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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1152.153827', 'ambient_counter': 153827, 'ambient_order': 1152, 'ambient_tex': 'C_3\\times C_2^5:A_4', 'central': False, 'central_factor': False, 'centralizer_order': 48, 'characteristic': False, 'core_order': 4, 'counter': 120, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1152.153827.72.o1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '72.o1', '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': 72, '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': '16.11', 'subgroup_hash': 11, 'subgroup_order': 16, 'subgroup_tex': 'C_2\\times D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1152.153827', 'aut_centralizer_order': 384, 'aut_label': '72.o1', 'aut_quo_index': None, 'aut_stab_index': 36, 'aut_weyl_group': '32.27', 'aut_weyl_index': 13824, 'centralizer': '24.a1', 'complements': None, 'conjugacy_class_count': 12, 'contained_in': ['24.q1', '36.c1', '36.d1', '36.l1', '36.l2'], 'contains': ['144.l1', '144.n1', '144.o1', '144.p1'], 'core': '288.b1', 'coset_action_label': None, 'count': 36, 'diagramx': [1280, -1, 1277, -1], 'generators': [381365832960, 635806042104, 25430670720, 211802935680], 'label': '1152.153827.72.o1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '9.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1', 'old_label': '72.o1', 'projective_image': '1152.153827', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '72.o1', 'subgroup_fusion': None, 'weyl_group': '8.5'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 2, 2, 2, 2, 2], 'aut_gens': [[126, 55, 289, 288], [127, 55, 289, 288], [126, 265, 1, 288], [54, 127, 289, 288], [414, 55, 289, 288], [127, 54, 289, 288], [414, 265, 289, 288]], 'aut_group': '64.138', 'aut_hash': 138, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 64, 'aut_permdeg': 8, 'aut_perms': [2309, 526, 5329, 3043, 12316, 18498], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 4, 1], [4, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\wr C_2^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr C_2', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 4], [4, 2, 2]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 11, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 4], [4, 2, 1, 2]], 'element_repr_type': 'Perm', 'elementary': 2, 'eulerian_function': 21, 'exponent': 4, 'exponents_of_order': [4], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '8.5', 'hash': 11, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 1, 1], 'inner_gens': [[126, 265, 289, 288], [414, 55, 289, 288], [126, 55, 289, 288], [126, 55, 289, 288]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 2]], 'label': '16.11', 'linC_count': 8, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 8, 'linQ_dim': 3, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*D4', 'ngens': 4, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 10, 'number_divisions': 10, 'number_normal_subgroups': 19, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 27, 'number_subgroups': 35, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 11], [4, 4]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 151, 0], 'outer_gens': [[54, 127, 289, 288], [127, 55, 289, 288], [415, 54, 1, 288], [127, 54, 289, 288]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [126, 55, 289, 288], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2]], 'representations': {'PC': {'code': 8772, 'gens': [1, 2, 3], 'pres': [4, -2, 2, 2, -2, 78, 34]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [16322, 16432, 3198]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [8912, 8156, 13286, 14044]}, 'Perm': {'d': 6, 'gens': [126, 55, 289, 288]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times D_4', 'transitive_degree': 8, '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': '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': [12, 8, 12], 'aut_gens': [[89704339200, 3, 519321747, 199348169064, 294111338640, 744, 1680, 381365832960, 211802935680], [623352401280, 4, 298940907024, 375138129144, 205575875304, 287883632400, 381365835384, 211802935680, 287883630720], [550057495680, 3, 373705073523, 375138126720, 300338398080, 381365835384, 287883632400, 211802935680, 381365832960], [623352401280, 3, 32176569744, 31658379384, 37885438824, 211802937360, 287883633144, 381365832960, 211802935680]], 'aut_group': None, 'aut_hash': 3813564387106945754, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 442368, 'aut_permdeg': 388, 'aut_perms': 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423395182556263546124684411021070247591427751529227987849319563459294298961447429847399793164605519632330083145823295008938725831486777206026604730940659400072937249526365318832382515248657860212415258413681537796577930006478651092800641785211708304026919501021049790946819980279768117004220307401872219000392452350058900754248221289854808335454667869316671820591281421024421671660693062747429733199404640501771208181657597060317500953665110022103326948988886930191218227184629708921827916022783273996820904204585754162155244970654945085429124452586898354231674658067525482070277422054948873223530362775248015281091428806599733385318068642123156984674902010048311983986493030573626478222242414608016762625376763114884894691977081595539156633722326642760796927028215637198179932490190413380573185811396251164937677443917242911342472645627, 3996693466554182390151067462487581153391798080180545099988001624237361268246316131570520010687225080759084522677201972209277593503505541873591590084686432715937800250942311702670842923319606219841443785979833756951238159596539091803136643878026095823961163163780245778267114759906159653314364857583891309281893012168918327706229652109813539038119863363882160702832999369179380002516095873465187462044654934841802778120034997111792113830048027382958207406689148925689764926772480217663913029371849704676779449232486461666227767175980808665258270181703000157571248332031089569424639755754962512926919699903334982196012370311020561931702923264452496216221215786287016617656195605317815502040962322309250029612989716577651351689355609029638372840535233272122939763829640677830978804974487823151493233235493332751788856405202852946329296676534], 'aut_phi_ratio': 1152.