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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1584.122', 'ambient_counter': 122, 'ambient_order': 1584, 'ambient_tex': 'D_{22}.D_{18}', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 36, 'counter': 51, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1584.122.22.g1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '22.g1.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': 22, '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': '72.4', 'subgroup_hash': 4, 'subgroup_order': 72, 'subgroup_tex': 'C_9:Q_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1584.122', 'aut_centralizer_order': 20, 'aut_label': '22.g1', 'aut_quo_index': None, 'aut_stab_index': 11, 'aut_weyl_group': '216.111', 'aut_weyl_index': 220, 'centralizer': '792.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.g1.a1', '11.a1.a1'], 'contains': ['44.c1.a1', '44.f1.a1', '44.g1.a1', '66.g1.a1'], 'core': '44.c1.a1', 'coset_action_label': None, 'count': 11, 'diagramx': [4089, -1, 2427, -1, 2583, -1, 3506, -1], 'generators': [73, 1056, 974, 4, 1408], 'label': '1584.122.22.g1.a1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.g1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '11.a1.a1', 'old_label': '22.g1.a1', 'projective_image': '792.40', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '22.g1.a1', 'subgroup_fusion': None, 'weyl_group': '72.17'}
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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': 2, 'aut_exponent': 36, 'aut_gen_orders': [2, 6, 36], 'aut_gens': [[1, 2], [37, 38], [1, 58], [11, 2]], 'aut_group': '432.362', 'aut_hash': 362, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 432, 'aut_permdeg': 13, 'aut_perms': [1, 91571887, 3093562217], 'aut_phi_ratio': 18.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 1], [4, 2, 1, 1], [4, 18, 2, 1], [6, 2, 1, 1], [9, 2, 3, 1], [12, 2, 2, 1], [18, 2, 3, 1], [36, 2, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_{36}:C_6', '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': 18, 'autcentquo_group': '108.26', 'autcentquo_hash': 26, 'autcentquo_nilpotent': False, 'autcentquo_order': 108, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{18}:C_6', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 2, 1], [4, 2, 1], [4, 18, 2], [6, 2, 1], [9, 2, 3], [12, 2, 2], [18, 2, 3], [36, 2, 6]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '36.4', 'commutator_count': 1, 'commutator_label': '18.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 5, '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], [3, 2, 1, 1], [4, 2, 1, 1], [4, 18, 1, 2], [6, 2, 1, 1], [9, 2, 3, 1], [12, 2, 2, 1], [18, 2, 3, 1], [36, 2, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 36, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, -1, 6]], 'familial': True, 'frattini_label': '6.2', 'frattini_quotient': '12.4', 'hash': 4, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [2, 18], 'inner_gens': [[1, 70], [5, 2]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 36, 'inner_split': True, 'inner_tex': 'D_{18}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 12, 'irrQ_dim': 24, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 17]], 'label': '72.4', 'linC_count': 6, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 2, 'linQ_dim': 10, 'linQ_dim_count': 2, 'linR_count': 6, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C9:Q8', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 21, 'number_divisions': 11, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 15, 'number_subgroup_classes': 18, 'number_subgroups': 48, 'old_label': None, 'order': 72, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 38], [6, 2], [9, 6], [12, 4], [18, 6], [36, 12]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[19, 38], [1, 58]], 'outer_group': '12.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [720, 27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 3], [4, 1], [6, 2], [12, 1]], 'representations': {'PC': {'code': 5632488865989664739, 'gens': [1, 2], 'pres': [5, -2, -2, -2, -3, -3, 180, 701, 26, 1022, 42, 1283, 78, 1204]}, 'Lie': [{'d': 2, 'q': 37, 'gens': [962409, 1823544, 911789, 8436], 'family': 'CSOPlus'}], 'GLFp': {'d': 2, 'p': 37, 'gens': [962409, 2701]}, 'Perm': {'d': 17, 'gens': [2965060190719, 12593, 18619, 25108353043200, 47450976612480]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_9:Q_8', 'transitive_degree': 72, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 990, 'aut_gen_orders': [30, 30, 30, 30, 90, 30], 'aut_gens': [[1, 2, 8], [1441, 6, 924], [581, 534, 760], [1301, 6, 252], [145, 6, 252], [869, 706, 1112], [1297, 534, 1352]], 'aut_group': None, 'aut_hash': 8058055515597410150, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 47520, 'aut_permdeg': 42, 'aut_perms': [869967285960884927124117865756680896309916748164614, 246118559555309919532341778604318213076072297282133, 45892894414582711174678626161428801312118533680005, 45892988921701632351690734665159793058755159242340, 577663689475475672815306654829365258494364235860385, 46201764127634687702165357300842172175832001588947], 'aut_phi_ratio': 99.