Formats: - HTML - YAML - JSON - 2026-07-24T20:25:42.357339
  • gps_subgroup_search • Show schema
    {'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '311040.j', 'ambient_counter': 10, 'ambient_order': 311040, 'ambient_tex': 'C_3^3:S_3.C_2^4:S_5', 'central': False, 'central_factor': False, 'centralizer_order': 48, 'characteristic': False, 'core_order': 1, 'counter': 762, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '311040.j.51840.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '51840.b1.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': 51840, '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': '6.1', 'subgroup_hash': 1, 'subgroup_order': 6, 'subgroup_tex': 'S_3', 'supersolvable': True, 'sylow': 0}
  • gps_subgroup_data • Show schema
    {'ambient': '311040.j', 'aut_centralizer_order': None, 'aut_label': '51840.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '6480.z1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['17280.a1.a1', '17280.b1.a1', '17280.b2.a1', '17280.b3.a1', '25920.g1.a1', '25920.h1.a1'], 'contains': ['103680.a1.a1', '155520.a1.a1'], 'core': '311040.a1.a1', 'coset_action_label': None, 'count': 1080, 'diagramx': [5107, -1, 4075, -1, 5354, -1, 5191, -1], 'generators': [4563710714067491935264040177344169864551888572486591018600540525227151423335270770090012696578125793858508085145946273883, 457204064046573758978307144795219438090255661314408443180818087904292671781693752127328358647603419136569127435979989166], 'label': '311040.j.51840.b1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1920.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '1080.a1.a1', 'old_label': '51840.b1.a1', 'projective_image': '311040.j', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '51840.b1.a1', 'subgroup_fusion': None, 'weyl_group': '6.1'}
  • gps_groups • Show schema
    {'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '2.1', '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, 4], [1, 3], [2, 4]], '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, 3, 1, 1], [3, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', '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': 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, 2, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 2, 1, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '6.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 3], [2, 4]], '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 2], [2, 1]], 'label': '6.1', 'linC_count': 1, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'S3', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 3, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 6, 'old_label': None, 'order': 6, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 3], [3, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 3, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 25, 'gens': [1, 2], 'pres': [2, -2, -3, 17]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [28, 45]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'GL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'SL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PSL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PGL'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'SO'}, {'d': 2, 'q': 2, 'gens': [20, 69], 'family': 'SU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'PSO'}, {'d': 2, 'q': 2, 'gens': [20, 69], 'family': 'PSU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'GO'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'Omega'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'PGO'}, {'d': 2, 'q': 2, 'gens': [20, 138], 'family': 'PGU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'POmega'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'CSO'}, {'d': 2, 'q': 4, 'gens': [20, 194], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [13, 14], 'family': 'CSOMinus'}, {'d': 2, 'q': 2, 'gens': [20, 69], 'family': 'CSU'}, {'d': 3, 'q': 2, 'gens': [84, 279], 'family': 'CO'}, {'d': 2, 'q': 2, 'gens': [13, 14], 'family': 'COMinus'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PGammaL'}, {'d': 2, 'q': 2, 'gens': [6, 11], 'family': 'PSigmaL'}, {'d': 2, 'q': 2, 'gens': [20, 138], 'family': 'PGammaU'}, {'d': 1, 'q': 3, 'gens': [1, 3], 'family': 'AGL'}, {'d': 1, 'q': 3, 'gens': [1, 3], 'family': 'AGammaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11, 6]}, 'Perm': {'d': 3, 'gens': [1, 4]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3', 'transitive_degree': 3, 'wreath_data': None, 'wreath_product': False}
