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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '129600.n', 'ambient_counter': 14, 'ambient_order': 129600, 'ambient_tex': 'A_5^2:S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 15, 'characteristic': False, 'core_order': 9, 'counter': 168, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '129600.n.288.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '288.c1', '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': 288, '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': '450.29', 'subgroup_hash': 29, 'subgroup_order': 450, 'subgroup_tex': 'C_{15}\\times D_{15}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '129600.n', 'aut_centralizer_order': None, 'aut_label': '288.c1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '8640.b1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['144.b1', '144.h1', '144.i1'], 'contains': ['576.a1', '864.e1', '864.f1', '1440.n1', '1440.o1'], 'core': '14400.a1', 'coset_action_label': None, 'count': 144, 'diagramx': None, 'generators': [648655297920, 676, 627, 8133091551178, 11863951954173], 'label': '129600.n.288.c1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '72.d1', 'old_label': '288.c1', 'projective_image': '129600.n', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '288.c1', 'subgroup_fusion': None, 'weyl_group': '120.42'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '30.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 60, 'aut_gen_orders': [2, 2, 4, 60], 'aut_gens': [[1, 30], [11, 30], [1, 330], [1, 240], [43, 30]], 'aut_group': '960.11024', 'aut_hash': 11024, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 960, 'aut_permdeg': 21, 'aut_perms': [43920661633114700310, 1346867225347685792, 1428885169286258465, 648174390504670936], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 15, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 8, 1], [6, 15, 2, 1], [10, 15, 4, 1], [15, 1, 8, 1], [15, 2, 4, 3], [15, 2, 8, 2], [15, 2, 16, 2], [15, 2, 32, 1], [30, 15, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_{30}:C_4^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '120.36', 'autcentquo_hash': 36, 'autcentquo_nilpotent': False, 'autcentquo_order': 120, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_5', 'cc_stats': [[1, 1, 1], [2, 15, 1], [3, 1, 2], [3, 2, 3], [5, 1, 4], [5, 2, 10], [6, 15, 2], [10, 15, 4], [15, 1, 8], [15, 2, 92], [30, 15, 8]], 'center_label': '15.1', 'center_order': 15, 'central_product': True, 'central_quotient': '30.3', 'commutator_count': 1, 'commutator_label': '15.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '5.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 29, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['30.3', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 15, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 4, 2], [6, 15, 2, 1], [10, 15, 4, 1], [15, 1, 8, 1], [15, 2, 4, 3], [15, 2, 8, 10], [30, 15, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72, 'exponent': 30, 'exponents_of_order': [2, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[2, 0, 32]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '450.29', 'hash': 29, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [2, 15], 'inner_gens': [[1, 420], [61, 30]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 30, 'inner_split': True, 'inner_tex': 'D_{15}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 4, 'irrep_stats': [[1, 30], [2, 105]], 'label': '450.29', 'linC_count': 32, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 12, 'linQ_dim': 12, 'linQ_dim_count': 12, 'linR_count': 48, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C15*D15', 'ngens': 5, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 20, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 135, 'number_divisions': 26, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 41, 'number_subgroup_classes': 48, 'number_subgroups': 144, 'old_label': None, 'order': 450, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 15], [3, 8], [5, 24], [6, 30], [10, 60], [15, 192], [30, 120]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4, 4], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[11, 30], [1, 390], [13, 30]], 'outer_group': '32.21', 'outer_hash': 21, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 10, 'outer_perms': [362880, 17, 5880], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 4], [8, 7], [16, 10]], 'representations': {'PC': {'code': 2077675066962947191, 'gens': [1, 4], 'pres': [5, -2, -3, -5, -3, -5, 10, 36, 8403, 78, 9004]}, 'Lie': [{'d': 2, 'q': 16, 'gens': [272, 36866, 8194], 'family': 'COPlus'}], 'GLFp': {'d': 2, 'p': 31, 'gens': [208546, 992, 268120]}, 'Perm': {'d': 16, 'gens': [87657655680, 3, 5904, 3991680, 1482509952000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [30], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{15}\\times D_{15}', 'transitive_degree': 30, 'wreath_data': ['C_{15}', '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [30, 12, 10, 20, 6], 'aut_gens': [[1502313402394, 2878959363997], [8570859857821, 10486913436948], [10748487501942, 8301625033117], [9502441366695, 2978595613084], [8128261981217, 9172562394664], [2833948011304, 10482631815757]], 'aut_group': None, 'aut_hash': 1122920581106867708, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1036800, 'aut_permdeg': 360, 'aut_perms': 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1768876242906651014807375879421125208933608571235976476639842568557079054149033412972268941579728553286416337134911043082398322137685660077102361000744633135834473101976110037917502331344573628171459560265752535083171156786110055104212369888700010668087564688358020250724146668499750642147029337320110759049471258553854125888207779746992813590463118765935395536610651440783059548877913305853840049577716391959724184218300953030125916387109574721496823682237451243968494496357437230662180823460284025690485786744159265263472517499530590666647906192977262760077828178155574791134576299017307044098082453086196903909403254244157314509568375413050282391587175642057477161560205494015292049738235862381466139672692303297679781708111713081512189206213496728760208963765170], 'aut_phi_ratio': 30.