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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '8748.fn', 'ambient_counter': 144, 'ambient_order': 8748, 'ambient_tex': 'C_3^5.S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 9, 'characteristic': False, 'core_order': 1, 'counter': 659, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '8748.fn.486.cf1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '486.cf1.a1', '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': 486, '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': '18.3', 'subgroup_hash': 3, 'subgroup_order': 18, 'subgroup_tex': 'C_3\\times S_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '8748.fn', 'aut_centralizer_order': None, 'aut_label': '486.cf1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '972.o1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['162.cx1.a1', '162.cz1.a1', '162.dd1.a1', '162.dd1.b1', '162.dd1.c1', '243.m1.a1'], 'contains': ['972.m1.a1', '1458.s1.a1', '1458.w1.a1'], 'core': '8748.a1.a1', 'coset_action_label': None, 'count': 81, 'diagramx': [6843, -1, 5047, -1, 6920, -1, 7368, -1], 'generators': [54, 36, 6804], 'label': '8748.fn.486.cf1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '18.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '81.j1.a1', 'old_label': '486.cf1.a1', 'projective_image': '8748.fn', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '486.cf1.a1', 'subgroup_fusion': None, 'weyl_group': '12.4'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 6], [5, 12], [17, 6]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 49], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [6, 3, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', '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, 3, 1], [3, 1, 2], [3, 2, 3], [6, 3, 2]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [6, 3, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '18.3', 'hash': 3, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 12], [13, 6]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 6], [2, 3]], 'label': '18.3', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 3, 'linQ_dim': 4, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*S3', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 9, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 9, 'number_subgroup_classes': 9, 'number_subgroups': 14, 'old_label': None, 'order': 18, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [3, 8], [6, 6]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[5, 6]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 1]], 'representations': {'PC': {'code': 5451, 'gens': [1, 3], 'pres': [3, -2, -3, -3, 6, 110]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [11780110, 20974441]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [20, 138], 'family': 'GU'}, {'d': 2, 'q': 4, 'gens': [20, 130, 194], 'family': 'COPlus'}, {'d': 2, 'q': 2, 'gens': [20, 130, 138], 'family': 'CU'}, {'d': 1, 'q': 9, 'gens': [93882, 62619, 1930], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 7, 'gens': [56, 687, 1374]}, 'Perm': {'d': 6, 'gens': [450, 147, 243]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times S_3', 'transitive_degree': 6, 'wreath_data': ['C_3', 'C_2', '2T1'], 'wreath_product': True}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 18, 'aut_gen_orders': [6, 6, 18], 'aut_gens': [[1, 6, 108, 324, 972, 2916], [8077, 3018, 5724, 4212, 7776, 2916], [8521, 1506, 3456, 2268, 4860, 2916], [8437, 3750, 4752, 2592, 7776, 5832]], 'aut_group': None, 'aut_hash': 9126939498597462977, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 26244, 'aut_permdeg': 171, 'aut_perms': [1104320998215467977570185323270928080580535360826614524953920166875093694180881644379612759680948515400645314737426664496478481619927166595480354906801246479049065461345569130248362636917540003470694328770945216099627414194815088890334007197782286251780252245170705517764375014388852308492391586469332553209887, 228374919994871278305545603316442923439909463468474179080261077730912160863485911201229509589673679362775003461231780277407640353369491413129972810167178923001921723280923036706533070403561468367219856665774487650918488998249487281328427632645903267096270706024639262342232755743914245957613277219856520243826, 485049340614363875601141125325824460515491508223994736980911459804316869479016153810894928183165558411611890013964290636709861152006662769798072233592192586436011502352982073860903843860155622969342968295281177545605197484989297023913892960895775329004015335986378703111709961033493257559529847170373336335094], 'aut_phi_ratio': 9.