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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '800.839', 'ambient_counter': 839, 'ambient_order': 800, 'ambient_tex': 'C_{10}^2.C_2^3', 'central': False, 'central_factor': False, 'centralizer_order': 400, 'characteristic': True, 'core_order': 25, 'counter': 92, 'cyclic': False, 'direct': False, 'hall': 5, 'label': '800.839.32.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '32.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '32.29', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 29, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 32, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^2:Q_8', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '25.2', 'subgroup_hash': 2, 'subgroup_order': 25, 'subgroup_tex': 'C_5^2', 'supersolvable': True, 'sylow': 5}
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gps_subgroup_data • Show schema
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{'ambient': '800.839', 'aut_centralizer_order': 640, 'aut_label': '32.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '16.2', 'aut_weyl_index': 640, 'centralizer': '2.c1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['16.a1.a1', '16.b1.a1', '16.c1.a1', '16.d1.a1', '16.d1.b1'], 'contains': ['160.a1.a1', '160.b1.a1', '160.c1.a1', '160.c1.b1'], 'core': '32.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [605, 656, 612, 8106, 2136, 3696, 1854, 6709], 'generators': [8, 160], 'label': '800.839.32.a1.a1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '32.a1.a1', 'normal_contained_in': ['16.c1.a1', '16.a1.a1', '16.b1.a1'], 'normal_contains': ['160.a1.a1', '160.b1.a1'], 'normalizer': '1.a1.a1', 'old_label': '32.a1.a1', 'projective_image': '160.167', 'quotient_action_image': '2.1', 'quotient_action_kernel': '16.12', 'quotient_action_kernel_order': 16, 'quotient_fusion': None, 'short_label': '32.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '25.2', '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], [4, 16], [17, 11]], 'aut_group': '480.218', 'aut_hash': 218, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 480, 'aut_permdeg': 24, 'aut_perms': [429137129243734377940845, 535103329646962732126054], 'aut_phi_ratio': 24.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [5, 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], [5, 1, 24]], 'center_label': '25.2', 'center_order': 25, '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': ['5.1', '5.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['5.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [5, 1, 4, 6]], 'element_repr_type': 'PC', 'elementary': 5, 'eulerian_function': 1, 'exponent': 5, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '25.2', 'hash': 2, '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, 25]], 'label': '25.2', 'linC_count': 240, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 15, 'linQ_dim': 8, 'linQ_dim_count': 15, 'linR_count': 60, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 25, 'number_divisions': 7, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 25, 'order_factorization_type': 2, 'order_stats': [[1, 1], [5, 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': [[4, 16], [17, 11]], 'outer_group': '480.218', 'outer_hash': 218, 'outer_nilpotent': False, 'outer_order': 480, 'outer_permdeg': 24, 'outer_perms': [429137129243734377940845, 535103329646962732126054], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,5)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 10, 'pgroup': 5, 'primary_abelian_invariants': [5, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 6]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -5, 5]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [3997, 5325]}, 'Perm': {'d': 10, 'gens': [1451520, 96]}}, 'schur_multiplier': [5], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [5, 5], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5^2', 'transitive_degree': 25, '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': '40.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [20, 20, 4, 4, 4, 4, 4, 4], 'aut_gens': [[1, 2, 40], [221, 306, 460], [601, 526, 40], [601, 142, 780], [201, 286, 760], [621, 606, 760], [201, 686, 140], [401, 258, 680], [21, 182, 520]], 'aut_group': None, 'aut_hash': 130960422074332702, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10240, 'aut_permdeg': 32, 'aut_perms': [136249891612819841779334465470748342, 60385655249514625039021316420290804, 60502122230329754262855313919651178, 109985513429195013645425770610594296, 89042001457241785685611946048970570, 114503513061641008530009019514715858, 225940436377206417977806791223443492, 44587938675712286456929849514237621], 'aut_phi_ratio': 32.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 10, 2, 1], [4, 2, 2, 1], [4, 4, 2, 1], [4, 10, 2, 1], [4, 20, 2, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 8, 1], [10, 1, 4, 3], [10, 2, 2, 3], [10, 2, 8, 3], [10, 10, 8, 1], [20, 2, 8, 1], [20, 4, 4, 1], [20, 4, 8, 2], [20, 4, 16, 1], [20, 4, 32, 1], [20, 10, 8, 1], [20, 20, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_5:(C_2^2\\times C_4^2\\times C_2^2\\wr C_2)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '256.56082', 'autcent_hash': 56082, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '40.