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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '95040.a', 'ambient_counter': 1, 'ambient_order': 95040, 'ambient_tex': 'M_{12}', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 1, 'counter': 82, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '95040.a.4752.b1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '4752.b1.b1', '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': 4752, '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': '20.3', 'subgroup_hash': 3, 'subgroup_order': 20, 'subgroup_tex': 'F_5', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '95040.a', 'aut_centralizer_order': None, 'aut_label': '4752.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '47520.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['132.b1.b1', '792.a1.b1', '792.c1.b1', '2376.a1.a1'], 'contains': ['9504.c1.a1', '23760.c1.b1'], 'core': '95040.a1.a1', 'coset_action_label': None, 'count': 2376, 'diagramx': [4795, -1, 6688, -1, 4411, -1, 5941, -1], 'generators': [12869049, 3933280, 16612361], 'label': '95040.a.4752.b1.b1', 'mobius_quo': None, 'mobius_sub': -2, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2376.a1.a1', 'old_label': '4752.b1.b1', 'projective_image': '95040.a', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4752.b1.b1', 'subgroup_fusion': None, 'weyl_group': '20.3'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [4, 5], 'aut_gens': [[1, 4], [1, 12], [5, 4]], 'aut_group': '20.3', 'aut_hash': 3, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20, 'aut_permdeg': 5, 'aut_perms': [18, 34], 'aut_phi_ratio': 2.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [4, 5, 1, 2], [5, 4, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'F_5', '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': 20, 'autcentquo_group': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 5, 1], [4, 5, 2], [5, 4, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '20.3', 'commutator_count': 1, 'commutator_label': '5.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [4, 5, 2, 1], [5, 4, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 20, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[4, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '20.3', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [4, 5], 'inner_gens': [[1, 12], [13, 4]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 20, 'inner_split': True, 'inner_tex': 'F_5', 'inner_used': [1, 2], 'irrC_degree': 4, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [4, 1]], 'label': '20.3', 'linC_count': 1, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'F5', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 5, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 14, 'old_label': None, 'order': 20, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 5], [4, 10], [5, 4]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 5, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 1]], 'representations': {'PC': {'code': 2551363, 'gens': [1, 3], 'pres': [3, -2, -2, -5, 6, 110, 77]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [23561683, 28172809]}, 'Lie': [{'d': 1, 'q': 5, 'gens': [33, 10], 'family': 'AGL'}, {'d': 1, 'q': 5, 'gens': [33, 10], 'family': 'AGammaL'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 127]}, 'Perm': {'d': 5, 'gens': [10, 23, 96]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'F_5', 'transitive_degree': 5, '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': '1.1', 'all_subgroups_known': True, 'almost_simple': True, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 1320, 'aut_gen_orders': [8, 12], 'aut_gens': [[407173912, 330985800], [123635636, 156831144], [127948579, 454128198]], 'aut_group': '190080.c', 'aut_hash': 6376759384993201448, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 190080, 'aut_permdeg': 396, 'aut_perms': [1261160693965387019534556708223600054928013576202146567821561679784367911149259213062343475318919617726454304290018868374424307225596014564561675666819592207899633567475629780179056245104310613609369849702537030927249962479959096545941110961227723804855619525486132440177409137348698082605302055915122735332587313200579164951136419945277959152018148369772958302770985574499221621063616872494168582305533708553301613664189857812644642409438510988329774463747278274699067243996477418831726568033178069004412790333750774266472052159765137729501525330346518065576574616441708789929687809009890836108944247972706186506953451826137088928920092398050649903274979595232590930465301514308432637411641679965471774261707722064232433525823398086212162679553772770520588303647355514788714377951804152389616001330542004742817955639777867130977901953333778842709505550324213, 