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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '11664.bf', 'ambient_counter': 32, 'ambient_order': 11664, 'ambient_tex': 'C_3^5:(C_2\\times S_4)', 'central': False, 'central_factor': False, 'centralizer_order': 9, 'characteristic': False, 'core_order': 81, 'counter': 148, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '11664.bf.48.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '48.b1', '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': 48, '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': '243.51', 'subgroup_hash': 51, 'subgroup_order': 243, 'subgroup_tex': 'C_3^4:C_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '11664.bf', 'aut_centralizer_order': None, 'aut_label': '48.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1296.c1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['12.b1', '16.a1', '24.b1', '24.i1', '24.k1'], 'contains': ['144.a1', '144.j1', '144.l1', '144.u1', '144.x1'], 'core': '144.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [8376, -1, 1050, -1], 'generators': [711374856237360, 43545604, 93448857748, 1510739033305434, 80788], 'label': '11664.bf.48.b1', 'mobius_quo': None, 'mobius_sub': 6, 'normal_closure': '12.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.b1', 'old_label': '48.b1', 'projective_image': '11664.bf', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '48.b1', 'subgroup_fusion': None, 'weyl_group': '324.122'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '27.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 6, 'aut_gen_orders': [2, 6, 6, 3, 3, 3, 3, 3], 'aut_gens': [[2920, 1270, 5134, 781], [2920, 3409, 2923, 988], [2920, 3220, 2923, 5665], [5110, 5353, 2977, 1237], [2920, 1243, 5134, 754], [2920, 1270, 5134, 2992], [2947, 1216, 5134, 808], [2920, 1270, 5134, 2935], [2920, 1270, 5134, 808]], 'aut_group': '17496.ry', 'aut_hash': 3015459247439121481, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 17496, 'aut_permdeg': 36, 'aut_perms': [2911159915070511651590889052422163200, 2434265848495029432072166212054057752593, 1487775970023571939815323226003657949467, 237364, 62113177563886912966867315310853941395200, 108060093727328449920328728820589910824367, 191304615744396667264844767454082374400, 146036060426579166800797573768622856192000], 'aut_phi_ratio': 108.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 1, 6, 1], [3, 3, 2, 1], [3, 3, 4, 1], [3, 3, 18, 1], [3, 9, 6, 1], [9, 9, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^4:S_3^3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '486.183', 'autcent_hash': 183, 'autcent_nilpotent': False, 'autcent_order': 486, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^4:S_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '36.10', 'autcentquo_hash': 10, 'autcentquo_nilpotent': False, 'autcentquo_order': 36, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3^2', 'cc_stats': [[1, 1, 1], [3, 1, 8], [3, 3, 24], [3, 9, 6], [9, 9, 12]], 'center_label': '9.2', 'center_order': 9, 'central_product': True, 'central_quotient': '27.3', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 51, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['81.7', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 4], [3, 3, 2, 12], [3, 9, 2, 3], [9, 9, 2, 6]], 'element_repr_type': 'GLZq', 'elementary': 3, 'eulerian_function': 468, 'exponent': 9, 'exponents_of_order': [5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '27.5', 'hash': 51, 'hyperelementary': 3, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [1, 3, 3, 3], 'inner_gens': [[2920, 1270, 5134, 781], [2920, 1270, 5134, 5149], [2920, 1270, 5134, 754], [2920, 3409, 5161, 781]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 27, 'inner_split': False, 'inner_tex': '\\He_3', 'inner_used': [2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 27], [3, 24]], 'label': '243.51', 'linC_count': 324, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 81, 'linQ_dim': 8, 'linQ_dim_count': 81, 'linR_count': 81, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^4:C3', 'ngens': 3, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 51, 'number_divisions': 26, 'number_normal_subgroups': 36, 'number_subgroup_autclasses': 34, 'number_subgroup_classes': 126, 'number_subgroups': 342, 'old_label': None, 'order': 243, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 134], [9, 108]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [3, 6, 3, 6, 2, 6, 6], 'outer_gen_pows': [730, 730, 730, 730, 2950, 730, 730], 'outer_gens': [[2947, 3406, 5134, 2962], [2920, 1027, 5161, 5119], [2974, 5623, 5134, 5152], [5164, 5407, 5161, 5119], [5110, 1000, 5161, 5119], [5164, 3190, 5161, 766], [2920, 1243, 2950, 1042]], 'outer_group': '648.608', 'outer_hash': 608, 'outer_nilpotent': False, 'outer_order': 648, 'outer_permdeg': 14, 'outer_perms': [44108920800, 47103286327, 2399484240, 643790191, 1, 15941363761, 46504604167], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3\\times C_3^2:D_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 12, 'pgroup': 3, 'primary_abelian_invariants': [3, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 13], [6, 12]], 'representations': {'PC': {'code': 1011272200, 'gens': [1, 2, 3, 4], 'pres': [5, 3, 3, 3, 3, 3, 248, 253, 78]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [24577758, 24507118, 2939606, 36731395, 35376673]}, 'GLZq': {'d': 2, 'q': 9, 'gens': [739, 757, 733, 3220, 2920]}, 'Perm': {'d': 12, 'gens': [47917563, 