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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '14792.f', 'ambient_counter': 6, 'ambient_order': 14792, 'ambient_tex': 'C_{43}:C_{344}', 'central': False, 'central_factor': False, 'centralizer_order': 7396, 'characteristic': False, 'core_order': 2, 'counter': 35, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '14792.f.172.c1.d1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '172.c1.d1', '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': 172, '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': '86.2', 'subgroup_hash': 2, 'subgroup_order': 86, 'subgroup_tex': 'C_{86}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '14792.f', 'aut_centralizer_order': None, 'aut_label': '172.c1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '2.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '86.c1.d1'], 'contains': ['344.c1.d1', '7396.a1.a1'], 'core': '7396.a1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [7034, -1, 5514, -1, 3068, -1, 4246, -1], 'generators': [172, 5512], 'label': '14792.f.172.c1.d1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.a1', 'old_label': '172.c1.d1', 'projective_image': '7396.c', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '172.c1.d1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '86.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 42, 'aut_gen_orders': [42], 'aut_gens': [[1], [3]], 'aut_group': '42.6', 'aut_hash': 6, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 42, 'aut_permdeg': 12, 'aut_perms': [40320873], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [43, 1, 42, 1], [86, 1, 42, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{42}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 42, 'autcent_group': '42.6', 'autcent_hash': 6, 'autcent_nilpotent': True, 'autcent_order': 42, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{42}', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 1], [43, 1, 42], [86, 1, 42]], 'center_label': '86.2', 'center_order': 86, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '43.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['43.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [43, 1, 42, 1], [86, 1, 42, 1]], 'element_repr_type': 'PC', 'elementary': 86, 'eulerian_function': 1, 'exponent': 86, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 43], 'faithful_reps': [[1, 0, 42]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '86.2', 'hash': 2, 'hyperelementary': 86, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 42, 'irrQ_dim': 42, 'irrR_degree': 2, 'irrep_stats': [[1, 86]], 'label': '86.2', 'linC_count': 42, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 42, 'linQ_degree_count': 1, 'linQ_dim': 42, 'linQ_dim_count': 1, 'linR_count': 21, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C86', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 86, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 86, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [43, 42], [86, 42]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 42, 'outer_gen_orders': [42], 'outer_gen_pows': [0], 'outer_gens': [[3]], 'outer_group': '42.6', 'outer_hash': 6, 'outer_nilpotent': True, 'outer_order': 42, 'outer_permdeg': 12, 'outer_perms': [40320873], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{42}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 45, 'pgroup': 0, 'primary_abelian_invariants': [2, 43], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [42, 2]], 'representations': {'PC': {'code': 8861, 'gens': [1], 'pres': [2, -2, -43, 4]}, 'GLFp': {'d': 2, 'p': 43, 'gens': [79551, 3339336]}, 'Perm': {'d': 45, 'gens': [2658271574788448768043625811014615890319638528000000000, 59010256945620955738811989462269485937328128000000000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [86], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{86}', 'transitive_degree': 86, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '344.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 1806, 'aut_gen_orders': [2, 2, 42, 42, 43], 'aut_gens': [[1, 344], [173, 14448], [259, 14448], [313, 14448], [1, 10320], [345, 344]], 'aut_group': None, 'aut_hash': 8820908445430199959, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 303408, 'aut_permdeg': 214, 'aut_perms': [2090864419793148863524221171870968128345246165358248269461871526661322121154314050207836556231154892267947651507996193052518860130720796002056699672329306485653623722992587058375963512923238247514484471445525105704134164036902284781270144510804681689606894572970144968803075679863445570125906675844824603726950234275075796203547966212135395977713254367529678812068374813409539800897300119737744276142719, 3661816244738265728875962866604164432006945434788349576293938832057885449429366642411508360838743689260571246033583340910529256624579828721362343069840595814975703213552182533678991127437688932007605884531180334737955951044331708664631213020276631599015864135707645377993539803296376993820745800752274822255096692004682005850958642584448149493047081363663359124702223054602268745078548119737744276142719, 