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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1944.3612', 'ambient_counter': 3612, 'ambient_order': 1944, 'ambient_tex': 'C_3^3:(C_3\\times S_4)', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 3, 'counter': 104, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1944.3612.54.g1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '54.g1', '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': 54, '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': '36.12', 'subgroup_hash': 12, 'subgroup_order': 36, 'subgroup_tex': 'C_6\\times S_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1944.3612', 'aut_centralizer_order': 648, 'aut_label': '54.g1', 'aut_quo_index': None, 'aut_stab_index': 81, 'aut_weyl_group': '24.14', 'aut_weyl_index': 52488, 'centralizer': '324.c1', 'complements': None, 'conjugacy_class_count': 3, 'contained_in': ['18.i1', '27.a1'], 'contains': ['108.a1', '108.i1', '162.d1', '162.e1'], 'core': '648.a1', 'coset_action_label': None, 'count': 81, 'diagramx': [7855, -1, 8265, -1], 'generators': [3, 972, 2, 756], 'label': '1944.3612.54.g1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '27.a1', 'old_label': '54.g1', 'projective_image': '1944.3612', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '54.g1', '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': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 2, 6], 'aut_gens': [[1, 6], [5, 6], [1, 30], [31, 6]], 'aut_group': '24.14', 'aut_hash': 14, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24, 'aut_permdeg': 7, 'aut_perms': [7, 127, 856], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 3, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_6', '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': 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, 1, 1], [2, 3, 2], [3, 1, 2], [3, 2, 3], [6, 1, 2], [6, 2, 3], [6, 3, 4]], 'center_label': '6.2', 'center_order': 6, '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', '2.1', '3.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 12, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 3, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 6, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '36.12', 'hash': 12, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 30], [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, 12], [2, 6]], 'label': '36.12', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 7, 'linQ_dim': 4, 'linQ_dim_count': 7, 'linR_count': 7, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C6*S3', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 18, 'number_divisions': 12, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 22, 'number_subgroups': 36, 'old_label': None, 'order': 36, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 7], [3, 8], [6, 20]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[19, 6], [5, 6]], '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': 2, 'perfect': False, 'permutation_degree': 8, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 2]], 'representations': {'PC': {'code': 403784533, 'gens': [1, 3], 'pres': [4, -2, -3, -2, -3, 8, 362, 34, 387]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16858573, 35931481]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [56, 687, 1718]}, 'Perm': {'d': 8, 'gens': [720, 1, 30, 5760]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6\\times S_3', 'transitive_degree': 12, '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': 36, 'aut_gen_orders': [6, 12, 6, 6, 6], 'aut_gens': [[1, 6, 18, 54, 324], [11, 660, 840, 54, 1620], [1289, 654, 1068, 270, 1890], [1589, 1302, 1368, 1242, 1458], [1405, 1308, 138, 270, 540], [443, 876, 1110, 54, 432]], 'aut_group': None, 'aut_hash': 5742461704192487224, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1259712, 'aut_permdeg': 168, 'aut_perms': [140893284666465992081439062528457917238398397625592550612859509683579789611263389433313090533556127398496119461248423177471620108163148509993257715905014795029767161537624487454158409698446732885706146261645264085552534393302132501556258226359087817943134187749723660390967052635776580896046329447298424, 71387769141066723981726075021926967368398716050235097378482207824524133222436624913635278867030667236037245343886388778067355859186642291475584534881199496105075677328786281042942501719976806105989093850984165084923831559160877190180853500037475351205295587055800450632807818713792237101692148459848370, 