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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1344.9827', 'ambient_counter': 9827, 'ambient_order': 1344, 'ambient_tex': 'C_{84}.C_2^4', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 112, 'counter': 31, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1344.9827.6.f1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '6.f1', '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': 6, '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': '224.185', 'subgroup_hash': 185, 'subgroup_order': 224, 'subgroup_tex': 'D_4:D_{14}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.9827', 'aut_centralizer_order': 4, 'aut_label': '6.f1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '8064.dl', 'aut_weyl_index': 12, 'centralizer': '672.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.f1', '3.a1'], 'contains': ['12.c1', '12.d1', '12.e1', '12.i1', '12.o1', '42.f1'], 'core': '12.c1', 'coset_action_label': None, 'count': 3, 'diagramx': [4821, -1, 3524, -1], 'generators': [7, 16, 56, 1008, 4, 672], 'label': '1344.9827.6.f1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.f1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1', 'old_label': '6.f1', 'projective_image': '672.1172', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.f1', 'subgroup_fusion': None, 'weyl_group': '224.176'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 84, 'aut_gen_orders': [2, 6, 2, 2, 28, 2, 2, 2, 3, 2], 'aut_gens': [[1, 2, 4, 112], [57, 86, 60, 168], [168, 170, 92, 141], [1, 58, 4, 112], [141, 142, 52, 168], [141, 214, 60, 168], [112, 2, 4, 1], [57, 58, 60, 168], [57, 58, 4, 112], [197, 31, 4, 57], [1, 58, 4, 168]], 'aut_group': '8064.dl', 'aut_hash': 109975980354635622, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 8064, 'aut_permdeg': 15, 'aut_perms': [40320, 6709651525, 3669120, 479002172, 274000568698, 180583603200, 11652480, 87657292800, 181062604800, 274467916800], 'aut_phi_ratio': 84.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 3, 1], [2, 14, 6, 1], [4, 2, 1, 1], [4, 2, 3, 1], [4, 14, 2, 1], [7, 2, 3, 1], [14, 2, 3, 1], [14, 4, 9, 1], [28, 2, 6, 1], [28, 4, 9, 1]], 'aut_supersolvable': False, 'aut_tex': 'D_4\\times S_4\\times F_7', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '504.162', 'autcentquo_hash': 162, 'autcentquo_nilpotent': False, 'autcentquo_order': 504, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 3], [2, 14, 6], [4, 2, 4], [4, 14, 2], [7, 2, 3], [14, 2, 3], [14, 4, 9], [28, 2, 6], [28, 4, 9]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '112.42', 'commutator_count': 1, 'commutator_label': '14.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '7.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 185, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 14, 1, 6], [4, 2, 1, 4], [4, 14, 1, 2], [7, 2, 3, 1], [14, 2, 3, 1], [14, 4, 3, 3], [28, 2, 6, 1], [28, 4, 3, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 95760, 'exponent': 28, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[4, 1, 6]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '112.42', 'hash': 185, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 2, 14, 2], 'inner_gens': [[1, 2, 4, 168], [1, 2, 108, 112], [1, 10, 4, 112], [57, 2, 4, 112]], 'inner_hash': 42, 'inner_nilpotent': False, 'inner_order': 112, 'inner_split': True, 'inner_tex': 'C_2^2\\times D_{14}', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 4, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 4, 'irrep_stats': [[1, 16], [2, 24], [4, 7]], 'label': '224.185', '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': 'D4:D14', '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': 12, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 47, 'number_divisions': 26, 'number_normal_subgroups': 85, 'number_subgroup_autclasses': 50, 'number_subgroup_classes': 166, 'number_subgroups': 742, 'old_label': None, 'order': 224, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 91], [4, 36], [7, 6], [14, 42], [28, 48]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 6, 6], 'outer_gen_pows': [0, 0, 28], 'outer_gens': [[1, 2, 60, 112], [141, 142, 12, 112], [168, 170, 4, 141]], 'outer_group': '72.48', 'outer_hash': 48, 'outer_nilpotent': False, 'outer_order': 72, 'outer_permdeg': 10, 'outer_perms': [1, 41070, 403921], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6\\times D_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [4, 1], [6, 8], [24, 1]], 'representations': {'PC': {'code': 277389703025223574752030, 'gens': [1, 2, 3, 6], 'pres': [6, 2, 2, 2, 2, 7, 2, 980, 50, 1257, 69, 1450, 6053]}, 'Perm': {'d': 15, 'gens': [88698798847, 1960277760, 199435259520, 264009715200, 369350876160, 973]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4:D_{14}', 