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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '3332.b', 'ambient_counter': 2, 'ambient_order': 3332, 'ambient_tex': 'C_7:C_{476}', 'central': False, 'central_factor': True, 'centralizer_order': 238, 'characteristic': True, 'core_order': 28, 'counter': 21, 'cyclic': False, 'direct': True, 'hall': 0, 'label': '3332.b.119.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '119.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '119.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 119, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{119}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '28.1', 'subgroup_hash': 1, 'subgroup_order': 28, 'subgroup_tex': 'C_7:C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '3332.b', 'aut_centralizer_order': 96, 'aut_label': '119.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '84.7', 'aut_weyl_index': 96, 'centralizer': '14.a1.a1', 'complements': ['28.a1.a1', '28.c1.c1', '28.c1.a1', '28.c1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['7.a1.a1', '17.a1.a1'], 'contains': ['238.b1.a1', '833.a1.a1'], 'core': '119.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [9184, 2041, 9712, 8034, 8649, 8789, 8981, 8789], 'generators': [119, 476, 238], 'label': '3332.b.119.a1.a1', 'mobius_quo': 0, 'mobius_sub': 1, 'normal_closure': '119.a1.a1', 'normal_contained_in': ['7.a1.a1', '17.a1.a1'], 'normal_contains': ['238.b1.a1'], 'normalizer': '1.a1.a1', 'old_label': '119.a1.a1', 'projective_image': '1666.6', 'quotient_action_image': '1.1', 'quotient_action_kernel': '119.1', 'quotient_action_kernel_order': 119, 'quotient_fusion': None, 'short_label': '119.a1.a1', 'subgroup_fusion': None, 'weyl_group': '14.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '4.1', '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, 4], [1, 12], [23, 4]], 'aut_group': '84.7', 'aut_hash': 7, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 84, 'aut_permdeg': 9, 'aut_perms': [25567, 103711], 'aut_phi_ratio': 7.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 7, 2, 1], [7, 2, 3, 1], [14, 2, 3, 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, 1, 1], [4, 7, 2], [7, 2, 3], [14, 2, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '14.1', 'commutator_count': 1, 'commutator_label': '7.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '7.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 7, 2, 1], [7, 2, 3, 1], [14, 2, 3, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 28, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[2, -1, 3]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '14.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 7], 'inner_gens': [[1, 24], [9, 4]], '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': 6, 'irrQ_dim': 12, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 6]], 'label': '28.1', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 8, 'linQ_dim_count': 1, 'linR_count': 6, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C7:C4', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 10, 'number_divisions': 5, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 12, 'old_label': None, 'order': 28, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [4, 14], [7, 6], [14, 6]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[3, 12]], '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': 11, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [6, 2]], 'representations': {'PC': {'code': 1018013, 'gens': [1, 3], 'pres': [3, -2, -2, -7, 6, 218]}, 'Lie': [{'d': 2, 'q': 13, 'gens': [26617, 12345, 26376], 'family': 'CSOMinus'}], 'GLFp': {'d': 2, 'p': 13, 'gens': [14073, 19977]}, 'Perm': {'d': 11, 'gens': [368049, 16, 4364760]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_7:C_4', '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': False, 'abelian_quotient': '476.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 336, 'aut_gen_orders': [2, 16, 6, 6, 7], 'aut_gens': [[1, 476], [239, 476], [197, 476], [101, 476], [1, 2380], [477, 476]], 'aut_group': None, 'aut_hash': 7087796913980897322, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 8064, 'aut_permdeg': 36, 'aut_perms': [5369294567177440245248902493766, 50558214050644341596017027784150715488937, 94814222811333944784643434394566150389003, 54878713961339233958975227110, 327085533608139701346679303854], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 7, 2, 1], [7, 1, 6, 1], [7, 2, 3, 1], [7, 2, 18, 1], [14, 1, 6, 1], [14, 2, 3, 1], [14, 2, 18, 1], [17, 1, 16, 1], [28, 7, 12, 1], [34, 1, 16, 1], [68, 7, 32, 1], [119, 1, 96, 1], [119, 2, 48, 1], [119, 2, 288, 1], [238, 1, 96, 1], [238, 2, 48, 1], [238, 2, 288, 1], [476, 7, 192, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times C_{48}\\times F_7', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 48, 'autcent_group': '192.935', 'autcent_hash': 935, 'autcent_nilpotent': True, 'autcent_order': 192, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_{48}', '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, 1, 1], [4, 7, 2], [7, 1, 6], [7, 2, 21], [14, 1, 6], [14, 2, 21], [17, 1, 16], [28, 7, 12], [34, 1, 16], [68, 7, 32], [119, 1, 96], [119, 2, 336], [238, 1, 96], [238, 2, 336], [476, 7, 192]], 'center_label': '238.4', 'center_order': 238, 'central_product': True, 'central_quotient': '14.1', 'commutator_count': 1, 'commutator_label': '7.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '7.1', '7.1', '17.