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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1760.56', 'ambient_counter': 56, 'ambient_order': 1760, 'ambient_tex': '(Q_8\\times C_{11}):C_{20}', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 440, 'counter': 7, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1760.56.4.b1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '4.b1.b1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '4.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '440.14', 'subgroup_hash': 14, 'subgroup_order': 440, 'subgroup_tex': 'C_{44}.C_{10}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1760.56', 'aut_centralizer_order': 4, 'aut_label': '4.b1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '880.118', 'aut_weyl_index': 8, 'centralizer': '440.a1.a1', 'complements': ['440.e1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['8.b1.a1', '8.d1.b1', '20.b1.b1', '44.d1.b1'], 'core': '4.b1.b1', 'coset_action_label': None, 'count': 1, 'diagramx': [5212, 5466, 5422, 6754, 4042, 4621, 6132, 4856], 'generators': [1655, 1322, 440, 1100, 80], 'label': '1760.56.4.b1.b1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '4.b1.b1', 'normal_contained_in': ['2.a1.a1'], 'normal_contains': ['8.b1.a1', '20.b1.b1'], 'normalizer': '1.a1.a1', 'old_label': '4.b1.b1', 'projective_image': '880.24', 'quotient_action_image': '2.1', 'quotient_action_kernel': '2.1', 'quotient_action_kernel_order': 2, 'quotient_fusion': None, 'short_label': '4.b1.b1', 'subgroup_fusion': None, 'weyl_group': '440.13'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 660, 'aut_gen_orders': [20, 30, 2, 22], 'aut_gens': [[1, 10], [211, 410], [251, 65], [221, 10], [21, 230]], 'aut_group': '2640.bv', 'aut_hash': 5844094576270439126, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2640, 'aut_permdeg': 17, 'aut_perms': [19494048661727, 207791732463532, 704176305845, 144672623319248], 'aut_phi_ratio': 16.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 3, 1], [5, 11, 1, 4], [10, 11, 1, 4], [11, 5, 2, 1], [20, 22, 3, 4], [22, 5, 2, 1], [44, 10, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_4\\times F_{11}', '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': 330, 'autcentquo_group': '660.15', 'autcentquo_hash': 15, 'autcentquo_nilpotent': False, 'autcentquo_order': 660, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 2, 3], [5, 11, 4], [10, 11, 4], [11, 5, 2], [20, 22, 12], [22, 5, 2], [44, 10, 6]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '220.8', 'commutator_count': 1, 'commutator_label': '22.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '5.1', '11.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 14, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['55.1', 1], ['8.4', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 3], [5, 11, 4, 1], [10, 11, 4, 1], [11, 5, 2, 1], [20, 22, 4, 3], [22, 5, 2, 1], [44, 10, 2, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 220, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [[10, 0, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '220.8', 'hash': 14, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 110, 'inner_gen_orders': [10, 22], 'inner_gens': [[1, 150], [301, 10]], 'inner_hash': 8, 'inner_nilpotent': False, 'inner_order': 220, 'inner_split': True, 'inner_tex': 'C_{22}:C_{10}', 'inner_used': [1, 2], 'irrC_degree': 10, 'irrQ_degree': 20, 'irrQ_dim': 20, 'irrR_degree': 20, 'irrep_stats': [[1, 20], [2, 5], [5, 8], [10, 2]], 'label': '440.14', 'linC_count': 40, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 4, 'linQ_dim': 14, 'linQ_dim_count': 4, 'linR_count': 12, 'linR_degree': 14, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C44.C10', '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': 18, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 35, 'number_divisions': 15, 'number_normal_subgroups': 18, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 24, 'number_subgroups': 84, 'old_label': None, 'order': 440, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 1], [4, 6], [5, 44], [10, 44], [11, 10], [20, 264], [22, 10], [44, 60]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [330, 0], 'outer_gens': [[111, 210], [111, 105]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [6, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 4], [8, 1], [10, 4], [20, 1]], 'representations': {'PC': {'code': 4055531876230862578582624868703009, 'gens': [1, 3], 'pres': [5, -2, -5, -2, -2, -11, 10, 1106, 2252, 382, 42, 6003, 1008, 58, 4004, 2509]}, 'GLZN': {'d': 2, 'p': 33, 'gens': [35962, 826574, 36037, 384086, 48302]}, 'Perm': {'d': 19, 'gens': [7039, 12593, 400434839844480, 18619, 7159885125120000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{44}.C_{10}', 'transitive_degree': 88, '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': '40.