72167474260809314003731089185131102248526174500994611432287404763119778969185502770667539507339558575628732489256864157425625611461822482461095810642070646801395565500971647949296613587284678553074918896356203158697997806065368483644182215162858829185535990628600664568300699735291769504769167398986473, 376833348755648505879483717702874368159727892196402127888894396118022242759863312404184757568529324222275357604106025378912365005796781044331482577230145740346766657736250259165917490001992863121614301696754809952710818983794846451694132577615435824878834350621446108804946766550060719637822403337125, 140197320457540781884805901315360456168741586412095615736860157543705303945466835012380994683668382644734146801433147911744174336119903616233323852915086170549037996018230321699665501108909245087621586116018981863540597847126576521067581010228653693776452221807929349318019487581208014522274983747335063], 'aut_phi_ratio': 1944.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 162, 1, 1], [3, 2, 1, 1], [3, 2, 3, 1], [3, 3, 2, 1], [3, 6, 2, 1], [3, 6, 3, 1], [3, 6, 6, 1], [3, 8, 9, 1], [3, 24, 6, 2], [3, 24, 12, 1], [4, 162, 1, 1], [6, 6, 1, 1], [6, 6, 3, 1], [6, 9, 2, 1], [6, 18, 2, 1], [6, 18, 3, 1], [6, 18, 6, 1], [6, 162, 2, 1], [12, 162, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4.C_6^2.S_3^3.C_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': 36, 'autcentquo_group': None, 'autcentquo_hash': 5742461704192487224, 'autcentquo_nilpotent': False, 'autcentquo_order': 1259712, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^4.C_6^2.C_3.D_6^2', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 162, 1], [3, 2, 4], [3, 3, 2], [3, 6, 11], [3, 8, 9], [3, 24, 24], [4, 162, 1], [6, 6, 4], [6, 9, 2], [6, 18, 11], [6, 162, 2], [12, 162, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1944.3612', 'commutator_count': 1, 'commutator_label': '324.171', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 3612, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 162, 1, 1], [3, 2, 1, 4], [3, 3, 2, 1], [3, 6, 1, 3], [3, 6, 2, 4], [3, 8, 1, 9], [3, 24, 1, 6], [3, 24, 2, 9], [4, 162, 1, 1], [6, 6, 1, 4], [6, 9, 2, 1], [6, 18, 1, 3], [6, 18, 2, 4], [6, 162, 2, 1], [12, 162, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 5896800, 'exponent': 12, 'exponents_of_order': [5, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '648.737', 'hash': 3612, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [6, 3, 3, 6, 6], 'inner_gens': [[1, 120, 252, 1242, 1620], [229, 6, 18, 54, 324], [145, 6, 18, 1188, 486], [1081, 6, 1152, 54, 324], [649, 6, 180, 54, 324]], 'inner_hash': 3612, 'inner_nilpotent': False, 'inner_order': 1944, 'inner_split': False, 'inner_tex': 'C_3^3:(C_3\\times S_4)', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 6], [2, 39], [3, 6], [6, 21], [18, 3]], 'label': '1944.3612', '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': True, 'name': 'C3^3:(C3*S4)', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 22, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 75, 'number_divisions': 54, 'number_normal_subgroups': 96, 'number_subgroup_autclasses': 239, 'number_subgroup_classes': 1005, 'number_subgroups': 20020, 'old_label': None, 'order': 1944, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 165], [3, 728], [4, 162], [6, 564], [12, 324]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [3, 6, 3, 3, 3, 2, 2], 'outer_gen_pows': [3, 0, 0, 3, 0, 0, 0], 'outer_gens': [[1, 12, 1440, 1242, 1620], [1, 12, 144, 1242, 324], [1, 1302, 1314, 54, 324], [1, 12, 684, 1242, 1620], [1, 114, 882, 54, 432], [5, 228, 36, 1026, 324], [1, 6, 150, 1026, 324]], 'outer_group': '648.555', 'outer_hash': 555, 'outer_nilpotent': False, 'outer_order': 648, 'outer_permdeg': 12, 'outer_perms': [120114720, 121214282, 787948, 2219904, 328544040, 9441506, 39937944], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3.S_3^3', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 15], [3, 2], [4, 13], [6, 15], [12, 4], [18, 3]], 'representations': {'PC': {'code': 615342338804487498494798894649800094238000819478679925, 'gens': [1, 3, 4, 5, 7], 'pres': [8, 2, 3, 3, 3, 2, 3, 2, 3, 16, 2882, 1378, 8067, 3755, 49684, 2668, 116, 10373, 90726, 1542, 166, 82951]}, 'Perm': {'d': 16, 'gens': [180587244289, 16313, 93448857600, 79833600, 43597532, 98499, 1313901388800, 2789705318400]}}, 'schur_multiplier': [3, 3, 3, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3:(C_3\\times S_4)', 'transitive_degree': 108, 'wreath_data': None, 'wreath_product': False}
