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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '108000.b', 'ambient_counter': 2, 'ambient_order': 108000, 'ambient_tex': 'D_5^3.C_3^2:D_6', 'central': False, 'central_factor': False, 'centralizer_order': 20, 'characteristic': False, 'core_order': 1, 'counter': 1125, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '108000.b.1080.bm1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '1080.bm1', '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': 1080, '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': '100.13', 'subgroup_hash': 13, 'subgroup_order': 100, 'subgroup_tex': 'D_5^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '108000.b', 'aut_centralizer_order': None, 'aut_label': '1080.bm1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '5400.d1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['216.l1', '360.eo1', '360.ep1', '360.eq1', '540.i1', '540.o1', '540.be1'], 'contains': ['2160.m1', '2160.v1', '5400.bm1'], 'core': '108000.a1', 'coset_action_label': None, 'count': 27, 'diagramx': None, 'generators': [92892, 864, 21600, 28152], 'label': '108000.b.1080.bm1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '12.d1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '27.a1', 'old_label': '1080.bm1', 'projective_image': '108000.b', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '1080.bm1', 'subgroup_fusion': None, 'weyl_group': '200.43'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 40, 'aut_gen_orders': [8, 4, 2, 5, 5, 4, 10], 'aut_gens': [[1, 2, 20], [10, 81, 4], [90, 73, 16], [5, 38, 80], [9, 42, 20], [13, 42, 20], [9, 54, 40], [9, 78, 20]], 'aut_group': '800.1191', 'aut_hash': 1191, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 800, 'aut_permdeg': 10, 'aut_perms': [370090, 2883024, 806992, 1999297, 2085339, 2015705, 2376125], 'aut_phi_ratio': 20.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 5, 2, 1], [2, 25, 1, 1], [5, 2, 4, 1], [5, 4, 4, 1], [10, 10, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_5\\wr C_2', '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': 40, 'autcentquo_group': '800.1191', 'autcentquo_hash': 1191, 'autcentquo_nilpotent': False, 'autcentquo_order': 800, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\wr C_2', 'cc_stats': [[1, 1, 1], [2, 5, 2], [2, 25, 1], [5, 2, 4], [5, 4, 4], [10, 10, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '100.13', 'commutator_count': 1, 'commutator_label': '25.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 13, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['10.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 5, 1, 2], [2, 25, 1, 1], [5, 2, 2, 2], [5, 4, 2, 2], [10, 10, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 10, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[4, 1, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '100.13', 'hash': 13, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 10, 5], 'inner_gens': [[1, 18, 20], [5, 2, 80], [1, 42, 20]], 'inner_hash': 13, 'inner_nilpotent': False, 'inner_order': 100, 'inner_split': False, 'inner_tex': 'D_5^2', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 8], [4, 4]], 'label': '100.13', 'linC_count': 20, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 6, 'linQ_dim': 8, 'linQ_dim_count': 6, 'linR_count': 20, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D5^2', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 16, 'number_divisions': 10, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 15, 'number_subgroup_classes': 24, 'number_subgroups': 124, 'old_label': None, 'order': 100, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 