11294644, 92348668, 15040224, 92348664]}}, 'schur_multiplier': [3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 3], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:C_3', 'transitive_degree': 27, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 36, 'aut_gen_orders': [6, 12, 2], 'aut_gens': [[21385030146934, 4890702311671707], [4536040554195985, 6380518299770853], [6024824948119439, 4556031078695067], [733430960191415, 5267312565943948]], 'aut_group': '23328.p', 'aut_hash': 1081498020297448794, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 23328, 'aut_permdeg': 171, 'aut_perms': [849990153799975059346275732007331704988470902739296865259872868885865243077880657357729317369948682710672182819349751917291962789064149138965335734103099029679249644284635302521821029674900359652237726906416987077524581549262442956401407349024130042606618293187287244183940993983781410764669698093634900147999, 504915607500595852406191609517625592176641363569734266581312718725708496041467293395310857592289055443555052470995931031021386955864238551178899799659179094520572517003896893918987686044581681946068599005172677652706315441511532041065872279142966059517822947078670301412590848848885284916552529077598068560997, 145956494680129583682776742501787071493184340175655526806422896630547039547162195375736752668728990064122668007044100355820592025573087506187383517518528481977517705138547292056919401756811751924596694341479737076300806750822304199706400782155089892126496022490677909355032824943326086124395714095548699515921], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 27, 1, 1], [2, 162, 1, 2], [2, 243, 1, 1], [3, 2, 1, 1], [3, 6, 1, 2], [3, 8, 1, 1], [3, 12, 1, 3], [3, 16, 1, 1], [3, 24, 1, 3], [3, 48, 1, 2], [3, 216, 1, 1], [3, 432, 1, 1], [4, 162, 1, 2], [6, 54, 1, 4], [6, 72, 1, 1], [6, 108, 1, 4], [6, 216, 1, 1], [6, 324, 1, 4], [6, 486, 1, 1], [6, 648, 1, 3], [9, 216, 2, 1], [9, 432, 2, 1], [12, 162, 2, 2], [12, 324, 1, 2], [12, 324, 2, 2], [18, 648, 2, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_3^2\\times S_3^3):D_6', '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': '23328.p', 'autcentquo_hash': 1081498020297448794, 'autcentquo_nilpotent': False, 'autcentquo_order': 23328, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '(C_3^2\\times S_3^3):D_6', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 27, 1], [2, 162, 2], [2, 243, 1], [3, 2, 1], [3, 6, 2], [3, 8, 1], [3, 12, 3], [3, 16, 1], [3, 24, 3], [3, 48, 2], [3, 216, 1], [3, 432, 1], [4, 162, 2], [6, 54, 4], [6, 72, 1], [6, 108, 4], [6, 216, 1], [6, 324, 4], [6, 486, 1], [6, 648, 3], [9, 216, 2], [9, 432, 2], [12, 162, 4], [12, 324, 6], [18, 648, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '11664.bf', 'commutator_count': 1, 'commutator_label': '2916.ei', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 32, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 27, 1, 1], [2, 162, 1, 2], [2, 243, 1, 1], [3, 2, 1, 1], [3, 6, 1, 2], [3, 8, 1, 1], [3, 12, 1, 3], [3, 16, 1, 1], [3, 24, 1, 3], [3, 48, 1, 2], [3, 216, 1, 1], [3, 432, 1, 1], [4, 162, 1, 2], [6, 54, 1, 4], [6, 72, 1, 1], [6, 108, 1, 4], [6, 216, 1, 1], [6, 324, 1, 4], [6, 486, 1, 1], [6, 648, 1, 3], [9, 216, 1, 2], [9, 432, 1, 2], [12, 162, 2, 2], [12, 324, 1, 2], [12, 324, 2, 2], [18, 648, 1, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 702, 'exponent': 36, 'exponents_of_order': [6, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 4], [24, 1, 4], [48, 1, 2]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '3888.bz', 'hash': 8976642636912607486, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [6, 12], 'inner_gens': [[21385030146934, 6045834656378758], [2288718064195919, 4890702311671707]], 'inner_hash': 8976642636912607486, 'inner_nilpotent': False, 'inner_order': 11664, 'inner_split': True, 'inner_tex': 'C_3^5:(C_2\\times S_4)', 'inner_used': [1, 2], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [2, 4], [3, 4], [4, 1], [6, 14], [8, 6], [12, 13], [16, 3], [24, 6], [48, 2]], 'label': '11664.bf', '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': 'C3^5:(C2*S4)', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 50, 'number_characteristic_subgroups': 19, 'number_conjugacy_classes': 57, 'number_divisions': 53, 'number_normal_subgroups': 19, 'number_subgroup_autclasses': 1173, 'number_subgroup_classes': 1226, 'number_subgroups': 96092, 'old_label': None, 'order': 11664, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 603], [3, 890], [4, 324], [6, 4662], [9, 1296], [12, 2592], [18, 1296]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[356056342831317, 4555937673428427]], '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': 7, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [3, 4], [4, 1], [6, 10], [8, 6], [12, 11], [16, 3], [24, 8], [48, 2]], 'representations': {'PC': {'code': '1868343570380846324220565108162779408692844796469816273277419315443303072011374563928786777964011236090613322206137325228307144467765685435567', 'gens': [1, 2, 4, 6, 8, 9, 10], 'pres': [10, 2, 2, 3, 2, 3, 2, 3, 3, 3, 3, 480, 146121, 51, 215882, 44302, 99373, 40223, 113, 10814, 6024, 488885, 315015, 78685, 42875, 175, 606486, 288976, 46226, 6756, 71047, 19217, 987, 233288, 32418, 87508, 3298, 72009, 64819, 18029, 64839, 10859]}, 'Perm': {'d': 18, 'gens': [4890702311671707, 21385030146934]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^5:(C_2\\times S_4)', 'transitive_degree': 18, 'wreath_data': None, 'wreath_product': False}