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 22, 1, 1], [2, 198, 1, 1], [3, 2, 1, 1], [4, 9, 2, 1], [4, 18, 1, 1], [4, 22, 1, 1], [4, 198, 1, 1], [6, 2, 1, 1], [6, 4, 1, 1], [6, 44, 1, 1], [9, 2, 3, 1], [11, 2, 5, 1], [12, 44, 1, 1], [18, 2, 3, 1], [18, 4, 3, 1], [18, 44, 3, 1], [22, 2, 5, 1], [22, 2, 10, 1], [33, 4, 5, 1], [36, 44, 3, 1], [44, 18, 10, 2], [66, 4, 5, 1], [66, 4, 10, 1], [99, 4, 15, 1], [198, 4, 15, 1], [198, 4, 30, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{99}.C_{30}.C_2^4', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 990, 'autcentquo_group': '5940.d', 'autcentquo_hash': 4717361004838527357, 'autcentquo_nilpotent': False, 'autcentquo_order': 5940, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_{99}:C_{30}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 22, 1], [2, 198, 1], [3, 2, 1], [4, 9, 2], [4, 18, 1], [4, 22, 1], [4, 198, 1], [6, 2, 1], [6, 4, 1], [6, 44, 1], [9, 2, 3], [11, 2, 5], [12, 44, 1], [18, 2, 3], [18, 4, 3], [18, 44, 3], [22, 2, 15], [33, 4, 5], [36, 44, 3], [44, 18, 20], [66, 4, 15], [99, 4, 15], [198, 4, 45]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '792.40', 'commutator_count': 1, 'commutator_label': '198.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 122, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 22, 1, 1], [2, 198, 1, 1], [3, 2, 1, 1], [4, 9, 2, 1], [4, 18, 1, 1], [4, 22, 1, 1], [4, 198, 1, 1], [6, 2, 1, 1], [6, 4, 1, 1], [6, 44, 1, 1], [9, 2, 3, 1], [11, 2, 5, 1], [12, 44, 1, 1], [18, 2, 3, 1], [18, 4, 3, 1], [18, 44, 3, 1], [22, 2, 5, 1], [22, 2, 10, 1], [33, 4, 5, 1], [36, 44, 3, 1], [44, 18, 10, 2], [66, 4, 5, 1], [66, 4, 10, 1], [99, 4, 15, 1], [198, 4, 15, 1], [198, 4, 30, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24192, 'exponent': 396, 'exponents_of_order': [4, 2, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 11], 'faithful_reps': [[4, 0, 30]], 'familial': False, 'frattini_label': '6.2', 'frattini_quotient': '264.34', 'hash': 122, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 198, 'inner_gen_orders': [2, 2, 198], 'inner_gens': [[1, 2, 876], [1, 2, 712], [725, 882, 8]], 'inner_hash': 40, 'inner_nilpotent': False, 'inner_order': 792, 'inner_split': True, 'inner_tex': 'D_9\\times D_{22}', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 120, 'irrQ_dim': 240, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 58], [4, 84]], 'label': '1584.122', 'linC_count': 270, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 20, 'linQ_degree_count': 32, 'linQ_dim': 20, 'linQ_dim_count': 16, 'linR_count': 120, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D22.D18', '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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 30, 'number_characteristic_subgroups': 41, 'number_conjugacy_classes': 150, 'number_divisions': 30, 'number_normal_subgroups': 41, 'number_subgroup_autclasses': 120, 'number_subgroup_classes': 120, 'number_subgroups': 1848, 'old_label': None, 'order': 1584, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 223], [3, 2], [4, 256], [6, 50], [9, 6], [11, 10], [12, 44], [18, 150], [22, 30], [33, 20], [36, 132], [44, 360], [66, 60], [99, 60], [198, 180]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 30, 'outer_gen_orders': [2, 30], 'outer_gen_pows': [1408, 0], 'outer_gens': [[1, 1410, 872], [1, 6, 808]], 'outer_group': '60.13', 'outer_hash': 13, 'outer_nilpotent': True, 'outer_order': 60, 'outer_permdeg': 12, 'outer_perms': [362880, 39922593], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{30}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 28, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 2], [6, 4], [10, 4], [12, 1], [20, 2], [40, 2], [60, 2], [120, 1]], 'representations': {'PC': {'code': 2407735622346666050835907763435875011745860000119, 'gens': [1, 2, 4], 'pres': [7, -2, -2, -2, 2, -3, -3, -11, 36, 24531, 9978, 80, 5604, 24931, 137, 20165, 23196, 166, 70566]}, 'Perm': {'d': 28, 'gens': [6446835113609647, 868708690492852967966208000, 1445242190515200, 20713823523993600, 12145280938112229701713920000, 4364893, 23454917710790789398855680000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{22}.D_{18}', 'transitive_degree': 792, 'wreath_data': None, 'wreath_product': False}