35], [5, 24], [10, 40]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 11], 'outer_gens': [[10, 81, 8], [10, 61, 8]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [2, 13], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [4, 4], [8, 2]], 'representations': {'PC': {'code': 413139638799, 'gens': [1, 2, 4], 'pres': [4, -2, -2, -5, -5, 145, 21, 194, 647]}, 'GLFp': {'d': 3, 'p': 5, 'gens': [25977, 1644884, 147908, 1594376]}, 'Perm': {'d': 10, 'gens': [2177183, 2045400, 409056, 1451616]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_5^2', 'transitive_degree': 10, '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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 60, 'aut_gen_orders': [12, 6, 12, 30, 10, 5], 'aut_gens': [[1, 4, 24, 144, 4320, 21600], [82225, 94796, 49128, 32688, 12960, 51840], [60497, 22324, 29520, 25320, 89856, 864], [32113, 29500, 27672, 61488, 34560, 43200], [22897, 66940, 53592, 60624, 82080, 21600], [63073, 67036, 80376, 80496, 4320, 21600], [2593, 66532, 3480, 144, 4320, 21600]], 'aut_group': '216000.d', 'aut_hash': 4125830444204071041, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 216000, 'aut_permdeg': 459, 'aut_perms': 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'aut_phi_ratio': 7.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 15, 1, 1], [2, 75, 1, 1], [2, 125, 1, 1], [2, 135, 1, 1], [2, 675, 1, 1], [2, 1125, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 600, 1, 1], [3, 1200, 1, 1], [4, 450, 1, 4], [4, 2250, 1, 4], [5, 12, 1, 1], [5, 12, 2, 1], [5, 16, 1, 1], [5, 24, 1, 1], [5, 48, 1, 1], [6, 30, 1, 2], [6, 60, 1, 1], [6, 150, 1, 2], [6, 250, 1, 1], [6, 300, 1, 1], [6, 750, 1, 1], [6, 1800, 1, 1], [6, 3000, 1, 1], [6, 6000, 1, 1], [6, 9000, 1, 1], [10, 60, 2, 2], [10, 108, 1, 1], [10, 108, 2, 1], [10, 120, 1, 1], [10, 144, 1, 1], [10, 216, 1, 1], [10, 300, 1, 1], [10, 432, 1, 1], [10, 540, 2, 2], [10, 1080, 1, 1], [10, 2700, 1, 1], [12, 900, 1, 4], [12, 4500, 1, 4], [15, 24, 1, 2], [15, 24, 2, 3], [15, 32, 1, 1], [15, 48, 1, 2], [15, 48, 2, 2], [15, 96, 1, 3], [15, 96, 2, 1], [15, 1200, 2, 1], [15, 2400, 2, 1], [20, 900, 2, 4], [30, 120, 2, 6], [30, 240, 1, 2], [30, 240, 2, 2], [30, 600, 1, 2], [30, 600, 2, 1], [30, 3600, 2, 1], [60, 1800, 2, 4]], 'aut_supersolvable': False, 'aut_tex': 'D_5^3:\\He_3.C_2^3', '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': 60, 'autcentquo_group': '216000.d', 'autcentquo_hash': 4125830444204071041, 'autcentquo_nilpotent': False, 'autcentquo_order': 216000, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'D_5^3:\\He_3.C_2^3', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 15, 1], [2, 75, 1], [2, 125, 1], [2, 135, 1], [2, 675, 1], [2, 1125, 1], [3, 2, 1], [3, 6, 1], [3, 600, 1], [3, 1200, 1], [4, 450, 4], [4, 2250, 4], [5, 12, 3], [5, 16, 1], [5, 24, 1], [5, 48, 1], [6, 30, 2], [6, 60, 1], [6, 150, 2], [6, 250, 1], [6, 300, 1], [6, 750, 1], [6, 1800, 1], [6, 3000, 1], [6, 6000, 1], [6, 9000, 1], [10, 60, 4], [10, 108, 3], [10, 120, 1], [10, 144, 1], [10, 216, 1], [10, 300, 1], [10, 432, 1], [10, 540, 4], [10, 1080, 1], [10, 2700, 1], [12, 900, 4], [12, 4500, 4], [15, 24, 8], [15, 32, 1], [15, 48, 6], [15, 96, 5], [15, 1200, 2], [15, 2400, 2], [20, 900, 8], [30, 120, 12], [30, 240, 6], [30, 600, 4], [30, 3600, 2], [60, 1800, 8]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '108000.b', 'commutator_count': 1, 'commutator_label': '13500.j', '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', '5.1', '5.1', '5.