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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1600.299', 'ambient_counter': 299, 'ambient_order': 1600, 'ambient_tex': '(C_2\\times C_4).D_{100}', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 20, 'counter': 128, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1600.299.40.d1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '40.d1.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': 40, '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': '40.13', 'subgroup_hash': 13, 'subgroup_order': 40, 'subgroup_tex': 'C_2\\times D_{10}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1600.299', 'aut_centralizer_order': None, 'aut_label': '40.d1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '200.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['8.b1.a1', '20.d1.a1', '20.h1.a1', '20.h1.b1'], 'contains': ['80.a1.a1', '80.h1.a1', '80.i1.a1', '80.o1.a1', '200.d1.a1'], 'core': '80.a1.a1', 'coset_action_label': None, 'count': 5, 'diagramx': [1196, -1, 5185, -1, 2910, -1, 2329, -1], 'generators': [1, 800, 320, 808], 'label': '1600.299.40.d1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '8.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '5.a1.a1', 'old_label': '40.d1.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '40.d1.a1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [4, 2, 2, 3, 5, 2, 2], 'aut_gens': [[1, 2, 4], [1, 2, 28], [1, 2, 5], [1, 2, 36], [20, 23, 5], [1, 10, 4], [1, 22, 4], [1, 23, 4]], 'aut_group': '480.1189', 'aut_hash': 1189, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 480, 'aut_permdeg': 9, 'aut_perms': [9, 5040, 16, 5760, 34, 41040, 90720], 'aut_phi_ratio': 30.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 5, 4, 1], [5, 2, 2, 1], [10, 2, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_5\\times S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.12', 'autcent_hash': 12, 'autcent_nilpotent': False, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 5, 4], [5, 2, 2], [10, 2, 6]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '10.1', 'commutator_count': 1, 'commutator_label': '5.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 13, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['10.1', 1], ['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 5, 1, 4], [5, 2, 2, 1], [10, 2, 2, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 42, 'exponent': 10, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '40.13', 'hash': 13, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [1, 2, 5], 'inner_gens': [[1, 2, 4], [1, 2, 36], [1, 10, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 10, 'inner_split': False, 'inner_tex': 'D_5', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 8]], 'label': '40.13', 'linC_count': 24, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 12, 'linQ_dim': 5, 'linQ_dim_count': 12, 'linR_count': 24, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*D10', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 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': 16, 'number_divisions': 12, 'number_normal_subgroups': 21, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 32, 'number_subgroups': 76, 'old_label': None, 'order': 40, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 23], [5, 4], [10, 12]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2], 'outer_gen_pows': [0, 2, 0, 0, 0], 'outer_gens': [[21, 2, 4], [1, 2, 28], [21, 23, 25], [1, 3, 4], [1, 23, 4]], 'outer_group': '48.48', 'outer_hash': 48, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 6, 'outer_perms': [143, 127, 576, 126, 121], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [4, 4]], 'representations': {'PC': {'code': 650298371, 'gens': [1, 2, 3], 'pres': [4, -2, -2, -2, -5, 222, 34, 263]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [706204575934, 141602720900, 705844438025]}, 'GLFp': {'d': 3, 'p': 5, 'gens': [1219033, 1692709, 497805, 1565004]}, 'Perm': {'d': 9, 'gens': [5167, 7, 16, 50520]}}, 'schur_multiplier': [2, 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': 'C_2\\times D_{10}', 'transitive_degree': 20, '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': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': None, 