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 220, 'aut_gen_orders': [10, 22, 10, 10, 44, 4, 10], 'aut_gens': [[1, 10, 880], [441, 70, 880], [961, 1330, 880], [1521, 1390, 880], [841, 690, 880], [781, 890, 880], [1501, 1530, 880], [281, 1570, 880]], 'aut_group': None, 'aut_hash': 7590525614158771080, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 7040, 'aut_permdeg': 27, 'aut_perms': [6072832759290494615706532566, 5990735184972846562447655304, 4672083359173636193083351942, 5235047575309097536759621745, 1280615578901530752440275872, 1255797180423332492874566138, 6042258855276829528603476041], 'aut_phi_ratio': 11.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 2], [4, 4, 2, 2], [5, 11, 1, 4], [8, 2, 4, 1], [10, 11, 1, 12], [11, 5, 2, 1], [20, 22, 1, 8], [20, 44, 2, 8], [22, 5, 2, 3], [40, 22, 4, 4], [44, 10, 2, 2], [44, 20, 4, 2], [88, 10, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{22}.(C_2^5\\times C_{10})', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 110, 'autcentquo_group': '440.42', 'autcentquo_hash': 42, 'autcentquo_nilpotent': False, 'autcentquo_order': 440, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 2], [4, 4, 4], [5, 11, 4], [8, 2, 4], [10, 11, 12], [11, 5, 2], [20, 22, 8], [20, 44, 16], [22, 5, 6], [40, 22, 16], [44, 10, 4], [44, 20, 8], [88, 10, 8]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '440.13', 'commutator_count': 1, 'commutator_label': '44.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 56, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['32.10', 1], ['55.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 2], [4, 4, 1, 2], [4, 4, 2, 1], [5, 11, 4, 1], [8, 2, 4, 1], [10, 11, 4, 3], [11, 5, 2, 1], [20, 22, 4, 2], [20, 44, 4, 2], [20, 44, 8, 1], [22, 5, 2, 3], [40, 22, 16, 1], [44, 10, 2, 2], [44, 20, 2, 2], [44, 20, 4, 1], [88, 10, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 144, 'exponent': 440, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '220.8', 'hash': 56, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 220, 'inner_gen_orders': [10, 44, 1], 'inner_gens': [[1, 1190, 880], [1461, 10, 880], [1, 10, 880]], 'inner_hash': 13, 'inner_nilpotent': False, 'inner_order': 440, 'inner_split': True, 'inner_tex': 'C_{44}:C_{10}', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 40], [2, 30], [5, 16], [10, 12]], 'label': '1760.56', 'linC_count': 160, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 24, 'linQ_dim': 16, 'linQ_dim_count': 12, 'linR_count': 44, 'linR_degree': 14, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': '(Q8*C11):C20', 'ngens': 7, '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': 54, 'number_characteristic_subgroups': 36, 'number_conjugacy_classes': 98, 'number_divisions': 30, 'number_normal_subgroups': 42, 'number_subgroup_autclasses': 76, 'number_subgroup_classes': 84, 'number_subgroups': 420, 'old_label': None, 'order': 1760, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 3], [4, 20], [5, 44], [8, 8], [10, 132], [11, 10], [20, 880], [22, 30], [40, 352], [44, 200], [88, 80]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[441, 650, 880], [1, 210, 880], [881, 650, 880], [1, 1530, 880]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 23, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 6], [8, 4], [10, 4], [16, 2], [20, 4], [40, 2]], 'representations': {'PC': {'code': 7123426626138595576458488976131725001395725249590300090, 'gens': [1, 3, 7], 'pres': [7, -2, -5, -2, -2, -2, -11, -2, 14, 9248, 24992, 8514, 58, 17363, 10370, 80, 12604, 10511, 102, 30245, 6732]}, 'Perm': {'d': 23, 'gens': [48036960, 12783009, 56341309687779916800, 16, 95242440, 139725360, 1233993572292415334400]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 20], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(Q_8\\times C_{11}):C_{20}', 'transitive_degree': 352, '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': '4.1', '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], [3]], '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], [2, 1, 1, 1], [4, 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], [2, 1, 1], [4, 1, 2]], 'center_label': '4.1', 'center_order': 4, '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': ['2.1', '2.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [2], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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, 4]], 'label': '4.1', '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': 'C4', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 1], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3]], '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': 4, 'pgroup': 2, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -2, -2, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [56, 15], 'family': 'CSOPlus'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [21]}, 'Perm': {'d': 4, 'gens': [22, 7]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
