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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '19440.bb', 'ambient_counter': 28, 'ambient_order': 19440, 'ambient_tex': 'C_3^2:C_6\\times A_6', 'central': False, 'central_factor': False, 'centralizer_order': 9, 'characteristic': False, 'core_order': 9, 'counter': 255, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '19440.bb.360.j1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '360.j1.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': 360, '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': '54.12', 'subgroup_hash': 12, 'subgroup_order': 54, 'subgroup_tex': 'S_3\\times C_3^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '19440.bb', 'aut_centralizer_order': None, 'aut_label': '360.j1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '2160.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['36.b1.b1', '90.o1.b1', '120.e1.b1', '120.f1.a1', '180.b1.b1'], 'contains': ['720.b1.b1', '1080.i1.a1', '1080.l1.b1', '1080.m1.b1'], 'core': '2160.a1.a1', 'coset_action_label': None, 'count': 60, 'diagramx': [7785, -1, 6701, -1, 1858, -1, 5384, -1], 'generators': [45600, 80884, 87261834384, 43626243], 'label': '19440.bb.360.j1.b1', 'mobius_quo': None, 'mobius_sub': 6, 'normal_closure': '6.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '60.b1.b1', 'old_label': '360.j1.b1', 'projective_image': '19440.bb', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '360.j1.b1', 'subgroup_fusion': None, 'weyl_group': '36.10'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '18.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [2, 6, 4, 12], 'aut_gens': [[1, 3, 18], [2, 4, 18], [12, 17, 36], [14, 4, 18], [6, 46, 18]], 'aut_group': '288.851', 'aut_hash': 851, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 288, 'aut_permdeg': 11, 'aut_perms': [1992360, 727201, 11753520, 5907604], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 8, 1], [3, 2, 1, 1], [3, 2, 8, 1], [6, 3, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times \\GL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '48.29', 'autcent_hash': 29, 'autcent_nilpotent': False, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(2,3)', '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, 8], [3, 2, 9], [6, 3, 8]], 'center_label': '9.2', 'center_order': 9, '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', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 12, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 2], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 1, 2, 4], [3, 2, 1, 1], [3, 2, 2, 4], [6, 3, 2, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '54.12', 'hash': 12, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [1, 2, 3], 'inner_gens': [[1, 3, 18], [1, 3, 36], [1, 39, 18]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 18], [2, 9]], 'label': '54.12', 'linC_count': 96, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 48, 'linQ_dim': 6, 'linQ_dim_count': 48, 'linR_count': 48, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'S3*C3^2', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 27, 'number_divisions': 15, 'number_normal_subgroups': 18, 'number_subgroup_autclasses': 14, 'number_subgroup_classes': 32, 'number_subgroups': 52, 'old_label': None, 'order': 54, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [3, 26], [6, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[2, 4, 18], [14, 11, 18]], 'outer_group': '48.29', 'outer_hash': 29, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 8, 'outer_perms': [36887, 21251], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,3)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 9], [4, 4]], 'representations': {'PC': {'code': 469035, 'gens': [1, 2, 4], 'pres': [4, -3, -2, -3, -3, 21, 199]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [108470321895114534, 125101736062734304]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [30419024, 36826717, 23068812, 32729290]}, 'Perm': {'d': 9, 'gens': [1, 144, 45600, 3]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times C_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': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 120, 'aut_gen_orders': [10, 24, 8, 24, 4], 'aut_gens': [[482987764, 80784, 243, 143405, 321844, 375098255851], [12936856197, 80784, 45364, 178825, 138933, 448824816479], [461841281280, 148, 80643, 40799, 276628, 100674161036], [268244708997, 45364, 243, 362636, 219474, 88139928956], [186810981364, 148, 80643, 40799, 357123, 362644445515], [13894859397, 45600, 80643, 178589, 219333, 287923789295]], 'aut_group': None, 'aut_hash': 2094941634879358500, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 155520, 'aut_permdeg': 60, 'aut_perms': [1942766464471054285980363528952979496622930700366527777971886073414047556281719859, 6491330948343710121067779207506969698100330792232775198983977300083925002809499153, 3763616582367394250566378064459610175106160326983311932026689517700051613312963032, 5349334641626781749807360899079595995783199573489769633343764394706106300514965914, 696379982475868673143422752154018570664081479169477463082466123793270380858600435], 'aut_phi_ratio': 30.