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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1280.1116277', 'ambient_counter': 1116277, 'ambient_order': 1280, 'ambient_tex': 'D_{10}.C_2^6', 'central': False, 'central_factor': False, 'centralizer_order': 32, 'characteristic': False, 'core_order': 160, 'counter': 38, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1280.1116277.8.n1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '8.n1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '8.5', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 5, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^3', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '160.236', 'subgroup_hash': 236, 'subgroup_order': 160, 'subgroup_tex': 'C_2^3\\times F_5', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1280.1116277', 'aut_centralizer_order': 4096, 'aut_label': '8.n1', 'aut_quo_index': 21, 'aut_stab_index': 14, 'aut_weyl_group': None, 'aut_weyl_index': 57344, 'centralizer': '40.c1', 'complements': [], 'conjugacy_class_count': 14, 'contained_in': ['4.c1', '4.f1', '4.n1'], 'contains': ['16.h1', '16.l1', '16.bf1', '40.p1'], 'core': '8.n1', 'coset_action_label': None, 'count': 14, 'diagramx': [4509, 7180, 5090, 7297], 'generators': [1729531, 64641, 1344021, 1888829, 64801, 576161], 'label': '1280.1116277.8.n1', 'mobius_quo': 0, 'mobius_sub': -8, 'normal_closure': '8.n1', 'normal_contained_in': ['4.c1', '4.f1', '4.n1'], 'normal_contains': ['16.h1', '16.l1'], 'normalizer': '1.a1', 'old_label': '8.n1', 'projective_image': '160.236', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.n1', 'subgroup_fusion': None, 'weyl_group': '40.12'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '32.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 420, 'aut_gen_orders': [4, 4], 'aut_gens': [[1, 2, 4, 16], [83, 1, 21, 34], [80, 81, 39, 114]], 'aut_group': '26880.c', 'aut_hash': 1014459643765653084, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 26880, 'aut_permdeg': 13, 'aut_perms': [1162793542, 2972763469], 'aut_phi_ratio': 420.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1], [2, 5, 1, 1], [2, 5, 7, 1], [4, 5, 8, 2], [5, 4, 1, 1], [10, 4, 7, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_5\\times C_2^3:\\GL(3,2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '1344.11686', 'autcent_hash': 11686, 'autcent_nilpotent': False, 'autcent_order': 1344, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^3:\\GL(3,2)', '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, 7], [2, 5, 8], [4, 5, 16], [5, 4, 1], [10, 4, 7]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '20.3', 'commutator_count': 1, 'commutator_label': '5.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 236, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['20.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 5, 1, 8], [4, 5, 2, 8], [5, 4, 1, 1], [10, 4, 1, 7]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 7440, 'exponent': 20, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '160.236', 'hash': 236, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [1, 1, 4, 5], 'inner_gens': [[1, 2, 4, 16], [1, 2, 4, 16], [1, 2, 4, 48], [1, 2, 132, 16]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 20, 'inner_split': True, 'inner_tex': 'F_5', 'inner_used': [3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [4, 8]], 'label': '160.236', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3*F5', 'ngens': 6, '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': 8, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 40, 'number_divisions': 32, 'number_normal_subgroups': 134, 'number_subgroup_autclasses': 30, 'number_subgroup_classes': 236, 'number_subgroups': 644, 'old_label': None, 'order': 160, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 47], [4, 80], [5, 4], [10, 28]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 84, 'outer_gen_orders': [4, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[81, 80, 5, 98], [83, 80, 7, 97]], 'outer_group': '1344.11686', 'outer_hash': 11686, 'outer_nilpotent': False, 'outer_order': 1344, 'outer_permdeg': 8, 'outer_perms': [5158, 14730], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^3:\\GL(3,2)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 4], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 8], [4, 8]], 'representations': {'PC': {'code': 1362374182334984437763, 'gens': [1, 2, 3, 5], 'pres': [6, -2, -2, -2, -2, -2, -5, 50, 376, 562, 88, 881, 599]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101740716535443, 125101729074815868, 24992906222183252, 25038633845686381]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [122109387504, 27617554165, 30690352864, 71446177916, 30746810364, 30619500416]}, 'Perm': {'d': 11, 'gens': [408366, 289, 127, 415, 806400, 4072320]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 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_2^3\\times F_5', 'transitive_degree': 40, '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': '128.2319', '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': [[1728421, 65151, 576429, 1248019, 1344021, 1888829], [1728421, 64671, 577229, 1248019, 1344021, 1888829], [1728421, 64511, 576429, 1248019, 1344021, 1888829], [1728901, 65151, 577229, 1248019, 1344021, 1888829], [1729381, 65151, 576429, 1248019, 1344021, 1888829], [1729221, 64351, 576429, 1248019, 1344021, 1888829], [1728421, 65151, 576429, 1248019, 1344021, 1888829], [2145239, 65151, 576429, 1248019, 1344021, 1888829], [865219, 65151, 576429, 1248019, 1344021, 1888829], [2144439, 65151, 576429, 1248019, 1344021, 1888829], [1088431, 65151, 576429, 1248019, 1344021, 1888829], [192009, 576899, 576429, 1248019, 1344021, 1888829], [1120011, 736721, 576429, 1248019, 1344021, 1888829], [1472029, 1856919, 576429, 1248019, 1344021, 1888829], [480821, 1376751, 576429, 1248019, 1344021, 1888829], [1728421, 65151, 