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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '3888.ek', 'ambient_counter': 115, 'ambient_order': 3888, 'ambient_tex': 'C_3^3:S_3^2:C_2^2', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 3, 'counter': 33, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '3888.ek.18.E', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '18.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': 18, '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': '216.102', 'subgroup_hash': 102, 'subgroup_order': 216, 'subgroup_tex': 'C_6.S_3^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '3888.ek', 'aut_centralizer_order': 2, 'aut_label': '18.E', 'aut_quo_index': None, 'aut_stab_index': 36, 'aut_weyl_group': '216.87', 'aut_weyl_index': 72, 'centralizer': '1944.A', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['6.a1'], 'contains': ['36.A', '36.K', '36.L', '36.N', '36.R', '54.A'], 'core': '1296.A', 'coset_action_label': None, 'count': 36, 'diagramx': [8841, -1, 6943, -1], 'generators': [1, 1296, 432, 2996, 504, 16], 'label': '3888.ek.18.E', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.A', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '18.E', 'old_label': '18.e1', 'projective_image': '3888.ek', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '18.E', 'subgroup_fusion': None, 'weyl_group': '108.17'}
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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': 4, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 6, 6, 6], 'aut_gens': [[7426, 8156, 8101, 13286, 15473, 13933], [12475, 16120, 7372, 13286, 13313, 8102], [7399, 8129, 8101, 13286, 17660, 13933], [8155, 7373, 7372, 13286, 18389, 14662], [6697, 6698, 8101, 13286, 15473, 9641], [7426, 13312, 8101, 13286, 15473, 6725]], 'aut_group': '864.2282', 'aut_hash': 2282, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 864, 'aut_permdeg': 22, 'aut_perms': [542917183727660355496, 7426804536364027687, 321277900978132279201, 462325431429391874401, 115245952960124049967], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 2, 1], [2, 9, 4, 1], [3, 2, 1, 1], [3, 6, 2, 1], [3, 12, 1, 1], [6, 2, 1, 1], [6, 6, 2, 1], [6, 12, 1, 1], [6, 18, 2, 1], [6, 18, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_2^2\\times \\He_3):D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '216.87', 'autcentquo_hash': 87, 'autcentquo_nilpotent': False, 'autcentquo_order': 216, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\He_3:D_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 9, 6], [3, 2, 1], [3, 6, 2], [3, 12, 1], [6, 2, 1], [6, 6, 2], [6, 12, 1], [6, 18, 6]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '108.17', 'commutator_count': 1, 'commutator_label': '27.3', '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': 102, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['108.17', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 6], [3, 2, 1, 1], [3, 6, 1, 2], [3, 12, 1, 1], [6, 2, 1, 1], [6, 6, 1, 2], [6, 12, 1, 1], [6, 18, 1, 6]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 3024, 'exponent': 6, 'exponents_of_order': [3, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 1, 2]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '72.46', 'hash': 102, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 2, 3, 1, 3, 2], 'inner_gens': [[7426, 7373, 8101, 13286, 16931, 14662], [8128, 8156, 7372, 13286, 16931, 13933], [7426, 6698, 8101, 13286, 15473, 13204], [7426, 8156, 8101, 13286, 15473, 13933], [6697, 7427, 8101, 13286, 15473, 18307], [8155, 8156, 7372, 13286, 17660, 13933]], 'inner_hash': 17, 'inner_nilpotent': False, 'inner_order': 108, 'inner_split': True, 'inner_tex': 'C_3^2:D_6', 'inner_used': [1, 2, 3, 5, 6], 'irrC_degree': 6, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 6, 'irrep_stats': [[1, 8], [2, 8], [4, 2], [6, 4]], 'label': '216.102', 'linC_count': 2, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 2, 'linQ_dim': 6, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C6.S3^2', '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': 12, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 22, 'number_divisions': 22, 'number_normal_subgroups': 30, 'number_subgroup_autclasses': 52, 'number_subgroup_classes': 122, 'number_subgroups': 682, 'old_label': None, 'order': 216, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 55], [3, 26], [6, 134]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [6643, 6643], 'outer_gens': [[7399, 14068, 8101, 13286, 17660, 13933], [11746, 16120, 8101, 13286, 13340, 14014]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [7, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [4, 2], [6, 4]], 'representations': {'PC': {'code': 69006255441088720861292059789, 'gens': [1, 2, 4, 5], 'pres': [6, -2, -2, -3, -3, -2, -3, 121, 31, 146, 1161, 519, 5404, 2710, 88, 5189, 2603]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [24970022465723971, 74634695064663880, 69485716379617366]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [7426, 