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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1344.10467', 'ambient_counter': 10467, 'ambient_order': 1344, 'ambient_tex': 'C_{84}.C_2^4', 'central': False, 'central_factor': False, 'centralizer_order': 12, 'characteristic': False, 'core_order': 24, 'counter': 104, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1344.10467.28.e1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '28.e1', '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': 28, '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': '48.46', 'subgroup_hash': 46, 'subgroup_order': 48, 'subgroup_tex': 'C_6\\times Q_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.10467', 'aut_centralizer_order': None, 'aut_label': '28.e1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '112.a1', 'complements': None, 'conjugacy_class_count': 3, 'contained_in': ['4.d1', '14.c1', '14.e1', '14.f1'], 'contains': ['56.a1', '56.i1', '56.n1', '56.q1', '84.e1'], 'core': '56.a1', 'coset_action_label': None, 'count': 21, 'diagramx': None, 'generators': [3, 16, 510, 8, 448], 'label': '1344.10467.28.e1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.d1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '7.a1', 'old_label': '28.e1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '28.e1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '24.15', '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, 2, 3, 2, 2, 2, 2], 'aut_gens': [[1, 2, 4], [1, 13, 43], [25, 2, 28], [1, 2, 20], [1, 39, 19], [1, 27, 4], [1, 2, 29], [1, 2, 28], [1, 26, 28]], 'aut_group': '384.17948', 'aut_hash': 17948, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 384, 'aut_permdeg': 10, 'aut_perms': [381247, 368209, 1, 21870, 1538646, 2044440, 1174446, 1681248], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [3, 1, 2, 1], [4, 2, 6, 1], [6, 1, 2, 1], [6, 1, 4, 1], [12, 2, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4:S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '64.202', 'autcent_hash': 202, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3:D_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 3], [3, 1, 2], [4, 2, 6], [6, 1, 6], [12, 2, 12]], 'center_label': '12.5', 'center_order': 12, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 46, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['8.4', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [4, 2, 1, 6], [6, 1, 2, 3], [12, 2, 2, 6]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 91, 'exponent': 12, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '24.15', 'hash': 46, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [1, 2, 2], 'inner_gens': [[1, 2, 4], [1, 2, 28], [1, 26, 4]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 24], [2, 6]], 'label': '48.46', 'linC_count': 64, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 8, 'linQ_dim': 5, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6*Q8', 'ngens': 5, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 8, 'number_conjugacy_classes': 30, 'number_divisions': 20, 'number_normal_subgroups': 38, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 38, 'number_subgroups': 38, 'old_label': None, 'order': 48, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 12], [6, 6], [12, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 6, 2, 2], 'outer_gen_pows': [0, 4, 0, 0, 0], 'outer_gens': [[25, 2, 4], [1, 14, 4], [1, 38, 10], [1, 3, 5], [1, 3, 4]], 'outer_group': '96.226', 'outer_hash': 226, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 8, 'outer_perms': [16, 127, 847, 16680, 5160], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^2\\times S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [4, 2]], 'representations': {'PC': {'code': 233213796801, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -2, -3, 126, 217, 42, 58]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [141602720900, 706104070413, 706117541838]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [33562260, 15426728, 34603218, 37508822, 5426393]}, 'Perm': {'d': 13, 'gens': [574480080, 1086554160, 24, 4, 1614937680]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6\\times Q_8', 'transitive_degree': 48, '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': '48.52', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 84, 'aut_gen_orders': [28, 12, 28, 6, 6, 12, 6], 'aut_gens': [[1, 2, 4, 32], [1, 2, 788, 931], [9, 673, 196, 859], [1, 682, 212, 32], [25, 689, 412, 563], [9, 26, 396, 736], [9, 10, 772, 352], [9, 674, 988, 1035]], 'aut_group': None, 'aut_hash': 1959763316369158610, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 32256, 'aut_permdeg': 40, 'aut_perms': [653335448885723743559344898499278841702712660175, 514117804819788542748034814288959733122190922378, 224965767413264859174446000122808331272944812164, 312809751291118709904143565973685231185742641727, 371238135270447544310427233223859323510877130061, 514103008047127898046985310834554379331814871154, 799361844092120823947169963985068674124854576327], 'aut_phi_ratio': 84.