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['17.1', 1], ['28.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 7, 2, 1], [7, 1, 6, 1], [7, 2, 3, 1], [7, 2, 6, 3], [14, 1, 6, 1], [14, 2, 3, 1], [14, 2, 6, 3], [17, 1, 16, 1], [28, 7, 12, 1], [34, 1, 16, 1], [68, 7, 32, 1], [119, 1, 96, 1], [119, 2, 48, 1], [119, 2, 96, 3], [238, 1, 96, 1], [238, 2, 48, 1], [238, 2, 96, 3], [476, 7, 192, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 864, 'exponent': 476, 'exponents_of_order': [2, 2, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7, 17], 'faithful_reps': [[2, 0, 288]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '1666.6', 'hash': 8817214371369575706, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 7], 'inner_gens': [[1, 2856], [953, 476]], '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': 192, 'irrQ_dim': 192, 'irrR_degree': None, 'irrep_stats': [[1, 476], [2, 714]], 'label': '3332.b', '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': True, 'monomial': True, 'name': 'C7:C476', '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, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 20, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 1190, 'number_divisions': 28, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 28, 'number_subgroup_classes': 36, 'number_subgroups': 72, 'old_label': None, 'order': 3332, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 1], [4, 14], [7, 48], [14, 48], [17, 16], [28, 84], [34, 16], [68, 224], [119, 768], [238, 768], [476, 1344]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 48, 'outer_gen_orders': [2, 6, 48], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[237, 476], [443, 476], [435, 2380]], 'outer_group': '576.4957', 'outer_hash': 4957, 'outer_nilpotent': True, 'outer_order': 576, 'outer_permdeg': 26, 'outer_perms': [25852016738884976640000, 15511261256275259472041760, 158517483605130545283], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6\\times C_{48}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 35, 'pgroup': 0, 'primary_abelian_invariants': [4, 7, 17], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [6, 4], [12, 7], [16, 2], [32, 1], [96, 4], [192, 7]], 'representations': {'PC': {'code': '3178983004042953754101', 'gens': [1, 5], 'pres': [5, -2, -2, -7, -17, -7, 10, 26, 117, 71404]}, 'GLFp': {'d': 2, 'p': 239, 'gens': [518773117, 95563440, 58794]}, 'Perm': {'d': 35, 'gens': [322098184976023927229722408679687320447, 625504723712914304190949742509424640007, 930440781530522261231320014558172063217]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [476], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_7:C_{476}', 'transitive_degree': 476, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '119.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 48, 'aut_gen_orders': [2, 48], 'aut_gens': [[1], [69], [3]], 'aut_group': '96.59', 'aut_hash': 59, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 96, 'aut_permdeg': 21, 'aut_perms': [2439325392333218664, 19984155209275203], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [7, 1, 6, 1], [17, 1, 16, 1], [119, 1, 96, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{48}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 48, 'autcent_group': '96.59', 'autcent_hash': 59, 'autcent_nilpotent': True, 'autcent_order': 96, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{48}', '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], [7, 1, 6], [17, 1, 16], [119, 1, 96]], 'center_label': '119.1', 'center_order': 119, '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': ['7.1', '17.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['17.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [7, 1, 6, 1], [17, 1, 16, 1], [119, 1, 96, 1]], 'element_repr_type': 'PC', 'elementary': 119, 'eulerian_function': 1, 'exponent': 119, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [7, 17], 'faithful_reps': [[1, 0, 96]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '119.1', 'hash': 1, 'hyperelementary': 119, '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': 96, 'irrQ_dim': 96, 'irrR_degree': 2, 'irrep_stats': [[1, 119]], 'label': '119.1', 'linC_count': 96, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 22, 'linQ_degree_count': 1, 'linQ_dim': 22, 'linQ_dim_count': 1, 'linR_count': 48, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C119', 'ngens': 2, '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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 119, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 119, 'order_factorization_type': 11, 'order_stats': [[1, 1], [7, 6], [17, 16], [119, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 48, 'outer_gen_orders': [2, 48], 'outer_gen_pows': [0, 0], 'outer_gens': [[69], [3]], 'outer_group': '96.59', 'outer_hash': 59, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 21, 'outer_perms': [2439325392333218664, 19984155209275203], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{48}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [7, 17], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [6, 1], [16, 1], [96, 1]], 'representations': {'PC': {'code': 1375, 'gens': [1], 'pres': [2, -7, -17, 14]}, 'GLFp': {'d': 2, 'p': 239, 'gens': [1092153667]}, 'Perm': {'d': 24, 'gens': [155112100433309859840000, 334764638208000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [119], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{119}', 'transitive_degree': 119, 'wreath_data': None, 'wreath_product': False}