'transitive_degree': 56, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 84, 'aut_gen_orders': [6, 12, 6, 6, 6, 12, 12], 'aut_gens': [[1, 2, 8, 112], [49, 674, 1060, 620], [53, 1122, 24, 1292], [709, 230, 696, 560], [677, 1126, 436, 172], [721, 450, 88, 560], [677, 230, 356, 956], [757, 674, 388, 620]], 'aut_group': None, 'aut_hash': 7051851194349206198, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96768, 'aut_permdeg': 46, 'aut_perms': [638881304812013605077721345306219725844008829497805796095, 5289325927106019875531847612675023499509275749736460826320, 2009429917190396401013594020724661842171962447379201682554, 3204460980097971763885437736805394103219192581110713331288, 4771220472604210885281362020301414842338388620174974607176, 3186017724290819919538708921232057025653873746850484902551, 2305314613255401631334919230421731256044154292770749330833], 'aut_phi_ratio': 252.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 3, 1], [2, 28, 1, 1], [2, 84, 3, 1], [3, 2, 1, 1], [4, 2, 1, 1], [4, 2, 3, 1], [4, 28, 3, 1], [4, 84, 1, 1], [6, 2, 1, 1], [6, 4, 3, 1], [6, 28, 2, 1], [7, 2, 3, 1], [8, 6, 2, 1], [8, 12, 3, 1], [12, 4, 1, 1], [12, 4, 3, 1], [12, 28, 6, 1], [14, 2, 3, 1], [14, 4, 9, 1], [21, 4, 3, 1], [28, 2, 6, 1], [28, 4, 9, 1], [42, 4, 3, 1], [42, 8, 9, 1], [56, 6, 12, 1], [56, 12, 18, 1], [84, 4, 6, 1], [84, 8, 9, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_{42}\\times A_4).C_6.C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': None, 'autcentquo_hash': 4137113322461592796, 'autcentquo_nilpotent': False, 'autcentquo_order': 6048, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times S_3^2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 3], [2, 28, 1], [2, 84, 3], [3, 2, 1], [4, 2, 4], [4, 28, 3], [4, 84, 1], [6, 2, 1], [6, 4, 3], [6, 28, 2], [7, 2, 3], [8, 6, 2], [8, 12, 3], [12, 4, 4], [12, 28, 6], [14, 2, 3], [14, 4, 9], [21, 4, 3], [28, 2, 6], [28, 4, 9], [42, 4, 3], [42, 8, 9], [56, 6, 12], [56, 12, 18], [84, 4, 6], [84, 8, 9]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '672.1172', 'commutator_count': 1, 'commutator_label': '84.6', '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': 9827, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 28, 1, 1], [2, 84, 1, 3], [3, 2, 1, 1], [4, 2, 1, 4], [4, 28, 1, 3], [4, 84, 1, 1], [6, 2, 1, 1], [6, 4, 1, 3], [6, 28, 2, 1], [7, 2, 3, 1], [8, 6, 2, 1], [8, 12, 1, 3], [12, 4, 1, 4], [12, 28, 2, 3], [14, 2, 3, 1], [14, 4, 3, 3], [21, 4, 3, 1], [28, 2, 6, 1], [28, 4, 3, 3], [42, 4, 3, 1], [42, 8, 3, 3], [56, 6, 12, 1], [56, 12, 6, 3], [84, 4, 6, 1], [84, 8, 3, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 9959040, 'exponent': 168, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[8, 1, 6]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '336.219', 'hash': 9827, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 4, 14, 6], 'inner_gens': [[1, 6, 776, 112], [677, 2, 8, 560], [689, 2, 8, 784], [1, 898, 680, 112]], 'inner_hash': 1172, 'inner_nilpotent': False, 'inner_order': 672, 'inner_split': True, 'inner_tex': 'D_{42}:C_2^3', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 48, 'irrQ_dim': 48, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 68], [4, 38], [8, 7]], 'label': '1344.9827', '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': 30, 'number_characteristic_subgroups': 32, 'number_conjugacy_classes': 129, 'number_divisions': 56, 'number_normal_subgroups': 152, 'number_subgroup_autclasses': 216, 'number_subgroup_classes': 516, 'number_subgroups': 4772, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 287], [3, 2], [4, 176], [6, 70], [7, 6], [8, 48], [12, 184], [14, 42], [21, 12], [28, 48], [42, 84], [56, 288], [84, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6, 6], 'outer_gen_pows': [0, 336, 0, 0], 'outer_gens': [[1, 2, 8, 560], [1, 674, 344, 112], [1, 678, 24, 112], [1, 6, 404, 172]], 'outer_group': '144.195', 'outer_hash': 195, 'outer_nilpotent': False, 'outer_order': 144, 'outer_permdeg': 12, 'outer_perms': [126, 40284846, 128177569, 168018768], '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], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 4], [6, 8], [8, 2], [12, 12], [48, 2]], 'representations': {'PC': {'code': 8020324940482114554518688159483986295271455365176930873, 'gens': [1, 2, 4, 6], 'pres': [8, -2, -2, -2, -2, -7, -2, -2, -3, 97, 41, 16226, 4050, 24835, 91, 3844, 13453, 4733, 141, 31374, 166, 28687]}, 'Perm': {'d': 26, 'gens': [621574835364392082572401, 358853980172015, 4579695696049, 5994942718867, 7217571701309, 8707980666344, 6758061133824000, 16754357281155028254720000]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 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_{84}.C_2^4', 'transitive_degree': 336, 'wreath_data': None, 'wreath_product': False}
