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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1568.264', 'ambient_counter': 264, 'ambient_order': 1568, 'ambient_tex': 'C_{14}^2.D_4', 'central': False, 'central_factor': False, 'centralizer_order': 784, 'characteristic': True, 'core_order': 196, 'counter': 15, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1568.264.8.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '8.b1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '8.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times C_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '196.7', 'subgroup_hash': 7, 'subgroup_order': 196, 'subgroup_tex': 'C_7\\times C_{28}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1568.264', 'aut_centralizer_order': 224, 'aut_label': '8.b1', 'aut_quo_index': 2, 'aut_stab_index': 1, 'aut_weyl_group': '72.50', 'aut_weyl_index': 224, 'centralizer': '2.c1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '4.b1.a1', '4.b1.b1'], 'contains': ['16.c1.a1', '56.c1.a1', '56.d1.a1', '56.j1.a1', '56.j1.b1', '56.j1.c1'], 'core': '8.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5795, 4472, 4264, 4578, 4772, 5846, 5453, 7518], 'generators': [812, 224, 56, 16], 'label': '1568.264.8.b1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '8.b1.a1', 'normal_contained_in': ['4.a1.a1', '4.b1.b1', '4.b1.a1'], 'normal_contains': ['16.c1.a1', '56.c1.a1', '56.d1.a1'], 'normalizer': '1.a1.a1', 'old_label': '8.b1.a1', 'projective_image': '112.13', 'quotient_action_image': '2.1', 'quotient_action_kernel': '4.1', 'quotient_action_kernel_order': 4, 'quotient_fusion': None, 'short_label': '8.b1.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': '196.7', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 336, 'aut_gen_orders': [6, 4, 6, 2, 6], 'aut_gens': [[1, 7], [3, 119], [115, 177], [2, 161], [6, 91], [5, 175]], 'aut_group': '4032.a', 'aut_hash': 8118585634924467994, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 4032, 'aut_permdeg': 21, 'aut_perms': [10374345680184501888, 20324222981930211840, 31, 10374345680184501840, 2689754449148132401], 'aut_phi_ratio': 48.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [7, 1, 48, 1], [14, 1, 48, 1], [28, 1, 96, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times \\GL(2,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 336, 'autcent_group': '4032.a', 'autcent_hash': 8118585634924467994, 'autcent_nilpotent': False, 'autcent_order': 4032, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2\\times \\GL(2,7)', '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], [2, 1, 1], [4, 1, 2], [7, 1, 48], [14, 1, 48], [28, 1, 96]], 'center_label': '196.7', 'center_order': 196, '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': ['2.1', '2.1', '7.1', '7.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 7, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['7.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [7, 1, 6, 8], [14, 1, 6, 8], [28, 1, 12, 8]], 'element_repr_type': 'PC', 'elementary': 7, 'eulerian_function': 6, 'exponent': 28, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '98.5', 'hash': 7, 'hyperelementary': 7, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 7], [1, 7]], '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, 196]], 'label': '196.7', 'linC_count': 12096, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 112, 'linQ_dim': 14, 'linQ_dim_count': 112, 'linR_count': 3024, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C7*C28', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 196, 'number_divisions': 27, 'number_normal_subgroups': 30, 'number_subgroup_autclasses': 9, 'number_subgroup_classes': 30, 'number_subgroups': 30, 'old_label': None, 'order': 196, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [4, 2], [7, 48], [14, 48], [28, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 336, 'outer_gen_orders': [6, 4, 6, 2, 6], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[3, 119], [115, 177], [2, 161], [6, 91], [5, 175]], 'outer_group': '4032.a', 'outer_hash': 8118585634924467994, 'outer_nilpotent': False, 'outer_order': 4032, 'outer_permdeg': 21, 'outer_perms': [10374345680184501888, 20324222981930211840, 31, 10374345680184501840, 2689754449148132401], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times \\GL(2,7)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [4, 7, 7], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [6, 16], [12, 8]], 'representations': {'PC': {'code': 134709917, 'gens': [1, 2], 'pres': [4, -7, -2, -2, -7, 21, 34]}, 'GLFp': {'d': 2, 'p': 29, 'gens': [365837, 390225]}, 'Perm': {'d': 18, 'gens': [1108907864064000, 37362124800, 4320, 356995102464000]}}, 'schur_multiplier': [7], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [7, 28], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_7\\times C_{28}', 'transitive_degree': 196, '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': '56.8', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [42, 12, 12, 6, 2, 12, 6, 6], 'aut_gens': [[1, 2, 112], [809, 786, 336], [53, 850, 1456], [109, 90, 560], [97, 74, 1456], [1, 810, 112], [869, 794, 1456], [881, 78, 112], [785, 794, 336]], 'aut_group': None, 'aut_hash': 98983681409546200, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 16128, 'aut_permdeg': 70, 'aut_perms': [9470335266968974959858229052594742014287953015707225639854108835902284471511260682514450464316311614, 11285920279961906508370524444251226854881724245470611682829568758526502291655388438804093640582620730, 