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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '336.72', 'ambient_counter': 72, 'ambient_order': 336, 'ambient_tex': 'C_{21}:Q_{16}', 'central': False, 'central_factor': False, 'centralizer_order': 42, 'characteristic': True, 'core_order': 8, 'counter': 26, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '336.72.42.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '42.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '42.4', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 4, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 42, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3\\times D_7', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '8.4', 'subgroup_hash': 4, 'subgroup_order': 8, 'subgroup_tex': 'Q_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '336.72', 'aut_centralizer_order': 168, 'aut_label': '42.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '8.3', 'aut_weyl_index': 168, 'centralizer': '8.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['6.a1.a1', '14.a1.a1', '21.a1.a1'], 'contains': ['84.a1.a1', '84.b1.a1'], 'core': '42.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5874, 6026, 6817, 5793, 7400, 7799, 3674, 7799], 'generators': [1, 12], 'label': '336.72.42.a1.a1', 'mobius_quo': 0, 'mobius_sub': -7, 'normal_closure': '42.a1.a1', 'normal_contained_in': ['6.a1.a1', '14.a1.a1'], 'normal_contains': ['84.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '42.a1.a1', 'projective_image': '168.28', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '42.a1.a1', 'subgroup_fusion': None, 'weyl_group': '8.3'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 3, 2, 2], 'aut_gens': [[1, 2], [2, 1], [3, 5], [5, 2], [1, 6]], 'aut_group': '24.12', 'aut_hash': 12, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24, 'aut_permdeg': 4, 'aut_perms': [2, 4, 16, 7], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 2, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', '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': 4, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [[2, -1, 1]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 4, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 6], [5, 2]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 1]], 'label': '8.4', 'linC_count': 1, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'Q8', 'ngens': 2, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 5, 'number_divisions': 5, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 1], [4, 6]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [2, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1]], 'representations': {'PC': {'code': 2343, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 12, 37, 16]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [25793872, 22766861]}, 'Lie': [{'d': 2, 'q': 5, 'gens': [90, 377, 504, 253], 'family': 'CSOPlus'}, {'d': 2, 'q': 3, 'gens': [56, 67, 15], 'family': 'CSOMinus'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [67, 21]}, 'Perm': {'d': 8, 'gens': [28776, 16577, 5167]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'Q_8', 'transitive_degree': 8, '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': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [2, 2, 2, 6, 28], 'aut_gens': [[1, 2, 16], [1, 10, 16], [1, 2, 128], [9, 14, 16], [9, 14, 160], [13, 290, 16]], 'aut_group': '1344.11434', 'aut_hash': 11434, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1344, 'aut_permdeg': 34, 'aut_perms': [823610741108647206640863271318010149, 264, 45219582810411889630048432473394714695, 49051805671462059125851800225714856957, 284877317209335388157846694440690580788], 'aut_phi_ratio': 14.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 2, 1, 1], [4, 4, 1, 1], [4, 28, 1, 1], [6, 1, 2, 1], [7, 2, 3, 1], [8, 14, 2, 1], [12, 2, 2, 1], [12, 4, 2, 1], [12, 28, 2, 1], [14, 2, 3, 1], [21, 2, 6, 1], [24, 14, 4, 1], [28, 4, 3, 1], [28, 4, 6, 1], [42, 2, 6, 1], [84, 4, 6, 1], [84, 4, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times D_4\\times F_7', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '168.47', 'autcentquo_hash': 47, 'autcentquo_nilpotent': False, 'autcentquo_order': 168, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 1, 2], [4, 2, 1], [4, 4, 1], [4, 28, 1], [6, 1, 2], [7, 2, 3], [8, 14, 2], [12, 2, 2], [12, 4, 2], [12, 28, 2], [14, 2, 3], [21, 2, 6], [24, 14, 4], [28, 4, 9], [42, 2, 6], [84, 4, 18]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '56.7', 'commutator_count': 1, 'commutator_label': '28.