '495.a1', 'aut_quo_index': None, 'aut_stab_index': 165, 'aut_weyl_group': '6.1', 'aut_weyl_index': 1320, 'centralizer': '495.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['165.a1.a1', '165.b1.a1', '165.c1.a1'], 'contains': ['990.a1.a1', '990.b1.a1', '990.c1.a1'], 'core': '3960.a1.a1', 'coset_action_label': None, 'count': 165, 'diagramx': [4789, -1, 6716, -1, 4659, -1, 4550, -1], 'generators': [12936673682, 2, 25470589082], 'label': '3960.o.495.a1.a1', 'mobius_quo': None, 'mobius_sub': -3, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '165.a1.a1', 'old_label': '495.a1.a1', 'projective_image': '3960.o', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '495.a1.a1', 'subgroup_fusion': None, 'weyl_group': '3.1'}
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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}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 660, 'aut_gen_orders': [10, 2], 'aut_gens': [[6785983586, 1089851041], [45274781045, 19160161225], [59730175585, 59422694642]], 'aut_group': '7920.q', 'aut_hash': 1157788818429169362, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 7920, 'aut_permdeg': 58, 'aut_perms': [587897353636398588644797072549007850204173044572507477404105747126651443728771, 23427148710332319059999225189100172835085593432025207827219142312268916422626], 'aut_phi_ratio': 8.25, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 55, 1, 1], [2, 165, 1, 1], [3, 2, 1, 1], [3, 110, 1, 1], [3, 220, 1, 1], [5, 132, 1, 2], [6, 110, 1, 2], [6, 220, 1, 1], [6, 330, 1, 2], [10, 396, 1, 2], [11, 60, 2, 1], [15, 264, 1, 2], [22, 180, 2, 1], [33, 120, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times \\PGL(2,11)', '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': 660, 'autcentquo_group': '7920.q', 'autcentquo_hash': 1157788818429169362, 'autcentquo_nilpotent': False, 'autcentquo_order': 7920, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\times \\PGL(2,11)', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 55, 1], [2, 165, 1], [3, 2, 1], [3, 110, 1], [3, 220, 1], [5, 132, 2], [6, 110, 2], [6, 220, 1], [6, 330, 2], [10, 396, 2], [11, 60, 2], [15, 264, 2], [22, 180, 2], [33, 120, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '3960.o', 'commutator_count': 1, 'commutator_label': '1980.57', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '660.13'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 15, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['6.1', 1], ['660.13', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 55, 1, 1], [2, 165, 1, 1], [3, 2, 1, 1], [3, 110, 1, 1], [3, 220, 1, 1], [5, 132, 2, 1], [6, 110, 1, 2], [6, 220, 1, 1], [6, 330, 1, 2], [10, 396, 2, 1], [11, 60, 2, 1], [15, 264, 2, 1], [22, 180, 2, 1], [33, 120, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 762, 'exponent': 330, 'exponents_of_order': [3, 2, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 5, 11], 'faithful_reps': [[10, 0, 2], [20, 1, 2], [22, 1, 1], [24, 1, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '3960.o', 'hash': 6655759197249528573, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 330, 'inner_gen_orders': [10, 6], 'inner_gens': [[6785983586, 37864835429], [19452708005, 1089851041]], 'inner_hash': 6655759197249528573, 'inner_nilpotent': False, 'inner_order': 3960, 'inner_split': True, 'inner_tex': 'S_3\\times \\PSL(2,11)', 'inner_used': [1, 2], 'irrC_degree': 10, 'irrQ_degree': 20, 'irrQ_dim': 20, 'irrR_degree': 20, 'irrep_stats': [[1, 2], [2, 1], [5, 4], [10, 6], [11, 2], [12, 4], [20, 2], [22, 1], [24, 2]], 'label': '3960.o', 'linC_count': 4, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 6, 'linQ_dim': 12, 'linQ_dim_count': 6, 'linR_count': 6, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'S3*PSL(2,11)', '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': 21, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 24, 'number_divisions': 18, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 75, 'number_subgroup_classes': 88, 'number_subgroups': 7504, 'old_label': None, 'order': 3960, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 223], [3, 332], [5, 264], [6, 1100], [10, 792], [11, 120], [15, 528], [22, 360], [33, 240]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [53460381024], 'outer_gens': [[58530145082, 56723690425]], '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': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [10, 6], [11, 2], [20, 3], [22, 1], [24, 2], [48, 1]], 'representations': {'Perm': {'d': 14, 'gens': [6785983586, 1089851041]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times \\PSL(2,11)', 'transitive_degree': 33, 'wreath_data': None, 'wreath_product': False}
