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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '20250.f', 'ambient_counter': 6, 'ambient_order': 20250, 'ambient_tex': 'C_{15}\\wr S_3', 'central': False, 'central_factor': False, 'centralizer_order': 450, 'characteristic': False, 'core_order': 5, 'counter': 147, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '20250.f.405.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '405.a1', 'outer_equivalence': True, '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': 405, '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': '50.5', 'subgroup_hash': 5, 'subgroup_order': 50, 'subgroup_tex': 'C_5\\times C_{10}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '20250.f', 'aut_centralizer_order': None, 'aut_label': '405.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '45.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['81.a1', '135.a1', '135.b1', '135.c1'], 'contains': ['810.b1', '2025.a1', '2025.b1', '2025.c1'], 'core': '4050.a1', 'coset_action_label': None, 'count': 45, 'diagramx': [8332, -1, 2791, -1], 'generators': [15, 16740, 6], 'label': '20250.f.405.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '3.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '45.a1', 'old_label': '405.a1', 'projective_image': '4050.l', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '405.a1', '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': '50.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 120, 'aut_gen_orders': [4, 3], 'aut_gens': [[1, 5], [12, 5], [41, 19]], 'aut_group': '480.218', 'aut_hash': 218, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 480, 'aut_permdeg': 24, 'aut_perms': [463694601489386759858004, 613368993823319198212950], 'aut_phi_ratio': 24.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [5, 1, 24, 1], [10, 1, 24, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(2,5)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 120, 'autcent_group': '480.218', 'autcent_hash': 218, 'autcent_nilpotent': False, 'autcent_order': 480, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(2,5)', '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], [5, 1, 24], [10, 1, 24]], 'center_label': '50.5', 'center_order': 50, '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', '5.1', '5.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['5.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [5, 1, 4, 6], [10, 1, 4, 6]], 'element_repr_type': 'PC', 'elementary': 5, 'eulerian_function': 3, 'exponent': 10, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '50.5', 'hash': 5, 'hyperelementary': 5, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 5], [1, 5]], '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, 50]], 'label': '50.5', 'linC_count': 720, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 45, 'linQ_dim': 8, 'linQ_dim_count': 45, 'linR_count': 180, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5*C10', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 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': 50, 'number_divisions': 14, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 16, 'number_subgroups': 16, 'old_label': None, 'order': 50, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [5, 24], [10, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 120, 'outer_gen_orders': [4, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[12, 5], [41, 19]], 'outer_group': '480.218', 'outer_hash': 218, 'outer_nilpotent': False, 'outer_order': 480, 'outer_permdeg': 24, 'outer_perms': [463694601489386759858004, 613368993823319198212950], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,5)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 5, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 12]], 'representations': {'PC': {'code': 2227, 'gens': [1, 2], 'pres': [3, -5, -2, -5, 16]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [2663, 3997]}, 'Perm': {'d': 12, 'gens': [39916800, 1451520, 96]}}, 'schur_multiplier': [5], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [5, 10], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times C_{10}', 'transitive_degree': 50, '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': '30.