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 81, 1, 3], [3, 2, 1, 1], [3, 6, 1, 2], [3, 6, 3, 1], [3, 9, 1, 2], [3, 12, 1, 1], [3, 18, 1, 2], [3, 18, 3, 2], [3, 36, 1, 1], [3, 54, 1, 3], [3, 108, 1, 3], [6, 81, 1, 2], [6, 162, 1, 5], [6, 162, 3, 3], [6, 243, 1, 4], [6, 486, 1, 3], [9, 162, 1, 3], [9, 324, 1, 3], [18, 486, 1, 3]], 'aut_supersolvable': True, 'aut_tex': 'C_3^4.C_3^4.C_2^2', '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': 18, 'autcentquo_group': None, 'autcentquo_hash': 9126939498597462977, 'autcentquo_nilpotent': False, 'autcentquo_order': 26244, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^4.C_3^4.C_2^2', 'cc_stats': [[1, 1, 1], [2, 81, 3], [3, 2, 1], [3, 6, 5], [3, 9, 2], [3, 12, 1], [3, 18, 8], [3, 36, 1], [3, 54, 3], [3, 108, 3], [6, 81, 2], [6, 162, 14], [6, 243, 4], [6, 486, 3], [9, 162, 3], [9, 324, 3], [18, 486, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '8748.fn', 'commutator_count': 1, 'commutator_label': '729.65', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 144, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 81, 1, 3], [3, 2, 1, 1], [3, 6, 1, 5], [3, 9, 2, 1], [3, 12, 1, 1], [3, 18, 2, 4], [3, 36, 1, 1], [3, 54, 1, 1], [3, 54, 2, 1], [3, 108, 1, 1], [3, 108, 2, 1], [6, 81, 2, 1], [6, 162, 1, 6], [6, 162, 2, 4], [6, 243, 2, 2], [6, 486, 1, 1], [6, 486, 2, 1], [9, 162, 1, 1], [9, 162, 2, 1], [9, 324, 1, 1], [9, 324, 2, 1], [18, 486, 1, 1], [18, 486, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 432, 'exponent': 18, 'exponents_of_order': [7, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[18, 0, 12], [18, 1, 6]], 'familial': False, 'frattini_label': '81.15', 'frattini_quotient': '108.38', 'hash': 6834645965991007480, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [6, 18, 3, 3, 3, 3], 'inner_gens': [[1, 3054, 4104, 324, 1944, 2916], [4513, 6, 7668, 5508, 7776, 5832], [7993, 5406, 108, 324, 972, 2916], [1, 7458, 108, 324, 972, 2916], [1945, 4866, 108, 324, 972, 2916], [1, 5838, 108, 324, 972, 2916]], 'inner_hash': 6834645965991007480, 'inner_nilpotent': False, 'inner_order': 8748, 'inner_split': True, 'inner_tex': 'C_3^5.S_3^2', 'inner_used': [1, 2], 'irrC_degree': 18, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 18, 'irrep_stats': [[1, 12], [2, 12], [4, 3], [6, 8], [12, 4], [18, 20], [36, 1]], 'label': '8748.fn', 'linC_count': 18, 'linC_degree': 18, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 6, 'linQ_dim': 18, 'linQ_dim_count': 6, 'linR_count': 6, 'linR_degree': 18, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^5.S3^2', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 48, 'number_characteristic_subgroups': 29, 'number_conjugacy_classes': 60, 'number_divisions': 42, 'number_normal_subgroups': 29, 'number_subgroup_autclasses': 577, 'number_subgroup_classes': 837, 'number_subgroups': 36532, 'old_label': None, 'order': 8748, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 243], [3, 728], [6, 4860], [9, 1458], [18, 1458]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 3, 'outer_gen_orders': [3], 'outer_gen_pows': [0], 'outer_gens': [[4861, 6, 1080, 3240, 972, 2916]], 'outer_group': '3.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 3, 'outer_permdeg': 3, 'outer_perms': [3], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 5], [6, 4], [8, 1], [12, 4], [18, 8], [24, 1], [36, 7]], 'representations': {'PC': {'code': '1621912307305717130572241985556822339052337438322098832930009293315936869075625237551807', 'gens': [1, 3, 6, 7, 8, 9], 'pres': [9, -2, -3, -2, -3, -3, -3, -3, -3, -3, 18, 82460, 44318, 74, 258771, 89004, 138, 134464, 221621, 29174, 69035, 12182, 8951, 57858, 42558, 139975, 93337, 40858, 78758]}, 'Perm': {'d': 27, 'gens': [8824017166890336963030130106, 8808423751645320656558839414]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^5.S_3^2', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}