1500725537165574936229534458886610626079018171151967287255890872373692760786984166723824795387397240599321337838204387140661669511150300828915338239464944744691978489578659466407198714467156457558863248746660615440131598495671696332942427024699400362862657545653867286649783710407412460789422211966610117562424404137931271596916594578883982173694828483075864367790976357675725305318725603062020876148741240132970770004754510095498659492362851947579245031328712441046713233531292985424206674801108571381011902899496241669351291540800573332507534699150684604794909239846429398795259448484296716602930561430093461992170290508621627992985621018009503445000043841688419985313039266066502037797253248665421904681618798649745800620207809278364005304292965327754281347223342347610714470983827767971550508528898429146412353669706639269434820132027403111652097566687104], 'aut_phi_ratio': 8.25, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 396, 1, 1], [2, 495, 1, 1], [3, 1760, 1, 1], [3, 2640, 1, 1], [4, 2970, 2, 1], [5, 9504, 1, 1], [6, 7920, 1, 1], [6, 15840, 1, 1], [8, 11880, 2, 1], [10, 9504, 1, 1], [11, 8640, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'M_{12}: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': 1320, 'autcentquo_group': '190080.c', 'autcentquo_hash': 6376759384993201448, 'autcentquo_nilpotent': False, 'autcentquo_order': 190080, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'M_{12}:C_2', 'cc_stats': [[1, 1, 1], [2, 396, 1], [2, 495, 1], [3, 1760, 1], [3, 2640, 1], [4, 2970, 2], [5, 9504, 1], [6, 7920, 1], [6, 15840, 1], [8, 11880, 2], [10, 9504, 1], [11, 8640, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '95040.a', 'commutator_count': 1, 'commutator_label': '95040.a', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['95040.a'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 396, 1, 1], [2, 495, 1, 1], [3, 1760, 1, 1], [3, 2640, 1, 1], [4, 2970, 1, 2], [5, 9504, 1, 1], [6, 7920, 1, 1], [6, 15840, 1, 1], [8, 11880, 1, 2], [10, 9504, 1, 1], [11, 8640, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 38664, 'exponent': 1320, 'exponents_of_order': [6, 3, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 5, 11], 'faithful_reps': [[11, 1, 2], [16, 0, 2], [45, 1, 1], [54, 1, 1], [55, 1, 3], [66, 1, 1], [99, 1, 1], [120, 1, 1], [144, 1, 1], [176, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '95040.a', 'hash': 4946793380810623143, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 1320, 'inner_gen_orders': [5, 8], 'inner_gens': [[407173912, 435547359], [352046079, 330985800]], 'inner_hash': 4946793380810623143, 'inner_nilpotent': False, 'inner_order': 95040, 'inner_split': True, 'inner_tex': 'M_{12}', 'inner_used': [1, 2], 'irrC_degree': 11, 'irrQ_degree': 11, 'irrQ_dim': 11, 'irrR_degree': 11, 'irrep_stats': [[1, 1], [11, 2], [16, 2], [45, 1], [54, 1], [55, 3], [66, 1], [99, 1], [120, 1], [144, 1], [176, 1]], 'label': '95040.a', 'linC_count': 2, 'linC_degree': 11, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 11, 'linQ_degree_count': 2, 'linQ_dim': 11, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 11, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'M12', '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, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 12, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 15, 'number_divisions': 14, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 107, 'number_subgroup_classes': 147, 'number_subgroups': 214871, 'old_label': None, 'order': 95040, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 891], [3, 4400], [4, 5940], [5, 9504], [6, 23760], [8, 23760], [10, 9504], [11, 17280]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [68854328], 'outer_gens': [[122352811, 89308026]], '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': None, 'perfect': True, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [], 'quasisimple': True, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [11, 2], [32, 1], [45, 1], [54, 1], [55, 3], [66, 1], [99, 1], [120, 1], [144, 1], [176, 1]], 'representations': {'Perm': {'d': 12, 'gens': [330985800, 407173912]}}, 'schur_multiplier': [2], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'M_{12}', 'transitive_degree': 12, 'wreath_data': None, 'wreath_product': False}
