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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1458.1107', 'ambient_counter': 1107, 'ambient_order': 1458, 'ambient_tex': 'C_3^5:C_6', 'central': False, 'central_factor': False, 'centralizer_order': 243, 'characteristic': False, 'core_order': 1, 'counter': 114, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1458.1107.54.bt1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '54.bt1', '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': 54, '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': '27.5', 'subgroup_hash': 5, 'subgroup_order': 27, 'subgroup_tex': 'C_3^3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1458.1107', 'aut_centralizer_order': 81, 'aut_label': '54.bt1', 'aut_quo_index': None, 'aut_stab_index': 54, 'aut_weyl_group': '4.2', 'aut_weyl_index': 4374, 'centralizer': '6.a1', 'complements': None, 'conjugacy_class_count': 9, 'contained_in': ['18.q1', '18.w1', '18.y1'], 'contains': ['162.bc1', '162.be1', '162.bf1', '162.bl1', '162.bp1', '162.bp2', '162.bp3', '162.bp4'], 'core': '1458.a1', 'coset_action_label': None, 'count': 54, 'diagramx': [7327, -1, 993, -1], 'generators': [487, 57, 162], 'label': '1458.1107.54.bt1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '6.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '6.a1', 'old_label': '54.bt1', 'projective_image': '1458.1107', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '54.bt1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '27.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 312, 'aut_gen_orders': [2, 13], 'aut_gens': [[1, 3, 9], [6, 2, 9], [5, 17, 11]], 'aut_group': '11232.a', 'aut_hash': 778507202365856770, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 11232, 'aut_permdeg': 15, 'aut_perms': [6452863345, 297839648544], 'aut_phi_ratio': 624.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [3, 1, 26, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(3,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 312, 'autcent_group': '11232.a', 'autcent_hash': 778507202365856770, 'autcent_nilpotent': False, 'autcent_order': 11232, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(3,3)', '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], [3, 1, 26]], 'center_label': '27.5', 'center_order': 27, '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': ['3.1', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 3]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 13]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3, 13], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '27.5', 'hash': 5, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 3, 9], [1, 3, 9], [1, 3, 9]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 27]], 'label': '27.5', 'linC_count': 1872, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 234, 'linQ_dim': 6, 'linQ_dim_count': 234, 'linR_count': 234, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^3', 'ngens': 3, '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': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 27, 'number_divisions': 14, 'number_normal_subgroups': 28, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 28, 'number_subgroups': 28, 'old_label': None, 'order': 27, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 26]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 312, 'outer_gen_orders': [2, 13], 'outer_gen_pows': [0, 0], 'outer_gens': [[6, 2, 9], [5, 17, 11]], 'outer_group': '11232.a', 'outer_hash': 778507202365856770, 'outer_nilpotent': False, 'outer_order': 11232, 'outer_permdeg': 15, 'outer_perms': [6452863345, 297839648544], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\GL(3,3)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 3, 'primary_abelian_invariants': [3, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 13]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3], 'pres': [3, -3, 3, 3]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [41624334336939899, 125101725607167719, 125101750005091125]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [10475581, 9534373, 23068812]}, 'Perm': {'d': 9, 'gens': [80640, 240, 4]}}, 'schur_multiplier': [3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 3], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3', 'transitive_degree': 27, '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': 3, 'aut_exponent': 6, 'aut_gen_orders': [3, 6, 6, 6, 3, 2, 3, 6, 3, 2], 'aut_gens': [[1, 3, 9, 162, 486], [1, 3, 819, 162, 486], [1, 3, 66, 324, 972], [59, 114, 64, 324, 972], [1, 3, 63, 810, 972], [1, 3, 9, 1134, 486], [1, 114, 50, 324, 486], [115, 57, 15, 162, 486], [55, 3, 63, 1296, 972], [1, 3, 495, 162, 486], [1, 3, 9, 324, 972]], 