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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '95040.a', 'ambient_counter': 1, 'ambient_order': 95040, 'ambient_tex': 'M_{12}', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 1, 'counter': 59, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '95040.a.2640.b1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '2640.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': 2640, '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': '36.9', 'subgroup_hash': 9, 'subgroup_order': 36, 'subgroup_tex': 'C_3^2:C_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '95040.a', 'aut_centralizer_order': None, 'aut_label': '2640.b1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '95040.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['264.a1.b1', '1320.b1.b1', '1320.c1.b1', '1320.d1.b1'], 'contains': ['5280.c1.a2', '23760.c1.b1'], 'core': '95040.a1.a1', 'coset_action_label': None, 'count': 660, 'diagramx': [7871, -1, 1826, -1, 8000, -1, 1820, -1], 'generators': [3933280, 38260841, 23875433, 19210264], 'label': '95040.a.2640.b1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '660.a1.b1', 'old_label': '2640.b1.b1', 'projective_image': '95040.a', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '2640.b1.b1', 'subgroup_fusion': None, 'weyl_group': '144.182'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [8, 2, 4, 2, 3, 3], 'aut_gens': [[474, 25, 244, 3], [474, 25, 3, 243], [451, 25, 148, 3], [474, 25, 243, 144], [474, 25, 147, 4], [477, 125, 244, 3], [573, 265, 244, 3]], 'aut_group': '144.182', 'aut_hash': 182, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 144, 'aut_permdeg': 9, 'aut_perms': [150716, 86194, 47170, 96371, 173037, 109253], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 4, 2, 1], [4, 9, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_9: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': 24, 'autcentquo_group': '144.182', 'autcentquo_hash': 182, 'autcentquo_nilpotent': False, 'autcentquo_order': 144, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_9:C_2', 'cc_stats': [[1, 1, 1], [2, 9, 1], [3, 4, 2], [4, 9, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '36.9', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 9, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 4, 1, 2], [4, 9, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 6, 'exponent': 12, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 1, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '36.9', 'hash': 9, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [4, 2, 3, 3], 'inner_gens': [[474, 25, 243, 144], [474, 25, 147, 4], [570, 122, 244, 3], [574, 29, 244, 3]], 'inner_hash': 9, 'inner_nilpotent': False, 'inner_order': 36, 'inner_split': False, 'inner_tex': 'C_3^2:C_4', 'inner_used': [1, 3], 'irrC_degree': 4, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [4, 2]], 'label': '36.9', 'linC_count': 2, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 2, 'linQ_dim': 4, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^2:C4', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 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': 6, 'number_divisions': 5, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 10, 'number_subgroups': 38, 'old_label': None, 'order': 36, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 9], [3, 8], [4, 18]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 474], 'outer_gens': [[451, 25, 148, 3], [474, 25, 3, 243]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 6, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 2]], 'representations': {'PC': {'code': 14948871955, 'gens': [1, 3, 4], 'pres': [4, -2, -2, -3, 3, 8, 194, 54, 451, 199]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16858733, 20971525]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [11017, 11777, 8101, 8183]}, 'Perm': {'d': 6, 'gens': [474, 25, 244, 3]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2:C_4', 'transitive_degree': 6, '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}
