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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '960.10958', 'ambient_counter': 10958, 'ambient_order': 960, 'ambient_tex': '\\GL(2,3):D_{10}', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 10, 'counter': 116, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '960.10958.24.g1.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '24.g1.c1', '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': 24, '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': '960.10958', 'aut_centralizer_order': None, 'aut_label': '24.g1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '120.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['12.c1.a1', '12.k1.a1', '12.l1.c1'], 'contains': ['48.c1.b1', '48.d1.a1', '48.e1.a1', '48.k1.c1', '48.k1.c2', '120.g1.c1'], 'core': '96.a1.a1', 'coset_action_label': None, 'count': 6, 'diagramx': None, 'generators': [5, 480, 192, 30], 'label': '960.10958.24.g1.c1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '2.d1.c1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '6.b1.a1', 'old_label': '24.g1.c1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '24.g1.c1', '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': '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': [2, 10, 4, 12, 4, 4], 'aut_gens': [[1, 2, 4, 24, 48], [729, 358, 764, 504, 936], [1, 290, 20, 240, 792], [757, 74, 268, 504, 624], [17, 486, 724, 240, 648], [741, 262, 260, 24, 840], [13, 590, 740, 720, 696]], 'aut_group': None, 'aut_hash': 7817645015147102572, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 15360, 'aut_permdeg': 44, 'aut_perms': [2319956062001913913470609418683530943542392405096036836, 58647457146653079567595143003666163553525559322218152, 434988749724240729042242382246917430166673135296812461, 2011249744069118130147105813025653228303884029846692884, 2513207139795620719847040852011263882652227507174915811, 548051428762891821178635311959237391472506259546159308], 'aut_phi_ratio': 60.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 12, 2, 1], [2, 30, 2, 1], [2, 60, 2, 1], [3, 8, 1, 1], [4, 6, 1, 2], [4, 10, 2, 1], [5, 2, 2, 1], [6, 8, 1, 1], [6, 8, 2, 1], [8, 12, 2, 1], [8, 60, 2, 1], [10, 2, 2, 1], [10, 2, 4, 1], [10, 24, 4, 1], [12, 40, 4, 1], [15, 16, 2, 1], [20, 12, 2, 2], [30, 16, 2, 1], [30, 16, 4, 1], [40, 12, 8, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_2^2\\times D_5\\times A_4).C_2^5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '480.1189', 'autcentquo_hash': 1189, 'autcentquo_nilpotent': False, 'autcentquo_order': 480, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\times S_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 12, 2], [2, 30, 2], [2, 60, 2], [3, 8, 1], [4, 6, 2], [4, 10, 2], [5, 2, 2], [6, 8, 3], [8, 12, 2], [8, 60, 2], [10, 2, 6], [10, 24, 4], [12, 40, 4], [15, 16, 2], [20, 12, 4], [30, 16, 6], [40, 12, 8]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '240.194', 'commutator_count': 1, 'commutator_label': '120.15', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 10958, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['480.970', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 12, 1, 2], [2, 30, 1, 2], [2, 60, 1, 2], [3, 8, 1, 1], [4, 6, 1, 2], [4, 10, 1, 2], [5, 2, 2, 1], [6, 8, 1, 3], [8, 12, 1, 2], [8, 60, 1, 2], [10, 2, 2, 3], [10, 24, 2, 2], [12, 40, 2, 2], [15, 16, 2, 1], [20, 12, 2, 2], [30, 16, 2, 3], [40, 12, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 15120, 'exponent': 120, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '480.1193', 'hash': 10958, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 2, 3, 2, 10], 'inner_gens': [[1, 482, 20, 240, 792], [481, 2, 20, 240, 216], [9, 10, 4, 264, 312], [265, 266, 724, 24, 528], [745, 362, 268, 504, 48]], 'inner_hash': 194, 'inner_nilpotent': False, 'inner_order': 240, 'inner_split': False, 'inner_tex': 'D_5\\times S_4', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 12], [3, 8], [4, 18], [6, 8], [8, 4]], 'label': '960.10958', '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': False, 'name': 'GL(2,3):D10', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 25, 'number_characteristic_subgroups': 21, 'number_conjugacy_classes': 58, 'number_divisions': 38, 'number_normal_subgroups': 41, 'number_subgroup_autclasses': 186, 'number_subgroup_classes': 338, 'number_subgroups': 3806, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 207], [3, 8], [4, 32], [5, 4], [6, 24], [8, 144], [10, 108], [12, 160], [15, 32], [20, 48], [30, 96], [40, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4, 4], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[493, 14, 4, 24, 48], [1, 14, 20, 240, 792], [1, 14, 508, 504, 528], [481, 2, 4, 24, 816]], 'outer_group': '64.206', 'outer_hash': 206, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 12, 'outer_perms': [40682887, 40723343, 135732257, 167087527], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4^2:C_2^2', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 23, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [3, 8], [4, 6], [8, 4], [12, 4], [16, 4]], 'representations': {'PC': {'code': 7279037444920539331405562878677333343172816959333571397551523717553848579, 'gens': [1, 2, 3, 5, 6], 'pres': [8, -2, -2, -2, -3, -2, 2, -2, -5, 7713, 482, 250, 66, 515, 267, 9604, 4812, 2660, 3628, 836, 38021, 5197, 3765, 3341, 1093, 141, 24206, 166, 24591]}, 'Perm': {'d': 23, 'gens': [1287511036623805634046, 219831968956052678401, 1, 3412873460737802880, 966, 2511775413347690568960, 3695978114415007182720, 4720559899960086376320]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\GL(2,3):D_{10}', 'transitive_degree': 160, 'wreath_data': None, 'wreath_product': False}