  • gps_groups • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 120, 'aut_gen_orders': [6, 8], 'aut_gens': [[3938632625685295871943746791959711692177342846391569385563424687439219660033259498372983731343195785497578337558334344355, 5512661209498819967020280990014347681011400779742950735797833264947559131608145782689824052719088350943896887434650941636], [4546827718076786023655309830207872385692149554473375856732050145234214376831883042529265429873627070660882339718929858121, 3559041876905299444537647595027790458300888637494333199111682568957611471079791330646624059617719462449592486946480376470], [179434987654430445210000456310829771807909831560978592961204264964477555123081012567433866574151657732098666864394037488, 1682514694102698159110427440087637326392034045189880765895422001826171333057070931378083979672399706102127499895234226879]], 'aut_group': '311040.j', 'aut_hash': 6830784901558741135, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 311040, 'aut_permdeg': 1080, 'aut_perms': 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'aut_phi_ratio': 3.75, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [2, 90, 1, 1], [2, 1080, 1, 1], [3, 80, 1, 1], [3, 720, 1, 1], [3, 5760, 1, 1], [4, 540, 1, 1], [4, 1620, 1, 1], [4, 3240, 1, 1], [4, 4860, 1, 1], [4, 9720, 1, 1], [5, 31104, 1, 1], [6, 720, 1, 1], [6, 6480, 1, 1], [6, 8640, 1, 2], [6, 17280, 1, 1], [8, 1620, 1, 2], [8, 6480, 1, 2], [8, 9720, 1, 1], [8, 19440, 1, 1], [8, 38880, 1, 1], [10, 31104, 1, 1], [12, 4320, 1, 1], [12, 12960, 1, 1], [12, 25920, 1, 1], [24, 12960, 1, 4]], 'aut_supersolvable': False, 'aut_tex': 'C_3^3:S_3.C_2^4:S_5', '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': '311040.j', 'autcentquo_hash': 6830784901558741135, 'autcentquo_nilpotent': False, 'autcentquo_order': 311040, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^3:S_3.C_2^4:S_5', 'cc_stats': [[1, 1, 1], [2, 81, 1], [2, 90, 1], [2, 1080, 1], [3, 80, 1], [3, 720, 1], [3, 5760, 1], [4, 540, 1], [4, 1620, 1], [4, 3240, 1], [4, 4860, 1], [4, 9720, 1], [5, 31104, 1], [6, 720, 1], [6, 6480, 1], [6, 8640, 2], [6, 17280, 1], [8, 1620, 2], [8, 6480, 2], [8, 9720, 1], [8, 19440, 1], [8, 38880, 1], [10, 31104, 1], [12, 4320, 1], [12, 12960, 1], [12, 25920, 1], [24, 12960, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '311040.j', 'commutator_count': 1, 'commutator_label': None, 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '60.5'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 10, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [2, 90, 1, 1], [2, 1080, 1, 1], [3, 80, 1, 1], [3, 720, 1, 1], [3, 5760, 1, 1], [4, 540, 1, 1], [4, 1620, 1, 1], [4, 3240, 1, 1], [4, 4860, 1, 1], [4, 9720, 1, 1], [5, 31104, 1, 1], [6, 720, 1, 1], [6, 6480, 1, 1], [6, 8640, 1, 2], [6, 17280, 1, 1], [8, 1620, 2, 1], [8, 6480, 2, 1], [8, 9720, 1, 1], [8, 19440, 1, 1], [8, 38880, 1, 1], [10, 31104, 1, 1], [12, 4320, 1, 1], [12, 12960, 1, 1], [12, 25920, 1, 1], [24, 12960, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 136800, 'exponent': 120, 'exponents_of_order': [8, 5, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[80, 1, 2], [160, 0, 2], [160, 1, 1], [240, 1, 2], [320, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '311040.j', 'hash': 6830784901558741135, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 120, 'inner_gen_orders': [6, 8], 'inner_gens': [[3938632625685295871943746791959711692177342846391569385563424687439219660033259498372983731343195785497578337558334344355, 1204688212998658619000334793945242109098409003317537044283164954829151548204437370340377386909595854753169830364257962917], [1369703252167805278818951658925534974015989240540526343408472167481828725179465657638062489175226389158015207098897864557, 5512661209498819967020280990014347681011400779742950735797833264947559131608145782689824052719088350943896887434650941636]], 'inner_hash': 6830784901558741135, 'inner_nilpotent': False, 'inner_order': 311040, 'inner_split': True, 'inner_tex': 'C_3^3:S_3.C_2^4:S_5', 'inner_used': [1, 2], 'irrC_degree': 80, 'irrQ_degree': 80, 'irrQ_dim': 80, 'irrR_degree': 80, 'irrep_stats': [[1, 2], [4, 4], [5, 4], [6, 1], [10, 5], [15, 2], [16, 2], [20, 4], [24, 1], [80, 2], [160, 3], [240, 2], [320, 1]], 'label': '311040.j', 'linC_count': 2, 'linC_degree': 80, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 80, 'linQ_degree_count': 2, 'linQ_dim': 80, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 80, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'C3^3:S3.C2^4:S5', 'ngens': 2, '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, 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': 33, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 33, 'number_divisions': 29, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 789, 'number_subgroup_classes': 789, 'number_subgroups': 1048248, 'old_label': None, 'order': 311040, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 1251], [3, 6560], [4, 19980], [5, 31104], [6, 41760], [8, 84240], [10, 31104], [12, 43200], [24, 51840]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': None, 'perfect': False, 'permutation_degree': 81, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 2], [5, 4], [6, 1], [8, 1], [10, 5], [15, 2], [20, 2], [24, 1], [32, 1], [40, 1], [80, 2], [160, 1], [240, 2], [320, 2]], 'representations': {'Perm': {'d': 81, 'gens': [3938632625685295871943746791959711692177342846391569385563424687439219660033259498372983731343195785497578337558334344355, 5512661209498819967020280990014347681011400779742950735797833264947559131608145782689824052719088350943896887434650941636]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3:S_3.C_2^4:S_5', 'transitive_degree': 81, 'wreath_data': None, 'wreath_product': False}