8156, 8101, 13286, 15473, 13933]}, 'Perm': {'d': 11, 'gens': [167887, 889560, 1, 4478784, 1150590, 8891526]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6.S_3^2', 'transitive_degree': 18, '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': 4, 'aut_exponent': 24, 'aut_gen_orders': [6, 4, 8, 4, 8], 'aut_gens': [[1, 2, 8, 48, 144, 432, 1296], [497, 646, 88, 3504, 1440, 3456, 1296], [481, 722, 3484, 304, 2160, 2736, 2592], [2179, 1346, 3224, 3024, 1456, 912, 2592], [3779, 3334, 1496, 3072, 1616, 3504, 2592], [1255, 3730, 232, 960, 304, 96, 1296]], 'aut_group': None, 'aut_hash': 641057888679208724, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 15552, 'aut_permdeg': 90, 'aut_perms': [687745654424180225897910730915222618616214195503785820560680125645002984165656216504447309810632755999276932650242387620641451770274257020, 1339197206830937299567864408558339193403082176271238153088509134573405703799158190796376926876536316457114698540538504817326306260781277261, 1169775886111392012462602504979559283938680000869746091329354507656136739113068484934972244277436747545780506891196963412440012417820658330, 1437578482329493225675724344266581534011653277773776211027843316046017893642540045816779289444252138542199912623531193826083909782600791843, 1443998081487990826866780060549434144895175052876218835063155790155277059408631824177080095573183697189132373337918485096184772797548533417], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 18, 2, 1], [2, 27, 2, 1], [2, 54, 2, 1], [2, 81, 1, 1], [3, 2, 1, 1], [3, 12, 4, 1], [3, 24, 2, 2], [3, 48, 2, 1], [4, 162, 1, 1], [4, 486, 1, 1], [6, 36, 2, 1], [6, 108, 4, 3], [6, 162, 1, 1], [6, 216, 2, 2], [12, 324, 1, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^3:C_3^2.C_4.C_2^4', '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': 24, 'autcentquo_group': None, 'autcentquo_hash': 641057888679208724, 'autcentquo_nilpotent': False, 'autcentquo_order': 15552, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^3:C_3^2.D_4.C_2^3', 'cc_stats': [[1, 1, 1], [2, 18, 2], [2, 27, 2], [2, 54, 2], [2, 81, 1], [3, 2, 1], [3, 12, 4], [3, 24, 4], [3, 48, 2], [4, 162, 1], [4, 486, 1], [6, 36, 2], [6, 108, 12], [6, 162, 1], [6, 216, 4], [12, 324, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '3888.ek', 'commutator_count': 1, 'commutator_label': '486.249', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 115, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 18, 1, 2], [2, 27, 1, 2], [2, 54, 1, 2], [2, 81, 1, 1], [3, 2, 1, 1], [3, 12, 1, 4], [3, 24, 1, 4], [3, 48, 1, 2], [4, 162, 1, 1], [4, 486, 1, 1], [6, 36, 1, 2], [6, 108, 1, 12], [6, 162, 1, 1], [6, 216, 1, 4], [12, 324, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1209600, 'exponent': 12, 'exponents_of_order': [5, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[18, 1, 4], [36, 1, 1]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '1296.3531', 'hash': 3272242732043843418, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 12, 6, 3, 3, 3, 3], 'inner_gens': [[1, 2998, 3592, 528, 2736, 432, 2592], [2997, 2, 1896, 3120, 2608, 480, 1296], [1793, 2482, 8, 1392, 144, 3456, 2592], [961, 2258, 2696, 48, 2736, 432, 1296], [1297, 1474, 8, 1344, 144, 432, 1296], [1, 98, 2168, 48, 144, 432, 1296], [2593, 2, 2600, 48, 144, 432, 1296]], 'inner_hash': 3272242732043843418, 'inner_nilpotent': False, 'inner_order': 3888, 'inner_split': False, 'inner_tex': 'C_3^3:S_3^2:C_2^2', 'inner_used': [1, 2, 3], 'irrC_degree': 18, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 18, 'irrep_stats': [[1, 8], [2, 2], [4, 16], [8, 8], [16, 2], [18, 4], [36, 1]], 'label': '3888.ek', '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': None, 'name': 'C3^3:S3^2:C2^2', 'ngens': 9, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 20, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 41, 'number_divisions': 41, 'number_normal_subgroups': 25, 'number_subgroup_autclasses': 281, 'number_subgroup_classes': 702, 'number_subgroups': 25451, 'old_label': None, 'order': 3888, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 279], [3, 242], [4, 648], [6, 2394], [12, 324]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [982, 0], 'outer_gens': [[2307, 1810, 248, 1728, 1456, 2208, 2592], [1, 2, 2460, 1472, 3456, 2880, 1296]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 7, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 16], [8, 8], [16, 2], [18, 4], [36, 1]], 'representations': {'PC': {'code': '9974425099924765025408242293674641170267230417236872111368710084358760681946798562664221108795633689304095', 'gens': [1, 2, 4, 6, 7, 8, 9], 'pres': [9, -2, -2, -2, -2, -3, 3, 3, 3, -3, 53965, 46, 13934, 8768, 129315, 34140, 8589, 102, 1444, 6493, 14962, 28517, 84254, 1319, 9428, 4577, 172374, 82167, 45384, 3642, 17296, 62233, 31138, 13651, 209960, 26279]}, 'Perm': {'d': 27, 'gens': [7549724245233350948573836948, 10888626160339536217045552585, 403587579360117356426414155]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3:S_3^2:C_2^2', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}