157954224253733954501802275463052746057971047383839578588222909715526908761344944260152140515846376726746156923873420006718578749265764625903553532852959864707458600963388093907819239123195113594177938138715918094501394623112727724138580399896979533118059603261500903261232443818218648007079957185692664928780014142839609257041559007430034714591794920053545846515506403918683469874365368159109864573159127039, 3492724417145172876652590273557727444586968322547622082591481008392416764905442859935787576105862692477425311893011973020704471187546716508548984240744596933668953675981046704076070210899368067634756267050276396103880929401567543571483014797961407437069235817983693210198130971731599754431775006321425046559204033276453013441856826184384607265437368013288804121036583043485877755416372700149762178853, 22540937467006253869516509729315942935509106176060598621310742231168707161569243525791057694507496688172815359824728638861820797697017662883476952918077630161612599922656056891065897768490443720240345425796280275078841201764049276090538574995973596666072995268941819027023475070755607921293590699990571118593416548279294709391763587421015191472926495916872783845920678974966108017067974579700719594393], 'aut_phi_ratio': 42.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [8, 43, 4, 1], [43, 1, 42, 1], [43, 2, 21, 1], [43, 2, 882, 1], [86, 1, 42, 1], [86, 2, 21, 1], [86, 2, 882, 1], [172, 1, 84, 1], [172, 2, 42, 1], [172, 2, 1764, 1], [344, 43, 168, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{43}.C_{21}^2.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 42, 'autcent_group': '168.57', 'autcent_hash': 57, 'autcent_nilpotent': True, 'autcent_order': 168, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_{42}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 1806, 'autcentquo_group': '1806.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 1806, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{43}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 1, 2], [8, 43, 4], [43, 1, 42], [43, 2, 903], [86, 1, 42], [86, 2, 903], [172, 1, 84], [172, 2, 1806], [344, 43, 168]], 'center_label': '172.2', 'center_order': 172, 'central_product': True, 'central_quotient': '86.1', 'commutator_count': 1, 'commutator_label': '43.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '43.1', '43.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 6, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['344.1', 1], ['43.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [8, 43, 4, 1], [43, 1, 42, 1], [43, 2, 21, 1], [43, 2, 42, 21], [86, 1, 42, 1], [86, 2, 21, 1], [86, 2, 42, 21], [172, 1, 84, 1], [172, 2, 42, 1], [172, 2, 84, 21], [344, 43, 168, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 528, 'exponent': 344, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3, 7, 43], 'factors_of_order': [2, 43], 'faithful_reps': [[2, 0, 1764]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '3698.a', 'hash': 1092357742861769304, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 86, 'inner_gen_orders': [2, 43], 'inner_gens': [[1, 14448], [689, 344]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 86, 'inner_split': True, 'inner_tex': 'D_{43}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 168, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 344], [2, 3612]], 'label': '14792.f', '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': True, 'monomial': True, 'name': 'C43:C344', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 14, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 3956, 'number_divisions': 74, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 19, 'number_subgroup_classes': 79, 'number_subgroups': 226, 'old_label': None, 'order': 14792, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 1], [4, 2], [8, 172], [43, 1848], [86, 1848], [172, 3696], [344, 7224]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 42, 'outer_gen_orders': [42, 42, 42, 42], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[21, 11696], [12963, 10664], [311, 6192], [1551, 4472]], 'outer_group': None, 'outer_hash': 2654381139783726931, 'outer_nilpotent': True, 'outer_order': 3528, 'outer_permdeg': 26, 'outer_perms': [25852144786386074961120, 15511261377585622505476200, 15537062188139589039321216, 15537113279081753998871520], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{42}^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 94, 'pgroup': 0, 'primary_abelian_invariants': [8, 43], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 2], [2, 1], [4, 1], [42, 4], [84, 44], [168, 22]], 'representations': {'PC': {'code': '284159266578894161611565891', 'gens': [1, 5], 'pres': [5, -2, -2, -2, -43, -43, 10, 26, 42, 361204]}, 'GLFp': {'d': 2, 'p': 173, 'gens': [673103214, 30275, 10355436]}, 'Perm': {'d': 94, 'gens': [30439438282041184191625789792993741528654834851957557642056517332, 1181922740319528978634082297209309151094926879348071073784668400973865471703759827366550909781947546313851950693680364165316508687269888000000000, 2350999930694452174824789524415262328897304023133671150446337216810022497000836721771783188694922908243137757910091816292945926931525199084775996]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [344], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{43}:C_{344}', 'transitive_degree': 344, 'wreath_data': None, 'wreath_product': False}
