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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1814400.a', 'ambient_counter': 1, 'ambient_order': 1814400, 'ambient_tex': 'A_{10}', 'central': False, 'central_factor': False, 'centralizer_order': 3, 'characteristic': False, 'core_order': 1, 'counter': 76, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1814400.a.8400.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '8400.a1.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': 8400, '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': '216.164', 'subgroup_hash': 164, 'subgroup_order': 216, 'subgroup_tex': 'C_3^2:S_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1814400.a', 'aut_centralizer_order': None, 'aut_label': '8400.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '604800.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4200.b1.a1'], 'contains': ['16800.a1.a1', '25200.b1.a1', '25200.c1.a1', '25200.d1.a1', '25200.f1.a1', '33600.b1.a1'], 'core': '1814400.a1.a1', 'coset_action_label': None, 'count': 4200, 'diagramx': None, 'generators': [45866, 80640, 806400, 367920, 408655, 725863], 'label': '1814400.a.8400.a1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4200.b1.a1', 'old_label': '8400.a1.a1', 'projective_image': '1814400.a', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8400.a1.a1', 'subgroup_fusion': None, 'weyl_group': '144.183'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 6, 6, 6], 'aut_gens': [[40456, 325, 45708, 435, 367920, 806400], [40456, 458, 45708, 435, 367920, 806400], [40456, 325, 81075, 348, 806400, 367920], [5612, 325, 45708, 348, 806400, 1174320], [1134572, 325, 448908, 435, 367920, 806400], [40456, 325, 725760, 435, 367920, 806400]], 'aut_group': '864.4000', 'aut_hash': 4000, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 864, 'aut_permdeg': 15, 'aut_perms': [316, 13898309095, 467233805464, 100602351629, 1009612882], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 18, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 8, 3, 1], [3, 8, 6, 1], [4, 18, 1, 1], [6, 3, 2, 1], [6, 6, 1, 1], [6, 6, 2, 1], [6, 18, 2, 1], [12, 18, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_6^2:D_6', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '432.523', 'autcentquo_hash': 523, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_6^2:D_6', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 18, 1], [3, 1, 2], [3, 2, 3], [3, 8, 9], [4, 18, 1], [6, 3, 2], [6, 6, 3], [6, 18, 2], [12, 18, 2]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '72.43', 'commutator_count': 1, 'commutator_label': '36.11', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 164, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['72.43', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 18, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 8, 1, 3], [3, 8, 2, 3], [4, 18, 1, 1], [6, 3, 2, 1], [6, 6, 1, 1], [6, 6, 2, 1], [6, 18, 2, 1], [12, 18, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 5460, 'exponent': 12, 'exponents_of_order': [3, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 0, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '216.164', 'hash': 164, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 1, 3, 3, 2, 2], 'inner_gens': [[40456, 325, 81075, 348, 806400, 367920], [40456, 325, 45708, 435, 367920, 806400], [5612, 325, 45708, 435, 806400, 1174320], [40979, 325, 45708, 435, 367920, 806400], [1134136, 325, 1129308, 435, 367920, 806400], [1134136, 325, 448908, 435, 367920, 806400]], 'inner_hash': 43, 'inner_nilpotent': False, 'inner_order': 72, 'inner_split': True, 'inner_tex': 'C_3:S_4', 'inner_used': [1, 3, 4, 5], 'irrC_degree': 6, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 6], [2, 12], [3, 6], [6, 3]], 'label': '216.164', 'linC_count': 48, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 7, 'linQ_degree_count': 18, 'linQ_dim': 7, 'linQ_dim_count': 18, 'linR_count': 18, 'linR_degree': 7, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^2:S4', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 14, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 27, 'number_divisions': 18, 'number_normal_subgroups': 18, 'number_subgroup_autclasses': 53, 'number_subgroup_classes': 77, 'number_subgroups': 360, 'old_label': None, 'order': 216, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 21], [3, 80], [4, 18], [6, 60], [12, 36]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[40456, 325, 45708, 348, 367920, 806400], [40456, 458, 80640, 348, 806400, 367920]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [6, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 6], [3, 2], [4, 4], [6, 3], [12, 1]], 'representations': {'PC': {'code': 190962431056117192051989, 'gens': [1, 2, 3, 5], 'pres': [6, 2, 3, 2, 3, 2, 3, 49, 1034, 50, 5944, 2440, 88, 5189]}, 'GLFq': {'d': 4, 'q': 4, 'gens': [1087120897, 21503875, 2150008898, 2149582850, 1075037441, 1083180289]}, 'Perm': {'d': 10, 'gens': [40456, 325, 45708, 435, 367920, 806400]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2:S_4', 'transitive_degree': 24, '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': '1.1', 'all_subgroups_known': True, 'almost_simple': True, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 2520, 'aut_gen_orders': [10, 2], 'aut_gens': [[368793, 403200], [1103025, 366], [1628673, 1492560]], 'aut_group': '3628800.a', 