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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '504.173', 'ambient_counter': 173, 'ambient_order': 504, 'ambient_tex': 'C_3^2\\times F_8', 'central': True, 'central_factor': False, 'centralizer_order': 504, 'characteristic': False, 'core_order': 3, 'counter': 31, 'cyclic': True, 'direct': True, 'hall': 0, 'label': '504.173.168.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '168.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '168.44', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 44, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 168, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': False, 'quotient_tex': 'C_3\\times F_8', 'simple': True, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '3.1', 'subgroup_hash': 1, 'subgroup_order': 3, 'subgroup_tex': 'C_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '504.173', 'aut_centralizer_order': 1008, 'aut_label': '168.a1', 'aut_quo_index': 1, 'aut_stab_index': 4, 'aut_weyl_group': '2.1', 'aut_weyl_index': 4032, 'centralizer': '1.a1.a1', 'complements': ['3.a1.b1', '3.a1.d1', '3.a1.c1'], 'conjugacy_class_count': 1, 'contained_in': ['24.a1.a1', '56.a1.a1', '84.a1.a1'], 'contains': ['504.a1.a1'], 'core': '168.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3210, 5765, 3872, 6446, 2893, 6122, 3517, 8092], 'generators': [168], 'label': '504.173.168.a1.a1', 'mobius_quo': -1, 'mobius_sub': -8, 'normal_closure': '168.a1.a1', 'normal_contained_in': ['21.a1.a1', '56.a1.a1'], 'normal_contains': ['504.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '168.a1.a1', 'projective_image': '168.44', 'quotient_action_image': '1.1', 'quotient_action_kernel': '168.44', 'quotient_action_kernel_order': 168, 'quotient_fusion': None, 'short_label': '168.a1.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': '3.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [2]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', '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], [3, 1, 2]], 'center_label': '3.1', 'center_order': 3, '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': ['3.1'], '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], [3, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [3], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '3.1', 'hash': 1, 'hyperelementary': 3, '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 3]], 'label': '3.1', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3', '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': 3, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 3, 'order_factorization_type': 1, 'order_stats': [[1, 1], [3, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[2]], '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': 1, 'perfect': False, 'permutation_degree': 3, 'pgroup': 3, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -3]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [7]}, 'Lie': [{'d': 1, 'q': 3, 'gens': [3], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [7]}, 'Perm': {'d': 3, 'gens': [4]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [3], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3', 'transitive_degree': 3, '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': '63.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 168, 'aut_gen_orders': [21, 6], 'aut_gens': [[1, 7, 14, 84], [1, 294, 231, 91], [305, 252, 273, 455]], 'aut_group': None, 'aut_hash': 5238263327796365768, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 8064, 'aut_permdeg': 32, 'aut_perms': [150631829715250634478129048936487997, 104528099079577337406884173977887], 'aut_phi_ratio': 56.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 7, 1, 1], [3, 1, 8, 1], [6, 7, 8, 1], [7, 8, 3, 2], [21, 8, 24, 2]], 'aut_supersolvable': False, 'aut_tex': 'F_8:C_3\\times \\GL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '48.29', 'autcent_hash': 29, 'autcent_nilpotent': False, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(2,3)', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '168.43', 'autcentquo_hash': 43, 'autcentquo_nilpotent': False, 'autcentquo_order': 168, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_8:C_3', 'cc_stats': [[1, 1, 1], [2, 7, 1], [3, 1, 8], [6, 7, 8], [7, 8, 6], [21, 8, 48]], 'center_label': '9.2', 'center_order': 9, 'central_product': True, 'central_quotient': '56.11', 'commutator_count': 1, 'commutator_label': '8.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '7.