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 40, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 10, 2], [4, 2, 2], [4, 4, 2], [4, 10, 2], [4, 20, 2], [5, 1, 4], [5, 2, 10], [10, 1, 12], [10, 2, 30], [10, 10, 8], [20, 2, 8], [20, 4, 68], [20, 10, 8], [20, 20, 8]], 'center_label': '20.5', 'center_order': 20, 'central_product': True, 'central_quotient': '40.13', 'commutator_count': 1, 'commutator_label': '20.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 839, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['160.167', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 10, 1, 2], [4, 2, 1, 2], [4, 4, 1, 2], [4, 10, 2, 1], [4, 20, 1, 2], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 4, 2], [10, 1, 4, 3], [10, 2, 2, 3], [10, 2, 4, 6], [10, 10, 4, 2], [20, 2, 4, 2], [20, 4, 2, 2], [20, 4, 4, 16], [20, 10, 8, 1], [20, 20, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 15624, 'exponent': 20, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '200.50', 'hash': 839, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 2, 10], 'inner_gens': [[1, 422, 440], [421, 2, 760], [401, 82, 40]], 'inner_hash': 13, 'inner_nilpotent': False, 'inner_order': 40, 'inner_split': False, 'inner_tex': 'C_2\\times D_{10}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 40], [2, 110], [4, 20]], 'label': '800.839', 'linC_count': 768, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 16, 'linQ_dim': 12, 'linQ_dim_count': 24, 'linR_count': 448, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C10^2.C2^3', 'ngens': 7, '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': 30, 'number_characteristic_subgroups': 38, 'number_conjugacy_classes': 170, 'number_divisions': 54, 'number_normal_subgroups': 70, 'number_subgroup_autclasses': 115, 'number_subgroup_classes': 186, 'number_subgroups': 524, 'old_label': None, 'order': 800, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 23], [4, 72], [5, 24], [10, 152], [20, 528]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4, 2, 4], 'outer_gen_pows': [0, 0, 0, 0, 200, 0], 'outer_gens': [[421, 2, 40], [401, 2, 460], [401, 18, 40], [21, 18, 520], [221, 638, 40], [401, 34, 40]], 'outer_group': '256.55632', 'outer_hash': 55632, 'outer_nilpotent': True, 'outer_order': 256, 'outer_permdeg': 16, 'outer_perms': [1313941673970, 1313988493075, 87091200, 87129797, 33046, 2795932339200], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4^2.C_2^4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 13], [8, 24], [16, 5]], 'representations': {'PC': {'code': 886923609722522662083860525158881935555, 'gens': [1, 2, 5], 'pres': [7, -2, -2, -2, -5, -2, -2, -5, 2800, 5909, 36, 58, 15404, 13311, 102, 15132, 124, 15693]}, 'GLZN': {'d': 2, 'p': 33, 'gens': [826574, 898450, 406252, 28083, 359380, 794398, 35962]}, 'Perm': {'d': 22, 'gens': [5165, 56462890705917696000, 109750244591699712007, 6749568000, 16, 163613728419837696000, 50520]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}^2.C_2^3', 'transitive_degree': 160, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 2, 2, 2, 2, 4], 'aut_gens': [[1, 2, 8], [17, 18, 12], [21, 22, 12], [17, 6, 8], [21, 18, 28], [1, 2, 28], [1, 2, 31]], 'aut_group': '128.2163', 'aut_hash': 2163, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 128, 'aut_permdeg': 12, 'aut_perms': [101693656, 175351807, 94434600, 22182600, 138989520, 255522967], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 2, 1], [4, 2, 2, 2], [4, 4, 2, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_2^4:D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [4, 2, 4], [4, 4, 4]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '8.5', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 29, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [4, 2, 1, 2], [4, 2, 2, 1], [4, 4, 1, 4]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 84, 'exponent': 4, 'exponents_of_order': [5], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '8.5', 'hash': 29, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 2], 'inner_gens': [[1, 2, 28], [1, 2, 24], [21, 18, 8]], 'inner_hash': 5, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'C_2^3', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 6]], 'label': '32.29', 'linC_count': 12, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 4, 'linQ_dim': 6, 'linQ_dim_count': 6, 'linR_count': 6, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2:Q8', 'ngens': 3, 'nilpotency_class': 2, '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': 9, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 14, 'number_divisions': 13, 'number_normal_subgroups': 24, 'number_subgroup_autclasses': 26, 'number_subgroup_classes': 37, 'number_subgroups': 50, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7], [4, 24]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [3, 0, 0, 0], 'outer_gens': [[1, 2, 11], [1, 22, 8], [1, 18, 8], [17, 2, 8]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 12, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 1]], 'representations': {'PC': {'code': 621347402, 'gens': [1, 2, 4], 'pres': [5, 2, 2, 2, 2, 2, 26, 563, 248, 58]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [58301505957300808, 25038659807093174, 24992896919818108]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [16424869, 28816400, 14811013, 28233362, 35101929]}, 'Perm': {'d': 12, 'gens': [40286719, 87103793, 7639099, 18619, 87091200]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2:Q_8', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