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 45, 1, 1], [2, 405, 1, 1], [3, 2, 1, 1], [3, 3, 2, 1], [3, 6, 1, 1], [3, 6, 2, 1], [3, 40, 2, 1], [3, 80, 2, 1], [3, 120, 4, 1], [3, 240, 2, 1], [3, 240, 4, 1], [4, 90, 1, 1], [4, 810, 1, 1], [5, 72, 2, 1], [6, 9, 2, 1], [6, 90, 1, 1], [6, 135, 2, 1], [6, 270, 1, 1], [6, 270, 2, 1], [6, 360, 2, 1], [6, 360, 4, 1], [6, 405, 2, 1], [10, 648, 2, 1], [12, 180, 1, 1], [12, 270, 2, 1], [12, 540, 1, 1], [12, 540, 2, 1], [12, 810, 2, 1], [15, 144, 2, 1], [15, 216, 4, 1], [15, 432, 2, 1], [15, 432, 4, 1], [30, 648, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3.S_3^2.A_6.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': 120, 'autcentquo_group': None, 'autcentquo_hash': 2094941634879358500, 'autcentquo_nilpotent': False, 'autcentquo_order': 155520, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3.S_3^2.A_6.C_2^2', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 45, 1], [2, 405, 1], [3, 2, 1], [3, 3, 2], [3, 6, 3], [3, 40, 2], [3, 80, 2], [3, 120, 4], [3, 240, 6], [4, 90, 1], [4, 810, 1], [5, 72, 2], [6, 9, 2], [6, 90, 1], [6, 135, 2], [6, 270, 3], [6, 360, 6], [6, 405, 2], [10, 648, 2], [12, 180, 1], [12, 270, 2], [12, 540, 3], [12, 810, 2], [15, 144, 2], [15, 216, 4], [15, 432, 6], [30, 648, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '19440.bb', 'commutator_count': 1, 'commutator_label': '3240.cl', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '360.118'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 28, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['360.118', 1], ['54.5', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 45, 1, 1], [2, 405, 1, 1], [3, 2, 1, 1], [3, 3, 2, 1], [3, 6, 1, 1], [3, 6, 2, 1], [3, 40, 1, 2], [3, 80, 1, 2], [3, 120, 2, 2], [3, 240, 1, 2], [3, 240, 2, 2], [4, 90, 1, 1], [4, 810, 1, 1], [5, 72, 2, 1], [6, 9, 2, 1], [6, 90, 1, 1], [6, 135, 2, 1], [6, 270, 1, 1], [6, 270, 2, 1], [6, 360, 1, 2], [6, 360, 2, 2], [6, 405, 2, 1], [10, 648, 2, 1], [12, 180, 1, 1], [12, 270, 2, 1], [12, 540, 1, 1], [12, 540, 2, 1], [12, 810, 2, 1], [15, 144, 2, 1], [15, 216, 4, 1], [15, 432, 2, 1], [15, 432, 4, 1], [30, 648, 4, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 636, 'exponent': 60, 'exponents_of_order': [5, 4, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[30, 1, 2], [48, 1, 2], [54, 1, 1], [60, 1, 1]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '6480.bp', 'hash': 527354253826721014, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [6, 3, 3, 6, 3, 12], 'inner_gens': [[482987764, 45364, 80784, 362636, 276723, 461322310441], [482906880, 80784, 243, 143405, 357264, 375098175356], [482906880, 80784, 243, 223946, 357264, 375098255851], [482769090, 80784, 45364, 143405, 219474, 375098532331], [482906880, 45364, 80784, 362636, 321844, 375098313146], [186810843574, 45364, 243, 282095, 138933, 375098255851]], 'inner_hash': 527354253826721014, 'inner_nilpotent': False, 'inner_order': 19440, 'inner_split': False, 'inner_tex': 'C_3^2:C_6\\times A_6', 'inner_used': [1, 2, 4, 6], 'irrC_degree': 30, 'irrQ_degree': 30, 'irrQ_dim': 30, 'irrR_degree': 30, 'irrep_stats': [[1, 6], [2, 3], [5, 12], [6, 1], [8, 12], [9, 6], [10, 12], [16, 6], [18, 3], [20, 3], [30, 2], [48, 2], [54, 1], [60, 1]], 'label': '19440.bb', 'linC_count': 12, 'linC_degree': 11, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 11, 'linQ_degree_count': 4, 'linQ_dim': 11, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 11, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'C3^2:C6*A6', '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, 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, 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': 14, 'number_conjugacy_classes': 70, 'number_divisions': 42, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 398, 'number_subgroup_classes': 602, 'number_subgroups': 84929, 'old_label': None, 'order': 19440, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 459], [3, 2186], [4, 900], [5, 144], [6, 4158], [10, 1296], [12, 3960], [15, 3744], [30, 2592]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [200302548000, 7707651984, 0], 'outer_gens': [[443120302080, 80784, 243, 143405, 357264, 99635967356], [88176533187, 80784, 243, 143405, 276723, 435974959199], [482952387, 45600, 148, 178589, 357123, 375098255756]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': None, 'perfect': False, 'permutation_degree': 15, '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], [5, 4], [6, 1], [9, 2], [10, 8], [16, 2], [18, 3], [20, 5], [30, 2], [32, 3], [36, 1], [40, 1], [54, 1], [60, 1], [64, 1], [96, 1]], 'representations': {'Perm': {'d': 15, 'gens': [482987764, 80784, 243, 143405, 321844, 375098255851]}}, 'schur_multiplier': [3, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2:C_6\\times A_6', 'transitive_degree': 54, 'wreath_data': None, 'wreath_product': False}