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 15, 1, 1], [2, 75, 1, 1], [2, 125, 1, 1], [2, 135, 1, 1], [2, 675, 1, 1], [2, 1125, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 600, 1, 1], [3, 1200, 1, 1], [4, 450, 2, 2], [4, 2250, 2, 2], [5, 12, 1, 1], [5, 12, 2, 1], [5, 16, 1, 1], [5, 24, 1, 1], [5, 48, 1, 1], [6, 30, 1, 2], [6, 60, 1, 1], [6, 150, 1, 2], [6, 250, 1, 1], [6, 300, 1, 1], [6, 750, 1, 1], [6, 1800, 1, 1], [6, 3000, 1, 1], [6, 6000, 1, 1], [6, 9000, 1, 1], [10, 60, 2, 2], [10, 108, 1, 1], [10, 108, 2, 1], [10, 120, 1, 1], [10, 144, 1, 1], [10, 216, 1, 1], [10, 300, 1, 1], [10, 432, 1, 1], [10, 540, 2, 2], [10, 1080, 1, 1], [10, 2700, 1, 1], [12, 900, 2, 2], [12, 4500, 2, 2], [15, 24, 1, 2], [15, 24, 2, 3], [15, 32, 1, 1], [15, 48, 1, 2], [15, 48, 2, 2], [15, 96, 1, 3], [15, 96, 2, 1], [15, 1200, 2, 1], [15, 2400, 2, 1], [20, 900, 4, 2], [30, 120, 2, 6], [30, 240, 1, 2], [30, 240, 2, 2], [30, 600, 1, 2], [30, 600, 2, 1], [30, 3600, 2, 1], [60, 1800, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6696, 'exponent': 60, 'exponents_of_order': [5, 3, 3], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[24, 0, 4], [24, 1, 16], [48, 1, 10], [96, 1, 4]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '36000.b', 'hash': 2628370552740919509, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [60, 6, 6, 30, 5, 5], 'inner_gens': [[1, 59492, 7656, 49968, 34560, 51840], [102705, 4, 8520, 73464, 107136, 25920], [5329, 4468, 24, 54576, 82080, 21600], [84097, 99628, 79512, 144, 17280, 86400], [99361, 31108, 51864, 8784, 4320, 21600], [99361, 17284, 24, 43344, 4320, 21600]], 'inner_hash': 2628370552740919509, 'inner_nilpotent': False, 'inner_order': 108000, 'inner_split': True, 'inner_tex': 'D_5^3.C_3^2:D_6', 'inner_used': [1, 2], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 8], [2, 8], [3, 8], [4, 2], [6, 12], [12, 26], [16, 6], [24, 36], [32, 3], [48, 14], [96, 5]], 'label': '108000.b', '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': 'D5^3.C3^2:D6', 'ngens': 11, '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, 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': 96, 'number_characteristic_subgroups': 33, 'number_conjugacy_classes': 128, 'number_divisions': 84, 'number_normal_subgroups': 33, 'number_subgroup_autclasses': 1890, 'number_subgroup_classes': 1890, 'number_subgroups': 514680, 'old_label': None, 'order': 108000, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 2159], [3, 1808], [4, 10800], [5, 124], [6, 21520], [10, 7716], [12, 21600], [15, 8192], [20, 7200], [30, 12480], [60, 14400]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [21266], 'outer_gens': [[39169, 32596, 53592, 45072, 12960, 64800]], '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': 6, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [3, 4], [4, 3], [6, 8], [12, 9], [16, 2], [24, 14], [32, 3], [48, 18], [64, 1], [96, 9], [192, 1]], 'representations': {'PC': {'code': '1008558753212225743673803130054926491617560988528588920181225151186974751577803730642033227721856379200180120859619223645038986172733357047246506002806261943757391498665657168457458362342551016280175657627641358386484303783543935', 'gens': [1, 3, 5, 7, 10, 11], 'pres': [11, 2, 2, 2, 3, 2, 3, 2, 3, 5, 5, 5, 22, 634668, 1963238, 601603, 90, 3980771, 1617454, 421084, 2234115, 117176, 64387, 158, 98213, 24579, 11918, 3847542, 2267513, 1414210, 314892, 175148, 226, 50695, 50706, 951485, 586648, 4275, 348, 171080, 171091, 534630, 320801, 14308, 3801609, 2946271, 1199922, 376253, 13275, 6272650, 784112, 52315, 72676]}, 'Perm': {'d': 24, 'gens': [1129129429352468595881, 28209890841556454636703]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'D_5^3.C_3^2:D_6', 'transitive_degree': 45, 'wreath_data': None, 'wreath_product': False}