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [12, 6, 12, 60, 12], 'aut_gens': [[1, 30, 90, 1350], [17467, 9150, 12510, 15030], [5669, 11940, 5310, 1350], [19987, 14160, 1170, 17550], [16087, 9420, 16290, 18900], [15167, 2370, 8820, 19620]], 'aut_group': None, 'aut_hash': 6505149385735295526, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 129600, 'aut_permdeg': 67, 'aut_perms': [14499139907892667952741172630372343361800962462394761242761466730557315673740011888500838229812, 34282396368304798301601812769755092881282383239782376175616865243245523884182039222143107800327, 25971875403696027615449504952292441244788872207235011379065161057314701157079707233602021313878, 34219400654457346908144313704691139635448271581851372839864054556262917027288813422107250362891, 12714602717120631981690664792253647499072431516353962837944607121870457650782084903345581361483], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 45, 1, 1], [3, 1, 2, 1], [3, 3, 6, 1], [3, 6, 1, 1], [3, 450, 1, 1], [5, 1, 4, 1], [5, 3, 4, 1], [5, 3, 16, 1], [5, 6, 2, 1], [5, 6, 8, 1], [6, 45, 2, 1], [6, 45, 6, 1], [9, 450, 2, 1], [10, 45, 4, 2], [10, 45, 16, 1], [15, 1, 8, 1], [15, 3, 8, 1], [15, 3, 24, 2], [15, 3, 32, 1], [15, 3, 96, 1], [15, 6, 4, 4], [15, 6, 8, 2], [15, 6, 12, 1], [15, 6, 16, 3], [15, 6, 24, 2], [15, 6, 32, 2], [15, 6, 48, 1], [15, 6, 96, 2], [15, 450, 4, 1], [30, 45, 8, 2], [30, 45, 24, 2], [30, 45, 32, 1], [30, 45, 96, 1], [45, 450, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{15}^2.C_6^2.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 12, 'autcent_group': '12.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 12, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{12}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '10800.t', 'autcentquo_hash': 664549956316970573, 'autcentquo_nilpotent': False, 'autcentquo_order': 10800, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_{15}^2:(C_4\\times D_6)', 'cc_stats': [[1, 1, 1], [2, 45, 1], [3, 1, 2], [3, 3, 6], [3, 6, 1], [3, 450, 1], [5, 1, 4], [5, 3, 20], [5, 6, 10], [6, 45, 8], [9, 450, 2], [10, 45, 24], [15, 1, 8], [15, 3, 184], [15, 6, 444], [15, 450, 4], [30, 45, 192], [45, 450, 8]], 'center_label': '15.1', 'center_order': 15, 'central_product': True, 'central_quotient': '1350.43', 'commutator_count': 1, 'commutator_label': '675.12', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '3.1', '5.1', '5.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 6, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['4050.l', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 45, 1, 1], [3, 1, 2, 1], [3, 3, 2, 3], [3, 6, 1, 1], [3, 450, 1, 1], [5, 1, 4, 1], [5, 3, 4, 5], [5, 6, 2, 1], [5, 6, 4, 2], [6, 45, 2, 4], [9, 450, 2, 1], [10, 45, 4, 6], [15, 1, 8, 1], [15, 3, 8, 23], [15, 6, 4, 11], [15, 6, 8, 50], [15, 450, 4, 1], [30, 45, 8, 24], [45, 450, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1296, 'exponent': 90, 'exponents_of_order': [4, 3, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[3, 0, 192], [6, 0, 240]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '2250.m', 'hash': 3995948840922571416, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [30, 3, 5, 15], 'inner_gens': [[1, 17610, 5220, 3960], [15091, 30, 5220, 2610], [16471, 16500, 90, 1350], [18991, 120, 90, 1350]], 'inner_hash': 43, 'inner_nilpotent': False, 'inner_order': 1350, 'inner_split': True, 'inner_tex': 'C_{15}^2:S_3', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': None, 'irrep_stats': [[1, 30], [2, 15], [3, 420], [6, 455]], 'label': '20250.f', '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': False, 'metacyclic': False, 'monomial': None, 'name': 'C15wrS3', 'ngens': 8, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 48, 'number_characteristic_subgroups': 24, 'number_conjugacy_classes': 920, 'number_divisions': 139, 'number_normal_subgroups': 24, 'number_subgroup_autclasses': 248, 'number_subgroup_classes': 632, 'number_subgroups': 7432, 'old_label': None, 'order': 20250, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 45], [3, 476], [5, 124], [6, 360], [9, 900], [10, 1080], [15, 5024], [30, 8640], [45, 3600]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 4, 12], 'outer_gen_pows': [0, 6750, 0], 'outer_gens': [[16751, 6630, 16380, 17640], [18811, 210, 1170, 17550], [6013, 7710, 4410, 14850]], 'outer_group': '96.161', 'outer_hash': 161, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 13, 'outer_perms': [479001600, 1704, 4032003], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4\\times C_{12}', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 3], [6, 7], [8, 3], [12, 11], [16, 1], [24, 59], [48, 50]], 'representations': {'PC': {'code': '263455016390904117307153948512773245157302429287487269118575250529255802740087', 'gens': [1, 4, 5, 7], 'pres': [8, 2, 3, 5, 3, 3, 5, 3, 5, 16, 57, 563523, 281291, 64979, 208804, 6988, 156, 622085, 20765, 221766, 4902, 222, 587527, 10975]}, 'Perm': {'d': 24, 'gens': [25961517332928799022568, 54108132655904139956416]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [30], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{15}\\wr S_3', 'transitive_degree': 45, 'wreath_data': ['C_{15}', 'S_3', '3T2'], 'wreath_product': True}