'aut_group': '17496.rs', 'aut_hash': 5827827197862549205, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 17496, 'aut_permdeg': 81, 'aut_perms': [353398522802987882633808833438952556639864618366170987794300784081393700342505936152190289387923083946262158569569252870, 64432631549894092301261951849891767170068413603544288468335150383958528989138519892963946197910560343827530127964951606, 36235025871400281530561398675934985197925408655782589122330045580853059983598278378223213952919866336366564997856255847, 60850263261227743866214071344136245957944343592613092230816487733080559757323330415224036022696269492338733629792820278, 22651080881560701085909056944764649670665949748381657471866007086251571062265567807265059808976214148909301760000, 62636860003421435067527540570475391985376271763688957836256488768356802391666155758606849378464534270697445700608011646, 30292747199639296909079962380431867753879935565279620914243568699146446098757786637938660063391303762885412813963036570, 10431353062943221976085427761755157484818610631220417057684416660301890439943756131821089072030987359013358714226082043, 716600680816211898273150686647302650813744822779027295587493823608870544070556311469099058611709507016714343811317760000, 9853657631155837247868817172187283056348376424211633795241770271785241708898219936141586681277167410079054874786175483], 'aut_phi_ratio': 36.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 3, 2, 1], [3, 3, 6, 1], [3, 6, 1, 1], [3, 6, 2, 2], [3, 6, 6, 2], [3, 6, 18, 1], [3, 27, 2, 1], [3, 54, 2, 1], [6, 9, 2, 1], [6, 27, 2, 1], [6, 27, 6, 1], [6, 81, 2, 1], [9, 27, 4, 1], [9, 54, 4, 1], [18, 81, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^4.S_3^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 3, 'autcent_group': '9.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 9, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '1944.3569', 'autcentquo_hash': 3569, 'autcentquo_nilpotent': False, 'autcentquo_order': 1944, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^2:S_3^3', 'cc_stats': [[1, 1, 1], [2, 9, 1], [3, 1, 2], [3, 2, 3], [3, 3, 8], [3, 6, 35], [3, 27, 2], [3, 54, 2], [6, 9, 2], [6, 27, 8], [6, 81, 2], [9, 27, 4], [9, 54, 4], [18, 81, 4]], 'center_label': '3.1', 'center_order': 3, 'central_product': False, 'central_quotient': '486.102', 'commutator_count': 1, 'commutator_label': '81.15', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1107, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 3, 2, 4], [3, 6, 1, 1], [3, 6, 2, 17], [3, 27, 2, 1], [3, 54, 2, 1], [6, 9, 2, 1], [6, 27, 2, 4], [6, 81, 2, 1], [9, 27, 2, 2], [9, 54, 2, 2], [18, 81, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 36, 'exponent': 18, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 0, 18]], 'familial': False, 'frattini_label': '27.5', 'frattini_quotient': '54.12', 'hash': 1107, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 3, 6, 3, 3], 'inner_gens': [[1, 3, 123, 162, 486], [1, 3, 63, 162, 486], [4, 111, 9, 1296, 972], [1, 3, 819, 162, 486], [1, 3, 981, 162, 486]], 'inner_hash': 102, 'inner_nilpotent': False, 'inner_order': 486, 'inner_split': True, 'inner_tex': 'C_3^4:C_6', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 6, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 18], [2, 9], [3, 16], [6, 35]], 'label': '1458.1107', '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': 'C3^5:C6', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 22, 'number_characteristic_subgroups': 22, 'number_conjugacy_classes': 78, 'number_divisions': 41, 'number_normal_subgroups': 28, 'number_subgroup_autclasses': 205, 'number_subgroup_classes': 650, 'number_subgroups': 4128, 'old_label': None, 'order': 1458, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 9], [3, 404], [6, 396], [9, 324], [18, 324]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [6, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[55, 114, 104, 324, 486], [8, 60, 118, 810, 972]], 'outer_group': '36.12', 'outer_hash': 12, 'outer_nilpotent': False, 'outer_order': 36, 'outer_permdeg': 8, 'outer_perms': [750, 10081], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6\\times S_3', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 9], [4, 4], [6, 9], [12, 17]], 'representations': {'PC': {'code': 701874495448638044126492850222843503, 'gens': [1, 2, 3, 6, 7], 'pres': [7, 3, 3, 2, 3, 3, 3, 3, 2585, 450, 58, 591, 1186, 108, 6067, 2672, 5312]}, 'Perm': {'d': 18, 'gens': [3402106385166720, 2267507354388480, 79833604, 93485145843, 174400128243, 376703666922484, 711549292688884]}}, 'schur_multiplier': [3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^5:C_6', 'transitive_degree': 18, 'wreath_data': None, 'wreath_product': False}