576429, 2528039, 1344021, 1888829], [1728421, 65151, 576429, 704811, 1344021, 1888829], [1728421, 65151, 576429, 1248019, 1344021, 608809], [1728421, 65151, 1856409, 1248019, 1344021, 1888829], [1728421, 65151, 576429, 2528839, 1344021, 1888829], [1728421, 65151, 576429, 1344021, 1248019, 704811], [1728421, 65151, 576429, 1248019, 704811, 1344021], [1728421, 65151, 576429, 1248019, 704811, 1888829], [1728421, 65151, 97221, 1248019, 1344021, 1888829], [1728421, 65151, 576429, 1248019, 1344021, 608009], [1728421, 65151, 1857209, 1248019, 1344021, 1888829], [1728421, 65151, 576429, 1248019, 1344821, 1888829], [1728421, 65151, 576429, 1248019, 608809, 1888829], [1728421, 65471, 576429, 1248019, 1344021, 1888829]], 'aut_group': None, 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 220200960, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': None, 'aut_solvable': None, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 14, 1], [2, 2, 16, 1], [2, 5, 1, 2], [2, 5, 14, 1], [2, 10, 16, 1], [4, 2, 8, 1], [4, 5, 16, 2], [4, 10, 8, 3], [4, 10, 16, 2], [5, 4, 1, 1], [10, 4, 1, 1], [10, 4, 14, 1], [10, 8, 16, 1], [20, 8, 8, 1]], '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, 15], [2, 2, 16], [2, 5, 16], [2, 10, 16], [4, 2, 8], [4, 5, 32], [4, 10, 56], [5, 4, 1], [10, 4, 15], [10, 8, 16], [20, 8, 8]], 'center_label': '16.14', 'center_order': 16, 'central_product': True, 'central_quotient': '80.50', 'commutator_count': 1, 'commutator_label': '10.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '5.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 1116277, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['20.3', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15], [2, 2, 1, 16], [2, 5, 1, 16], [2, 10, 1, 16], [4, 2, 1, 8], [4, 5, 2, 16], [4, 10, 1, 8], [4, 10, 2, 24], [5, 4, 1, 1], [10, 4, 1, 15], [10, 8, 1, 16], [20, 8, 1, 8]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 5857125120, 'exponent': 20, 'exponents_of_order': [8, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '640.21539', 'hash': 1116277, 'hyperelementary': 2, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': None, 'inner_gens': None, 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': 80, 'inner_split': None, 'inner_tex': 'C_2^2\\times F_5', 'inner_used': None, 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 128], [2, 32], [4, 32], [8, 8]], 'label': '1280.1116277', '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': True, 'metacyclic': False, 'monomial': True, 'name': 'D10.C2^6', 'ngens': 9, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 21, 'number_characteristic_subgroups': 23, 'number_conjugacy_classes': 200, 'number_divisions': 160, 'number_normal_subgroups': 6264, 'number_subgroup_autclasses': 284, 'number_subgroup_classes': 19998, 'number_subgroups': 81206, 'old_label': None, 'order': 1280, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 287], [4, 736], [5, 4], [10, 188], [20, 64]], 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': True, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': None, 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 2752512, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': None, 'pc_rank': None, 'perfect': False, 'permutation_degree': None, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 2, 4], 'quasisimple': False, 'rank': 6, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 64], [2, 48], [4, 40], [8, 8]], 'representations': {'PC': {'code': '268340497924671365978583158130947158994848768', 'gens': [1, 3, 5, 7, 8, 9], 'pres': [9, -2, -2, -2, -5, -2, -2, -2, -2, -2, 18, 2918, 497, 74, 579, 588, 5404, 1372, 130]}, 'GLZN': {'d': 2, 'p': 40, 'gens': [1248019, 192009, 1248829, 64109, 64801, 1344021, 576001, 1888829, 64321]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{10}.C_2^6', 'transitive_degree': 160, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 84, 'aut_gen_orders': [3, 3], 'aut_gens': [[1, 2, 4], [4, 5, 3], [2, 4, 1]], 'aut_group': '168.42', 'aut_hash': 42, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 168, 'aut_permdeg': 7, 'aut_perms': [4361, 244], 'aut_phi_ratio': 42.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1]], 'aut_supersolvable': False, 'aut_tex': '\\PSL(2,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '168.42', 'autcent_hash': 42, 'autcent_nilpotent': False, 'autcent_order': 168, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\PSL(2,7)', '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, 7]], 'center_label': '8.5', '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': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 3]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '8.5', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], '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': '8.5', 'linC_count': 28, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 28, 'linQ_dim': 3, 'linQ_dim_count': 28, 'linR_count': 28, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3', 'ngens': 3, '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': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 8, 'number_divisions': 8, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 16, 'number_subgroups': 16, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 84, 'outer_gen_orders': [3, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[4, 5, 3], [2, 4, 1]], 'outer_group': '168.42', 'outer_hash': 42, 'outer_nilpotent': False, 'outer_order': 168, 'outer_permdeg': 7, 'outer_perms': [4361, 244], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\PSL(2,7)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3], 'pres': [3, -2, 2, 2]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [16482, 16322, 3362]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [8156, 13286, 13933]}, 'Perm': {'d': 6, 'gens': [120, 6, 1]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}