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 3, 4, 1], [2, 4, 1, 1], [2, 6, 8, 1], [2, 12, 1, 1], [3, 1, 2, 1], [3, 64, 6, 1], [4, 12, 4, 1], [6, 3, 2, 1], [6, 3, 8, 1], [6, 4, 2, 1], [6, 6, 16, 1], [6, 12, 2, 1], [6, 64, 6, 1], [12, 12, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^6.C_2^6.C_3^2.D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': None, 'autcentquo_hash': 970489985340270929, 'autcentquo_nilpotent': False, 'autcentquo_order': 73728, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2^8.A_4^2.C_2', 'cc_stats': [[1, 1, 1], [2, 3, 5], [2, 4, 1], [2, 6, 8], [2, 12, 1], [3, 1, 2], [3, 64, 6], [4, 12, 4], [6, 3, 10], [6, 4, 2], [6, 6, 16], [6, 12, 2], [6, 64, 6], [12, 12, 8]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '384.18221', 'commutator_count': 1, 'commutator_label': '64.267', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 153827, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['384.18221', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 5], [2, 4, 1, 1], [2, 6, 1, 8], [2, 12, 1, 1], [3, 1, 2, 1], [3, 64, 2, 3], [4, 12, 1, 4], [6, 3, 2, 5], [6, 4, 2, 1], [6, 6, 2, 8], [6, 12, 2, 1], [6, 64, 2, 3], [12, 12, 2, 4]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 1820, 'exponent': 12, 'exponents_of_order': [7, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '288.1042', 'hash': 153827, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 1, 3, 2, 2, 2, 2, 2, 2], 'inner_gens': [[89704339200, 3, 519321747, 300338398824, 375138128400, 744, 1680, 381365832960, 211802935680], [89704339200, 3, 519321747, 199348169064, 294111338640, 744, 1680, 381365832960, 211802935680], [89704339200, 3, 519321747, 294111338640, 387592896504, 1680, 2424, 211802935680, 287883630720], [550057495680, 3, 387037286763, 199348169064, 294111338640, 744, 1680, 381365832960, 211802935680], [623352401280, 3, 200782271067, 199348169064, 294111338640, 744, 1680, 381365832960, 211802935680], [89704339200, 3, 519323883, 199348169064, 294111338640, 744, 1680, 381365832960, 211802935680], [89704339200, 3, 519322587, 199348169064, 294111338640, 744, 1680, 381365832960, 211802935680], [89704339200, 3, 286487268627, 199348169064, 294111338640, 744, 1680, 381365832960, 211802935680], [89704339200, 3, 380809900947, 199348169064, 294111338640, 744, 1680, 381365832960, 211802935680]], 'inner_hash': 18221, 'inner_nilpotent': False, 'inner_order': 384, 'inner_split': False, 'inner_tex': 'C_2^5:A_4', 'inner_used': [1, 3, 4, 6], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 18], [3, 30], [6, 24]], 'label': '1152.153827', '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': True, 'metacyclic': False, 'monomial': True, 'name': 'C3*C2^5:A4', 'ngens': 9, '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': 16, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 72, 'number_divisions': 46, 'number_normal_subgroups': 46, 'number_subgroup_autclasses': 192, 'number_subgroup_classes': 1728, 'number_subgroups': 10752, 'old_label': None, 'order': 1152, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 79], [3, 386], [4, 48], [6, 542], [12, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 3, 2, 2, 2, 2, 2], 'outer_gen_pows': [519321747, 0, 89704339200, 89704339200, 89704339200, 89704339200, 0, 0, 0], 'outer_gens': [[89704339200, 4, 519321748, 25430673144, 31658377704, 2424, 744, 287883630720, 381365832960], [89704339200, 4, 958729204, 294111336960, 199348168320, 1680, 744, 211802935680, 381365832960], [89704339200, 3, 519321747, 205575876240, 300338400504, 744, 1680, 287883630720, 381365832960], [89704339200, 3, 519321744, 375138128400, 205575876984, 1680, 2424, 211802935680, 287883630720], [89704339200, 3, 519321747, 355938107280, 180623522424, 211802936424, 287883632400, 381365832960, 211802935680], [89704339200, 3, 519321747, 199348168320, 294111336960, 381365833704, 211802937360, 381365832960, 211802935680], [89704339200, 3, 519321747, 300338399760, 375138129144, 744, 1680, 381365832960, 211802935680], [89704339200, 3, 519321747, 199348168320, 294111336960, 744, 1680, 381365832960, 211802935680], [89704339200, 3, 519321747, 355938106344, 180623521680, 744, 1680, 381365832960, 211802935680]], 'outer_group': '1152.157664', 'outer_hash': 6171274809820688557, 'outer_nilpotent': False, 'outer_order': 1152, 'outer_permdeg': 11, 'outer_perms': [8361601, 8726400, 788544, 3, 3729864, 13267704, 16080, 1134264, 8361600], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'S_3\\times C_2^3:S_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 8], [3, 10], [6, 18], [12, 8]], 'representations': {'PC': {'code': '8503859300144481920180003953617440095256166825935127186080', 'gens': [1, 3, 5, 6, 7, 8, 9], 'pres': [9, 2, 3, 2, 3, 2, 2, 2, 2, 2, 18, 23978, 11756, 74, 36454, 12568, 54437, 29660, 13614, 29499, 54439, 37600, 39374, 43028]}, 'Perm': {'d': 15, 'gens': [89704339200, 3, 519321747, 199348169064, 294111338640, 744, 1680, 381365832960, 211802935680]}}, 'schur_multiplier': [2, 2, 2, 2, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 6], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times C_2^5:A_4', 'transitive_degree': 72, 'wreath_data': None, 'wreath_product': False}