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 173, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 2], ['56.11', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 7, 1, 1], [3, 1, 2, 4], [6, 7, 2, 4], [7, 8, 6, 1], [21, 8, 12, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 16, 'exponent': 42, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '504.173', 'hash': 173, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [7, 2, 2, 2], 'inner_gens': [[1, 294, 63, 91], [302, 7, 14, 84], [50, 7, 14, 84], [8, 7, 14, 84]], 'inner_hash': 11, 'inner_nilpotent': False, 'inner_order': 56, 'inner_split': True, 'inner_tex': 'F_8', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 63], [7, 9]], 'label': '504.173', 'linC_count': 336, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 11, 'linQ_degree_count': 6, 'linQ_dim': 11, 'linQ_dim_count': 6, 'linR_count': 294, 'linR_degree': 11, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^2*F8', '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': 8, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 72, 'number_divisions': 15, 'number_normal_subgroups': 18, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 36, 'number_subgroups': 150, 'old_label': None, 'order': 504, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 7], [3, 8], [6, 56], [7, 48], [21, 384]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [6, 24, 3], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[2, 49, 280, 462], [4, 301, 462, 406], [1, 7, 14, 112]], 'outer_group': '144.122', 'outer_hash': 122, 'outer_nilpotent': False, 'outer_order': 144, 'outer_permdeg': 11, 'outer_perms': [3670803, 20544004, 98064], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_3\\times \\GL(2,3)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [3, 3, 7], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4], [6, 1], [7, 1], [12, 4], [14, 4]], 'representations': {'PC': {'code': 114016808767465777097989, 'gens': [1, 2, 3, 5], 'pres': [6, 7, 2, 2, 3, 2, 3, 3529, 1136, 50, 2734, 88]}, 'Perm': {'d': 14, 'gens': [147, 4, 526982400, 6314117040, 13902013440, 20757824640]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 21], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2\\times F_8', 'transitive_degree': 72, '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': '21.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 42, 'aut_gen_orders': [2, 3, 7, 2, 2, 2], 'aut_gens': [[1, 7, 14, 28], [1, 7, 14, 140], [2, 98, 84, 133], [1, 91, 105, 42], [15, 7, 14, 28], [106, 7, 14, 28], [85, 7, 14, 28]], 'aut_group': '336.210', 'aut_hash': 210, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 336, 'aut_permdeg': 10, 'aut_perms': [1, 255120, 91848, 2436168, 374406, 1270440], 'aut_phi_ratio': 7.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 7, 1, 1], [3, 1, 2, 1], [6, 7, 2, 1], [7, 8, 3, 2], [21, 8, 6, 2]], 'aut_supersolvable': False, 'aut_tex': 'F_8:C_6', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '168.43', 'autcentquo_hash': 43, 'autcentquo_nilpotent': False, 'autcentquo_order': 168, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_8:C_3', 'cc_stats': [[1, 1, 1], [2, 7, 1], [3, 1, 2], [6, 7, 2], [7, 8, 6], [21, 8, 12]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '56.11', 'commutator_count': 1, 'commutator_label': '8.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 44, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['56.11', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 7, 1, 1], [3, 1, 2, 1], [6, 7, 2, 1], [7, 8, 6, 1], [21, 8, 12, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 64, 'exponent': 42, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[7, 0, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '168.44', 'hash': 44, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [7, 2, 2, 2], 'inner_gens': [[1, 105, 7, 133], [99, 7, 14, 28], [22, 7, 14, 28], [106, 7, 14, 28]], 'inner_hash': 11, 'inner_nilpotent': False, 'inner_order': 56, 'inner_split': True, 'inner_tex': 'F_8', 'inner_used': [1, 2], 'irrC_degree': 7, 'irrQ_degree': 14, 'irrQ_dim': 14, 'irrR_degree': 14, 'irrep_stats': [[1, 21], [7, 3]], 'label': '168.44', 'linC_count': 2, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 9, 'linQ_degree_count': 1, 'linQ_dim': 9, 'linQ_dim_count': 1, 'linR_count': 7, 'linR_degree': 9, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3*F8', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 8, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 24, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 12, 'number_subgroups': 50, 'old_label': None, 'order': 168, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 7], [3, 2], [6, 14], [7, 48], [21, 96]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[4, 21, 7, 147]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [3, 7], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [6, 1], [7, 1], [12, 1], [14, 1]], 'representations': {'PC': {'code': 13354991580763729, 'gens': [1, 2, 3, 4], 'pres': [5, -7, -2, 2, 2, -3, 1051, 107, 2663, 58]}, 'GLFq': {'d': 2, 'q': 64, 'gens': [13789728, 2430985, 3670030, 12631600, 13616307]}, 'Perm': {'d': 11, 'gens': [3, 414960, 3719544, 8361600, 12501360]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [21], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times F_8', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
