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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '12960.bq', 'ambient_counter': 43, 'ambient_order': 12960, 'ambient_tex': 'F_5\\times S_3\\wr C_3', 'central': False, 'central_factor': False, 'centralizer_order': 96, 'characteristic': False, 'core_order': 1, 'counter': 849, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '12960.bq.1620.c1.a2', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '1620.c1.a2', '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': 1620, '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': '8.2', 'subgroup_hash': 2, 'subgroup_order': 8, 'subgroup_tex': 'C_2\\times C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '12960.bq', 'aut_centralizer_order': None, 'aut_label': '1620.c1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '135.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['324.c1.a2', '540.d1.b1', '540.e1.a1', '540.l1.b1', '810.b1.a1', '810.c1.a1', '810.d1.a1'], 'contains': ['3240.a1.a1', '3240.e1.a1', '3240.g1.a1'], 'core': '12960.a1.a1', 'coset_action_label': None, 'count': 135, 'diagramx': [9273, -1, 5655, -1, 8154, -1, 7888, -1], 'generators': [10800, 123752889, 16], 'label': '12960.bq.1620.c1.a2', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '3.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '135.b1.a1', 'old_label': '1620.c1.a2', 'projective_image': '12960.bq', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '1620.c1.a2', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 3], [5, 3]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [1, 22], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.3', 'autcent_hash': 3, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_4', '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, 3], [4, 1, 4]], 'center_label': '8.2', 'center_order': 8, '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': ['2.1', '2.1', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8]], 'label': '8.2', 'linC_count': 12, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 4, 'linQ_dim': 3, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C4', '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': 8, 'number_divisions': 6, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 4]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 3], [5, 3]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [1, 22], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2]], 'representations': {'PC': {'code': 34, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16426]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [251, 504]}, 'Perm': {'d': 6, 'gens': [22, 120, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_4', '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': '24.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 180, 'aut_gen_orders': [12, 20, 6], 'aut_gens': [[7265669795, 13057516096], [20847547519, 31618108147], [37612915218, 12066587], [50027050835, 19676176]], 'aut_group': '25920.bd', 'aut_hash': 6704104401456500954, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 25920, 'aut_permdeg': 86, 'aut_perms': [240420706233968873696370216417425301944932400350761967064944191520295502164039062953565095079963216596060625128548239882769252717, 14086839614373628907125349665268903155012651967222841201245009603476292429051450692061829060224831390717575840697695061314753915397, 1256353305942835464253328861985993937297739480037492187332499037461582992802134826796458615668754785272970314521438396464682039], 'aut_phi_ratio': 7.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [2, 9, 1, 1], [2, 27, 1, 2], [2, 45, 1, 1], [2, 135, 1, 2], [3, 6, 1, 1], [3, 8, 1, 1], [3, 12, 1, 1], [3, 36, 2, 1], [4, 5, 1, 2], [4, 45, 1, 2], [4, 135, 1, 4], [5, 4, 1, 1], [6, 18, 2, 1], [6, 30, 1, 1], [6, 36, 1, 1], [6, 40, 1, 1], [6, 54, 1, 1], [6, 60, 1, 1], [6, 90, 2, 1], [6, 108, 2, 1], [6, 180, 1, 1], [6, 180, 2, 1], [6, 270, 1, 1], [6, 540, 2, 1], [9, 72, 2, 1], [10, 36, 1, 1], [10, 108, 1, 2], [12, 30, 1, 2], [12, 40, 1, 2], [12, 60, 1, 2], [12, 90, 2, 2], [12, 180, 1, 2], [12, 180, 2, 2], [12, 270, 1, 2], [12, 540, 2, 2], [15, 24, 1, 1], [15, 32, 1, 1], [15, 48, 1, 1], [15, 144, 2, 1], [18, 360, 2, 1], [30, 72, 2, 1], [30, 144, 1, 1], [30, 216, 1, 1], [30, 432, 2, 1], [36, 360, 2, 2], [45, 288, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_5\\times S_3\\wr S_3', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 180, 