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 30, 1, 1], [2, 180, 2, 1], [2, 225, 1, 1], [2, 270, 1, 1], [2, 2025, 1, 1], [3, 2, 2, 1], [3, 4, 1, 1], [3, 40, 1, 1], [3, 80, 2, 2], [3, 400, 1, 1], [3, 800, 2, 1], [3, 1600, 1, 1], [4, 2700, 2, 1], [5, 24, 2, 1], [5, 144, 2, 1], [5, 288, 1, 1], [6, 60, 2, 2], [6, 360, 1, 1], [6, 360, 2, 1], [6, 450, 2, 1], [6, 600, 1, 1], [6, 900, 1, 1], [6, 1200, 2, 2], [6, 3600, 1, 1], [6, 3600, 2, 1], [6, 5400, 1, 1], [6, 7200, 2, 1], [10, 216, 2, 1], [10, 360, 2, 1], [10, 1296, 2, 1], [10, 2160, 4, 1], [10, 2592, 1, 1], [10, 3240, 2, 1], [12, 5400, 2, 1], [15, 48, 4, 2], [15, 288, 4, 1], [15, 480, 2, 1], [15, 576, 2, 3], [15, 960, 4, 2], [30, 720, 4, 2], [30, 4320, 2, 1], [30, 4320, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\wr C_2.A_5^2.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': 60, 'autcentquo_group': None, 'autcentquo_hash': 1122920581106867708, 'autcentquo_nilpotent': False, 'autcentquo_order': 1036800, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\wr C_2.A_5^2.C_2^2', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 30, 1], [2, 180, 2], [2, 225, 1], [2, 270, 1], [2, 2025, 1], [3, 2, 2], [3, 4, 1], [3, 40, 1], [3, 80, 4], [3, 400, 1], [3, 800, 2], [3, 1600, 1], [4, 2700, 2], [5, 24, 2], [5, 144, 2], [5, 288, 1], [6, 60, 4], [6, 360, 3], [6, 450, 2], [6, 600, 1], [6, 900, 1], [6, 1200, 4], [6, 3600, 3], [6, 5400, 1], [6, 7200, 2], [10, 216, 2], [10, 360, 2], [10, 1296, 2], [10, 2160, 4], [10, 2592, 1], [10, 3240, 2], [12, 5400, 2], [15, 48, 8], [15, 288, 4], [15, 480, 2], [15, 576, 6], [15, 960, 8], [30, 720, 8], [30, 4320, 6]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '129600.n', 'commutator_count': 1, 'commutator_label': None, 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '60.5', '60.5'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 14, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 30, 1, 1], [2, 180, 1, 2], [2, 225, 1, 1], [2, 270, 1, 1], [2, 2025, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [3, 40, 1, 1], [3, 80, 1, 4], [3, 400, 1, 1], [3, 800, 1, 2], [3, 1600, 1, 1], [4, 2700, 1, 2], [5, 24, 2, 1], [5, 144, 2, 1], [5, 288, 1, 1], [6, 60, 1, 4], [6, 360, 1, 3], [6, 450, 1, 2], [6, 600, 1, 1], [6, 900, 1, 1], [6, 1200, 1, 4], [6, 3600, 1, 3], [6, 5400, 1, 1], [6, 7200, 1, 2], [10, 216, 2, 1], [10, 360, 2, 1], [10, 1296, 2, 1], [10, 2160, 2, 2], [10, 2592, 1, 1], [10, 3240, 2, 1], [12, 5400, 1, 2], [15, 48, 2, 4], [15, 288, 2, 2], [15, 480, 2, 1], [15, 576, 1, 2], [15, 576, 2, 2], [15, 960, 2, 4], [30, 720, 2, 4], [30, 4320, 2, 3]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 2445, 'exponent': 60, 'exponents_of_order': [6, 4, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[12, 1, 4], [16, 1, 2], [20, 1, 2], [36, 1, 4], [48, 1, 4], [60, 1, 4], [64, 1, 1], [80, 1, 2], [100, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '129600.n', 'hash': 8643365490559466147, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [6, 12], 'inner_gens': [[1502313402394, 9610771645981], [1321697185624, 2878959363997]], 'inner_hash': 8643365490559466147, 'inner_nilpotent': False, 'inner_order': 129600, 'inner_split': True, 'inner_tex': 'A_5^2:S_3^2', 'inner_used': [1, 2], 'irrC_degree': 12, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [2, 4], [4, 1], [6, 4], [8, 2], [9, 8], [10, 2], [12, 8], [16, 8], [18, 10], [20, 4], [24, 4], [25, 4], [30, 4], [32, 4], [36, 6], [40, 2], [48, 8], [50, 4], [60, 8], [64, 1], [80, 4], [100, 1]], 'label': '129600.n', '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': False, 'name': 'A5^2:S3^2', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 52, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 105, 'number_divisions': 77, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 1328, 'number_subgroup_classes': 2115, 'number_subgroups': 1379108, 'old_label': None, 'order': 129600, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 2919], [3, 3968], [4, 5400], [5, 624], [6, 39120], [10, 21456], [12, 10800], [15, 13632], [30, 31680]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [5324103187200, 23, 0], 'outer_gens': [[9877227822228, 9415386898076], [1502313402604, 2878959364215], [1502313402615, 2878959364204]], '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': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 1], [8, 2], [10, 2], [12, 2], [16, 8], [18, 6], [20, 4], [24, 4], [25, 4], [32, 4], [36, 6], [40, 2], [48, 2], [50, 4], [60, 2], [64, 1], [72, 2], [80, 4], [96, 4], [100, 1], [120, 4]], 'representations': {'Perm': {'d': 16, 'gens': [1502313402394, 2878959363997]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'A_5^2:S_3^2', 'transitive_degree': 30, 'wreath_data': None, 'wreath_product': False}
