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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1344.7781', 'ambient_counter': 7781, 'ambient_order': 1344, 'ambient_tex': '(C_2\\times D_{14}):D_{12}', 'central': False, 'central_factor': False, 'centralizer_order': 48, 'characteristic': False, 'core_order': 1, 'counter': 667, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1344.7781.336.s1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '336.s1.b1', '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': 336, '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': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.7781', 'aut_centralizer_order': None, 'aut_label': '336.s1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '28.f1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['48.s1.b1', '112.s1.b1', '168.c1.a1', '168.c1.b1', '168.s1.b1'], 'contains': ['672.c1.a1', '672.e1.b1', '672.f1.b1'], 'core': '1344.a1.a1', 'coset_action_label': None, 'count': 28, 'diagramx': None, 'generators': [673, 2], 'label': '1344.7781.336.s1.b1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '24.f1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '28.f1.a1', 'old_label': '336.s1.b1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '336.s1.b1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', '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]], 'center_label': '4.2', 'center_order': 4, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', '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, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, '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': 2, 'aut_exponent': 84, 'aut_gen_orders': [12, 12, 6, 12, 6, 12, 12, 12], 'aut_gens': [[1, 2, 4, 16], [673, 2, 684, 754], [201, 674, 908, 1042], [297, 2, 1132, 208], [681, 682, 1132, 274], [385, 682, 228, 80], [585, 674, 4, 762], [105, 674, 908, 754], [961, 682, 452, 946]], 'aut_group': None, 'aut_hash': 4513743374577309144, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 129024, 'aut_permdeg': 48, 'aut_perms': [6237239441315307764108621998809576080212913719187615074728147, 8804440113147152411046795213086915807625496984147466125540106, 11803555266778060169750106227195151886048212342692526886555591, 9448300190904657327780677015646059588201755150661464375756080, 11798528690180565617454250734451088801282249046288737494014806, 8834354878161477783781806050906152986397646895923548252879115, 4149221764104339750511355329478490170203748011384344403460879, 5065715565392692291845148467148866202015831457835209647940795], 'aut_phi_ratio': 336.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 12, 1, 1], [2, 14, 2, 2], [2, 84, 1, 1], [2, 84, 2, 1], [3, 2, 1, 1], [4, 4, 2, 1], [4, 12, 1, 1], [4, 12, 2, 1], [4, 28, 2, 1], [4, 84, 1, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 4, 2, 1], [6, 14, 4, 1], [6, 28, 2, 1], [7, 2, 3, 1], [12, 4, 4, 1], [12, 28, 4, 1], [14, 2, 3, 1], [14, 2, 6, 1], [14, 4, 6, 1], [14, 24, 3, 1], [21, 4, 3, 1], [28, 8, 6, 1], [28, 12, 12, 1], [28, 24, 3, 1], [42, 4, 3, 1], [42, 4, 6, 1], [42, 8, 6, 1], [84, 8, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{42}.(C_2^5\\times C_6).C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '256.56092', 'autcent_hash': 56092, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8', '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, 3], [2, 2, 2], [2, 12, 1], [2, 14, 4], [2, 84, 3], [3, 2, 1], [4, 4, 2], [4, 12, 3], [4, 28, 2], [4, 84, 1], [6, 2, 3], [6, 4, 2], [6, 14, 4], [6, 28, 2], [7, 2, 3], [12, 4, 4], [12, 28, 4], [14, 2, 9], [14, 4, 6], [14, 24, 3], [21, 4, 3], [28, 8, 6], [28, 12, 12], [28, 24, 3], [42, 4, 9], [42, 8, 6], [84, 8, 12]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '336.219', 'commutator_count': 1, 'commutator_label': '84.15', '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': 7781, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 12, 1, 1], [2, 14, 1, 4], [2, 84, 1, 3], [3, 2, 1, 1], [4, 4, 1, 2], [4, 12, 1, 3], [4, 28, 1, 2], [4, 84, 1, 1], [6, 2, 1, 3], [6, 4, 1, 2], [6, 14, 2, 2], [6, 28, 1, 2], [7, 2, 3, 1], [12, 4, 2, 2], [12, 28, 2, 2], [14, 2, 3, 3], [14, 4, 3, 2], [14, 24, 3, 1], [21, 4, 3, 1], [28, 8, 3, 2], [28, 12, 6, 2], [28, 24, 3, 1], [42, 4, 3, 3], [42, 8, 3, 2], [84, 8, 6, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 7469280, 'exponent': 84, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '336.219', 'hash': 7781, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [2, 2, 2, 42], 'inner_gens': [[1, 2, 4, 216], [1, 2, 12, 24], [1, 10, 4, 1136], [1161, 10, 228, 16]], 'inner_hash': 219, 'inner_nilpotent': False, 'inner_order': 336, 'inner_split': True, 'inner_tex': 'D_6\\times D_{14}', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 44], [4, 48], [8, 6]], 'label': '1344.7781', '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': '(C2*D14):D12', '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': 35, 'number_characteristic_subgroups': 54, 'number_conjugacy_classes': 114, 'number_divisions': 56, 'number_normal_subgroups': 144, 'number_subgroup_autclasses': 384, 'number_subgroup_classes': 692, 'number_subgroups': 7652, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 327], [3, 2], [4, 184], [6, 126], [7, 6], [12, 128], [14, 114], [21, 12], [28, 264], [42, 84], [84, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 2, 2, 12], 'outer_gen_pows': [0, 0, 0, 0, 0, 0], 'outer_gens': [[673, 2, 4, 16], [1, 2, 676, 464], [1, 674, 4, 16], [9, 2, 12, 16], [9, 2, 4, 16], [1, 674, 4, 754]], 'outer_group': '384.17286', 'outer_hash': 17286, 'outer_nilpotent': True, 'outer_order': 384, 'outer_permdeg': 15, 'outer_perms': [270160934400, 180747302400, 19200343944, 636677354880, 288848891520, 127095092067], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^6:C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 22, '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, 6], [6, 8], [8, 2], [12, 6], [24, 6]], 'representations': {'PC': {'code': 18069263251470896178270622947116621514286602073144882187, 'gens': [1, 2, 3, 5], 'pres': [8, -2, -2, -2, -2, 2, -2, -3, -7, 154, 66, 8644, 492, 11380, 116, 19973, 11157, 141, 46598, 7190, 222, 73735]}, 'Perm': {'d': 22, 'gens': [2439325304713437960, 6268888921, 94434607, 134352137, 7, 178981920, 93405312000, 55969571858264064000]}}, '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_2\\times D_{14}):D_{12}', 'transitive_degree': 336, 'wreath_data': None, 'wreath_product': False}