'aut_hash': 1946414812177288349, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3628800, 'aut_permdeg': 240, 'aut_perms': [1892267407218635388185410327880875268839406969802759729238779136310400970943464787034286500617824505362418225194227777800429296888992217005781915659299246781491127976562570201812724457916584626931871219342917120808670263373949297976517639220899279561748331491932652769110571130799801171719218049761489598685981114389134582001519825812451059770514425656087240493727764142768547377550686214626854300692552013251454807016162168635447368941706009602409555436474972981996059, 189842308812466110203449554687060587514965674491390571712398971004326430809336558305270156328337290624287063142271206337422043707353101231601965828832798492654399182917686802482581038908727740750301767882217140115679574172057734552173137910728215431819771759896860257873844025171953998376194247626828464371523258011472172018254549718601434716390800084944968078237597180780809580097446200825406931277572793321522862173651024990811808762809530087131753612102438580521140], 'aut_phi_ratio': 8.75, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 630, 1, 1], [2, 4725, 1, 1], [3, 240, 1, 1], [3, 8400, 1, 1], [3, 22400, 1, 1], [4, 18900, 1, 2], [4, 56700, 1, 1], [5, 6048, 1, 1], [5, 72576, 1, 1], [6, 25200, 1, 2], [6, 151200, 1, 1], [7, 86400, 1, 1], [8, 226800, 1, 1], [9, 201600, 2, 1], [10, 90720, 1, 1], [12, 151200, 1, 2], [15, 120960, 1, 1], [21, 86400, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_{10}', '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': 2520, 'autcentquo_group': '3628800.a', 'autcentquo_hash': 1946414812177288349, 'autcentquo_nilpotent': False, 'autcentquo_order': 3628800, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_{10}', 'cc_stats': [[1, 1, 1], [2, 630, 1], [2, 4725, 1], [3, 240, 1], [3, 8400, 1], [3, 22400, 1], [4, 18900, 2], [4, 56700, 1], [5, 6048, 1], [5, 72576, 1], [6, 25200, 2], [6, 151200, 1], [7, 86400, 1], [8, 226800, 1], [9, 201600, 2], [10, 90720, 1], [12, 151200, 2], [15, 120960, 1], [21, 86400, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1814400.a', 'commutator_count': 1, 'commutator_label': '1814400.a', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['1814400.a'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 630, 1, 1], [2, 4725, 1, 1], [3, 240, 1, 1], [3, 8400, 1, 1], [3, 22400, 1, 1], [4, 18900, 1, 2], [4, 56700, 1, 1], [5, 6048, 1, 1], [5, 72576, 1, 1], [6, 25200, 1, 2], [6, 151200, 1, 1], [7, 86400, 1, 1], [8, 226800, 1, 1], [9, 201600, 1, 2], [10, 90720, 1, 1], [12, 151200, 1, 2], [15, 120960, 1, 1], [21, 86400, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 793827, 'exponent': 2520, 'exponents_of_order': [7, 4, 2, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 3, 5, 7], 'faithful_reps': [[9, 1, 1], [35, 1, 1], [36, 1, 1], [42, 1, 1], [75, 1, 1], [84, 1, 1], [90, 1, 1], [126, 1, 1], [160, 1, 1], [210, 1, 1], [224, 1, 2], [225, 1, 1], [252, 1, 1], [288, 1, 1], [300, 1, 1], [315, 1, 1], [350, 1, 1], [384, 1, 2], [450, 1, 1], [525, 1, 1], [567, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '1814400.a', 'hash': 9212489253922166078, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 2520, 'inner_gen_orders': [8, 3], 'inner_gens': [[368793, 1093680], [1175193, 403200]], 'inner_hash': 9212489253922166078, 'inner_nilpotent': False, 'inner_order': 1814400, 'inner_split': True, 'inner_tex': 'A_{10}', 'inner_used': [1, 2], 'irrC_degree': 9, 'irrQ_degree': 9, 'irrQ_dim': 9, 'irrR_degree': 9, 'irrep_stats': [[1, 1], [9, 1], [35, 1], [36, 1], [42, 1], [75, 1], [84, 1], [90, 1], [126, 1], [160, 1], [210, 1], [224, 2], [225, 1], [252, 1], [288, 1], [300, 1], [315, 1], [350, 1], [384, 2], [450, 1], [525, 1], [567, 1]], 'label': '1814400.a', 'linC_count': 1, 'linC_degree': 9, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 9, 'linQ_degree_count': 1, 'linQ_dim': 9, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 9, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'A10', 'ngens': 2, '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, 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, 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': 22, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 24, 'number_divisions': 23, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 423, 'number_subgroup_classes': 430, 'number_subgroups': 6469142, 'old_label': None, 'order': 1814400, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 5355], [3, 31040], [4, 94500], [5, 78624], [6, 201600], [7, 86400], [8, 226800], [9, 403200], [10, 90720], [12, 302400], [15, 120960], [21, 172800]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [552317], 'outer_gens': [[3162370, 2586395]], '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': None, 'perfect': True, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [], 'quasisimple': True, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [9, 1], [35, 1], [36, 1], [42, 1], [75, 1], [84, 1], [90, 1], [126, 1], [160, 1], [210, 1], [224, 2], [225, 1], [252, 1], [288, 1], [300, 1], [315, 1], [350, 1], [450, 1], [525, 1], [567, 1], [768, 1]], 'representations': {'Perm': {'d': 10, 'gens': [368793, 403200]}}, 'schur_multiplier': [2], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'A_{10}', 'transitive_degree': 10, 'wreath_data': None, 'wreath_product': False}