'autcentquo_group': '25920.bd', 'autcentquo_hash': 6704104401456500954, 'autcentquo_nilpotent': False, 'autcentquo_order': 25920, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\times S_3\\wr S_3', 'cc_stats': [[1, 1, 1], [2, 5, 1], [2, 9, 1], [2, 27, 2], [2, 45, 1], [2, 135, 2], [3, 6, 1], [3, 8, 1], [3, 12, 1], [3, 36, 2], [4, 5, 2], [4, 45, 2], [4, 135, 4], [5, 4, 1], [6, 18, 2], [6, 30, 1], [6, 36, 1], [6, 40, 1], [6, 54, 1], [6, 60, 1], [6, 90, 2], [6, 108, 2], [6, 180, 3], [6, 270, 1], [6, 540, 2], [9, 72, 2], [10, 36, 1], [10, 108, 2], [12, 30, 2], [12, 40, 2], [12, 60, 2], [12, 90, 4], [12, 180, 6], [12, 270, 2], [12, 540, 4], [15, 24, 1], [15, 32, 1], [15, 48, 1], [15, 144, 2], [18, 360, 2], [30, 72, 2], [30, 144, 1], [30, 216, 1], [30, 432, 2], [36, 360, 4], [45, 288, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '12960.bq', 'commutator_count': 1, 'commutator_label': '540.109', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '5.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 43, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['20.3', 1], ['648.705', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [2, 9, 1, 1], [2, 27, 1, 2], [2, 45, 1, 1], [2, 135, 1, 2], [3, 6, 1, 1], [3, 8, 1, 1], [3, 12, 1, 1], [3, 36, 2, 1], [4, 5, 2, 1], [4, 45, 2, 1], [4, 135, 2, 2], [5, 4, 1, 1], [6, 18, 1, 2], [6, 30, 1, 1], [6, 36, 1, 1], [6, 40, 1, 1], [6, 54, 1, 1], [6, 60, 1, 1], [6, 90, 1, 2], [6, 108, 2, 1], [6, 180, 1, 1], [6, 180, 2, 1], [6, 270, 1, 1], [6, 540, 2, 1], [9, 72, 2, 1], [10, 36, 1, 1], [10, 108, 1, 2], [12, 30, 2, 1], [12, 40, 2, 1], [12, 60, 2, 1], [12, 90, 2, 2], [12, 180, 2, 1], [12, 180, 4, 1], [12, 270, 2, 1], [12, 540, 4, 1], [15, 24, 1, 1], [15, 32, 1, 1], [15, 48, 1, 1], [15, 144, 2, 1], [18, 360, 2, 1], [30, 72, 1, 2], [30, 144, 1, 1], [30, 216, 1, 1], [30, 432, 2, 1], [36, 360, 4, 1], [45, 288, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 1248, 'exponent': 180, 'exponents_of_order': [5, 4, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[24, 1, 4], [32, 0, 2], [32, 1, 1], [48, 1, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '12960.bq', 'hash': 5911408896146137813, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 180, 'inner_gen_orders': [12, 6], 'inner_gens': [[7265669795, 1481689547], [26390206879, 13057516096]], 'inner_hash': 5911408896146137813, 'inner_nilpotent': False, 'inner_order': 12960, 'inner_split': True, 'inner_tex': 'F_5\\times S_3\\wr C_3', 'inner_used': [1, 2], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 24], [3, 8], [4, 6], [6, 16], [8, 12], [12, 10], [24, 4], [32, 3], [48, 2]], 'label': '12960.bq', 'linC_count': 96, 'linC_degree': 10, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 16, 'linQ_dim': 10, 'linQ_dim_count': 16, 'linR_count': 16, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'F5*S3wrC3', 'ngens': 2, '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, 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': 65, 'number_characteristic_subgroups': 34, 'number_conjugacy_classes': 85, 'number_divisions': 56, 'number_normal_subgroups': 34, 'number_subgroup_autclasses': 768, 'number_subgroup_classes': 920, 'number_subgroups': 44464, 'old_label': None, 'order': 12960, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 383], [3, 98], [4, 640], [5, 4], [6, 2542], [9, 144], [10, 252], [12, 4400], [15, 392], [18, 720], [30, 1368], [36, 1440], [45, 576]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[18920643875, 19676176]], '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': 5, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [3, 4], [4, 4], [6, 10], [8, 4], [12, 10], [16, 3], [24, 6], [32, 2], [48, 2], [64, 1]], 'representations': {'PC': {'code': '98224154899213930606546677184170932679389431293467915986623443818909926226690234005324131711235683042688586504863190853358664447418046197605687215577134809373', 'gens': [1, 4, 5, 7, 10], 'pres': [10, 2, 2, 3, 2, 2, 3, 2, 3, 5, 3, 20, 51, 283683, 37933, 7583, 25953, 275404, 127214, 158874, 36534, 144, 604805, 305295, 108025, 22115, 687126, 365836, 118046, 34476, 29866, 6566, 206, 418567, 449297, 209307, 21157, 10607, 317, 233288, 155538, 288009, 218419, 36029, 6069]}, 'Perm': {'d': 14, 'gens': [7265669795, 13057516096]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 12], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'F_5\\times S_3\\wr C_3', 'transitive_degree': 45, 'wreath_data': None, 'wreath_product': False}