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 72, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['112.17', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 2, 1, 1], [4, 4, 1, 1], [4, 28, 1, 1], [6, 1, 2, 1], [7, 2, 3, 1], [8, 14, 2, 1], [12, 2, 2, 1], [12, 4, 2, 1], [12, 28, 2, 1], [14, 2, 3, 1], [21, 2, 6, 1], [24, 14, 4, 1], [28, 4, 3, 1], [28, 4, 6, 1], [42, 2, 6, 1], [84, 4, 6, 1], [84, 4, 12, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 168, 'exponents_of_order': [4, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[4, 0, 6]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '84.12', 'hash': 72, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 28, 'inner_gen_orders': [2, 4, 7], 'inner_gens': [[1, 14, 16], [5, 2, 208], [1, 146, 16]], 'inner_hash': 7, 'inner_nilpotent': False, 'inner_order': 56, 'inner_split': True, 'inner_tex': 'C_7:D_4', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 8, 'irrep_stats': [[1, 12], [2, 45], [4, 9]], 'label': '336.72', 'linC_count': 198, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 8, 'linQ_dim': 14, 'linQ_dim_count': 2, 'linR_count': 12, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C21:Q16', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 20, 'number_characteristic_subgroups': 22, 'number_conjugacy_classes': 66, 'number_divisions': 20, 'number_normal_subgroups': 22, 'number_subgroup_autclasses': 36, 'number_subgroup_classes': 36, 'number_subgroups': 104, 'old_label': None, 'order': 336, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 34], [6, 2], [7, 6], [8, 28], [12, 68], [14, 6], [21, 12], [24, 56], [28, 36], [42, 12], [84, 72]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [0, 4, 4], 'outer_gens': [[1, 10, 16], [13, 2, 16], [13, 2, 272]], 'outer_group': '24.15', 'outer_hash': 15, 'outer_nilpotent': True, 'outer_order': 24, 'outer_permdeg': 9, 'outer_perms': [40320, 24, 723], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 5], [4, 2], [6, 2], [8, 1], [12, 4], [24, 2]], 'representations': {'PC': {'code': 118549964731740847521713403851, 'gens': [1, 2, 5], 'pres': [6, -2, -2, -2, -2, -3, -7, 48, 169, 31, 218, 50, 3130, 118, 5195]}, 'GLZN': {'d': 2, 'p': 91, 'gens': [3316452, 40371124, 16927702, 12057152, 29285190, 20346444]}, 'Perm': {'d': 26, 'gens': [621574835364635852792133, 2987351515484, 6758061133824000, 4404319792351, 5808408640626, 16754357281155028254720000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{21}:Q_{16}', 'transitive_degree': 336, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 42, 'aut_gen_orders': [6, 14], 'aut_gens': [[1, 2], [1, 20], [37, 16]], 'aut_group': '84.7', 'aut_hash': 7, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 84, 'aut_permdeg': 9, 'aut_perms': [6055, 203047], '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, 2, 3, 1], [21, 2, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times F_7', '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': '42.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 42, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_7', 'cc_stats': [[1, 1, 1], [2, 7, 1], [3, 1, 2], [6, 7, 2], [7, 2, 3], [21, 2, 6]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '14.1', 'commutator_count': 1, 'commutator_label': '7.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '7.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['14.1', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 7, 1, 1], [3, 1, 2, 1], [6, 7, 2, 1], [7, 2, 3, 1], [21, 2, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 42, 'exponents_of_order': [1, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[2, 0, 6]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '42.4', 'hash': 4, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 7], 'inner_gens': [[1, 26], [19, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 14, 'inner_split': True, 'inner_tex': 'D_7', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 4, 'irrep_stats': [[1, 6], [2, 9]], 'label': '42.4', 'linC_count': 6, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 2, 'linQ_dim': 8, 'linQ_dim_count': 2, 'linR_count': 9, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*D7', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 15, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 20, 'old_label': None, 'order': 42, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 7], [3, 2], [6, 14], [7, 6], [21, 12]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[1, 10]], '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': 2, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2], [6, 1], [12, 1]], 'representations': {'PC': {'code': 62135003, 'gens': [1, 2], 'pres': [3, -2, -3, -7, 157, 22, 326]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [351, 1717]}, 'Perm': {'d': 10, 'gens': [41064, 3, 444984]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times D_7', 'transitive_degree': 21, 'wreath_data': None, 'wreath_product': False}