9767089154765091039236262632281059547004794059035677520978453544604352215055801368021630859019211673, 7733080900456212022049996569882133908306239780381329156560680621265924426057839583584967013311981413, 449785614039509010741697526557666197322405678688029060010571216460616612202471092932597053072695, 3863287798692865399981716577008302399839085287720992920677844064426335483316235541562384293036146958, 8738232526328008770455822939913005859128605213224768930932110290730122595825424809747280306262178074, 1688059237742208558747289271777894736201423153881962800821738796671430050280855703852950009873176153], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 2], [4, 28, 2, 2], [7, 1, 6, 1], [7, 2, 3, 1], [7, 2, 18, 1], [8, 2, 4, 1], [14, 1, 6, 3], [14, 2, 3, 3], [14, 2, 18, 3], [28, 2, 6, 4], [28, 2, 36, 2], [28, 28, 12, 2], [56, 2, 24, 2], [56, 2, 144, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_6\\times C_7:C_3).C_2^6.C_2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '96.231', 'autcent_hash': 231, 'autcent_nilpotent': True, 'autcent_order': 96, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4\\times C_6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '168.47', 'autcentquo_hash': 47, 'autcentquo_nilpotent': False, 'autcentquo_order': 168, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 2], [4, 28, 4], [7, 1, 6], [7, 2, 21], [8, 2, 4], [14, 1, 18], [14, 2, 63], [28, 2, 96], [28, 28, 24], [56, 2, 192]], 'center_label': '28.4', 'center_order': 28, 'central_product': True, 'central_quotient': '56.5', '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', '7.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 264, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['224.22', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 2], [4, 28, 1, 2], [4, 28, 2, 1], [7, 1, 6, 1], [7, 2, 3, 1], [7, 2, 6, 3], [8, 2, 4, 1], [14, 1, 6, 3], [14, 2, 3, 3], [14, 2, 6, 9], [28, 2, 6, 4], [28, 2, 12, 6], [28, 28, 6, 2], [28, 28, 12, 1], [56, 2, 24, 8]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 48, 'exponent': 56, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '196.10', 'hash': 264, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 28, 'inner_gen_orders': [2, 28, 1], 'inner_gens': [[1, 838, 112], [845, 2, 112], [1, 2, 112]], 'inner_hash': 5, 'inner_nilpotent': False, 'inner_order': 56, 'inner_split': True, 'inner_tex': 'D_{28}', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 56], [2, 378]], 'label': '1568.264', 'linC_count': 4608, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 12, 'linQ_dim': 16, 'linQ_dim_count': 6, 'linR_count': 1296, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C14^2.D4', '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': 32, 'number_characteristic_subgroups': 42, 'number_conjugacy_classes': 434, 'number_divisions': 51, 'number_normal_subgroups': 46, 'number_subgroup_autclasses': 86, 'number_subgroup_classes': 117, 'number_subgroups': 414, 'old_label': None, 'order': 1568, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 3], [4, 116], [7, 48], [8, 8], [14, 144], [28, 864], [56, 384]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 6, 6], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 58, 112], [785, 2, 112], [1, 870, 112], [1, 786, 1232], [1, 790, 1456]], 'outer_group': '288.1045', 'outer_hash': 1045, 'outer_nilpotent': True, 'outer_order': 288, 'outer_permdeg': 16, 'outer_perms': [6227020800, 1307674368000, 5040, 39916944, 362883], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_6^2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 7], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 2], [6, 6], [12, 19], [24, 10], [48, 6]], 'representations': {'PC': {'code': 96176250732955190976096639203466690782009, 'gens': [1, 2, 6], 'pres': [7, -2, -2, -2, -2, -7, -2, -7, 392, 11733, 36, 2270, 58, 2915, 80, 3364, 124]}, 'GLZN': {'d': 2, 'p': 87, 'gens': [37976253, 46319827, 10536064, 491550, 658519, 18438112, 38851736]}, 'Perm': {'d': 26, 'gens': [15568541872220315641471320, 80288049, 32991990780808352400307200, 128193976, 128193960, 175795200, 49151793490183855357286400]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 28], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{14}^2.D_4', 'transitive_degree': 224, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 3], [5, 3]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [1, 22], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.3', 'autcent_hash': 3, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_4', '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], [2, 1, 3], [4, 1, 4]], 'center_label': '8.2', 'center_order': 8, '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': ['2.1', '2.1', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 8]], 'label': '8.2', 'linC_count': 12, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 4, 'linQ_dim': 3, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C4', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 8, 'number_divisions': 6, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 4]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 3], [5, 3]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [1, 22], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2]], 'representations': {'PC': {'code': 34, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16426]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [251, 504]}, 'Perm': {'d': 6, 'gens': [22, 120, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_4', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}