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 3, 1], [2, 4, 3, 1], [2, 28, 1, 1], [3, 1, 2, 1], [4, 2, 1, 1], [4, 2, 3, 1], [4, 4, 1, 1], [4, 28, 3, 1], [6, 1, 2, 1], [6, 2, 6, 1], [6, 4, 6, 1], [6, 28, 2, 1], [7, 2, 3, 1], [8, 14, 2, 1], [8, 28, 3, 1], [12, 2, 2, 1], [12, 2, 6, 1], [12, 4, 2, 1], [12, 28, 6, 1], [14, 2, 3, 1], [14, 4, 9, 1], [14, 4, 18, 1], [21, 2, 6, 1], [24, 14, 4, 1], [24, 28, 6, 1], [28, 4, 3, 1], [28, 4, 6, 1], [28, 4, 9, 1], [42, 2, 6, 1], [42, 4, 18, 1], [42, 4, 36, 1], [84, 4, 6, 1], [84, 4, 12, 1], [84, 4, 18, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2\\times D_4\\times S_4\\times F_7', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '32.51', 'autcent_hash': 51, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^5', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '1008.906', 'autcentquo_hash': 906, 'autcentquo_nilpotent': False, 'autcentquo_order': 1008, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_6\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 3], [2, 4, 3], [2, 28, 1], [3, 1, 2], [4, 2, 4], [4, 4, 1], [4, 28, 3], [6, 1, 2], [6, 2, 6], [6, 4, 6], [6, 28, 2], [7, 2, 3], [8, 14, 2], [8, 28, 3], [12, 2, 8], [12, 4, 2], [12, 28, 6], [14, 2, 3], [14, 4, 27], [21, 2, 6], [24, 14, 4], [24, 28, 6], [28, 4, 18], [42, 2, 6], [42, 4, 54], [84, 4, 36]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '224.188', 'commutator_count': 1, 'commutator_label': '28.2', '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', '7.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 10467, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['448.1289', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 4, 1, 3], [2, 28, 1, 1], [3, 1, 2, 1], [4, 2, 1, 4], [4, 4, 1, 1], [4, 28, 1, 3], [6, 1, 2, 1], [6, 2, 2, 3], [6, 4, 2, 3], [6, 28, 2, 1], [7, 2, 3, 1], [8, 14, 2, 1], [8, 28, 1, 3], [12, 2, 2, 4], [12, 4, 2, 1], [12, 28, 2, 3], [14, 2, 3, 1], [14, 4, 3, 3], [14, 4, 6, 3], [21, 2, 6, 1], [24, 14, 4, 1], [24, 28, 2, 3], [28, 4, 3, 4], [28, 4, 6, 1], [42, 2, 6, 1], [42, 4, 6, 3], [42, 4, 12, 3], [84, 4, 6, 4], [84, 4, 12, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 30643200, 'exponent': 168, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[8, 0, 6]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '336.225', 'hash': 10467, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 28, 'inner_gen_orders': [2, 2, 4, 14], 'inner_gens': [[1, 2, 12, 48], [1, 2, 12, 32], [25, 26, 4, 416], [17, 2, 964, 32]], 'inner_hash': 188, 'inner_nilpotent': False, 'inner_order': 224, 'inner_split': True, 'inner_tex': 'C_2^3:D_{14}', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 48, 'irrQ_dim': 96, 'irrR_degree': 16, 'irrep_stats': [[1, 48], [2, 156], [4, 6], [8, 9]], 'label': '1344.10467', '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': 'C84.C2^4', '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': 36, 'number_characteristic_subgroups': 40, 'number_conjugacy_classes': 219, 'number_divisions': 68, 'number_normal_subgroups': 214, 'number_subgroup_autclasses': 216, 'number_subgroup_classes': 516, 'number_subgroups': 1960, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 47], [3, 2], [4, 96], [6, 94], [7, 6], [8, 112], [12, 192], [14, 114], [21, 12], [24, 224], [28, 72], [42, 228], [84, 144]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6, 6], 'outer_gen_pows': [0, 3, 0, 0], 'outer_gens': [[1, 2, 20, 32], [1, 2, 20, 35], [1, 2, 20, 608], [17, 682, 4, 283]], 'outer_group': '144.195', 'outer_hash': 195, 'outer_nilpotent': False, 'outer_order': 144, 'outer_permdeg': 12, 'outer_perms': [127, 40284720, 128177694, 168018606], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6^2:C_2^2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 3], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 20], [4, 4], [6, 8], [8, 1], [12, 12], [16, 1], [24, 5], [48, 1]], 'representations': {'PC': {'code': 10029507548987958521519268669758115062430003271927368971, 'gens': [1, 2, 3, 6], 'pres': [8, -2, -2, -2, -2, -2, 2, -3, -7, 128, 290, 154, 66, 771, 395, 91, 2309, 5013, 141, 11670, 222, 18455]}, 'Perm': {'d': 26, 'gens': [15646626220754069735081064, 34879283502353817938611200, 51140911694862552588748800, 67267154475470722777804800, 3, 83246209896508891187155200, 99560785234615042078540800, 444984]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{84}.C_2^4', 'transitive_degree': 336, 'wreath_data': None, 'wreath_product': False}