'aut_cyclic': None, 'aut_derived_length': None, 'aut_exponent': None, 'aut_gen_orders': None, 'aut_gens': [[1, 2, 8, 16], [1581, 814, 8, 276], [65, 810, 8, 1012], [1409, 814, 8, 380], [1129, 14, 8, 788], [161, 802, 8, 1116], [289, 2, 8, 1172], [73, 810, 8, 220], [933, 6, 8, 696], [901, 810, 8, 688], [365, 810, 8, 444], [673, 6, 8, 372]], 'aut_group': None, 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 256000, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': 400.0, 'aut_solvable': None, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 100, 2, 1], [4, 2, 4, 1], [4, 4, 2, 2], [4, 50, 4, 2], [4, 100, 2, 1], [5, 2, 2, 1], [10, 2, 2, 7], [20, 4, 8, 3], [25, 2, 10, 1], [50, 2, 10, 7], [100, 4, 40, 3]], 'aut_supersolvable': None, 'aut_tex': None, 'autcent_abelian': None, 'autcent_cyclic': None, 'autcent_exponent': None, 'autcent_group': None, 'autcent_hash': None, 'autcent_nilpotent': None, 'autcent_order': None, 'autcent_solvable': None, 'autcent_split': None, 'autcent_supersolvable': None, 'autcent_tex': None, 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': None, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 100, 2], [4, 2, 4], [4, 4, 4], [4, 50, 8], [4, 100, 2], [5, 2, 2], [10, 2, 14], [20, 4, 24], [25, 2, 10], [50, 2, 70], [100, 4, 120]], 'center_label': '8.5', 'center_order': 8, 'central_product': False, 'central_quotient': '200.13', 'commutator_count': 1, 'commutator_label': '100.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 299, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 100, 1, 2], [4, 2, 2, 2], [4, 4, 1, 2], [4, 4, 2, 1], [4, 50, 2, 4], [4, 100, 2, 1], [5, 2, 2, 1], [10, 2, 2, 7], [20, 4, 4, 4], [20, 4, 8, 1], [25, 2, 10, 1], [50, 2, 10, 7], [100, 4, 20, 4], [100, 4, 40, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 5040, 'exponent': 100, 'exponents_of_order': [6, 2], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '40.14', 'frattini_quotient': '40.13', 'hash': 299, 'hyperelementary': 2, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': [2, 2, 1, 50], 'inner_gens': [[1, 810, 8, 792], [809, 2, 8, 824], [1, 2, 8, 16], [841, 810, 8, 16]], 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': 200, 'inner_split': None, 'inner_tex': 'C_2\\times D_{50}', 'inner_used': None, 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 204], [4, 48]], 'label': '1600.299', 'linC_count': 5120, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 26, 'linQ_degree_count': 64, 'linQ_dim': 26, 'linQ_dim_count': 64, 'linR_count': 320, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': '(C2*C4).D100', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 37, 'number_characteristic_subgroups': 71, 'number_conjugacy_classes': 268, 'number_divisions': 46, 'number_normal_subgroups': 93, 'number_subgroup_autclasses': 225, 'number_subgroup_classes': 285, 'number_subgroups': 3591, 'old_label': None, 'order': 1600, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 207], [4, 624], [5, 4], [10, 28], [20, 96], [25, 20], [50, 140], [100, 480]], 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': False, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': '1280.56082', 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 1280, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': None, 'pc_rank': 4, 'perfect': False, 'permutation_degree': 37, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [4, 8], [8, 8], [16, 1], [20, 4], [40, 8], [80, 1]], 'representations': {'PC': {'code': 21200014464347714264260929435620181336046829583, 'gens': [1, 2, 4, 5], 'pres': [8, -2, -2, -2, -2, 2, -2, -5, -5, 12961, 41, 31684, 16492, 116, 75269, 141, 86022, 334, 81927]}, 'Perm': {'d': 37, 'gens': [20961792128206979198188043012391118421616, 382621970951438941074596907262279680000000, 764653492485494940855952486318473216000000, 263396721347933845490279645184000000, 764652949512574724795332198976716800000000, 764652949513206279223456463402041344000000, 701391835124898593479300, 1350198335266800302106816]}}, 'schur_multiplier': [2, 2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_2\\times C_4).D_{100}', 'transitive_degree': 800, 'wreath_data': None, 'wreath_product': False}
