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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1376.5', 'ambient_counter': 5, 'ambient_order': 1376, 'ambient_tex': 'C_2^2:C_{344}', 'central': False, 'central_factor': False, 'centralizer_order': 688, 'characteristic': False, 'core_order': 172, 'counter': 10, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '1376.5.4.e1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '4.e1.a1', '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': 4, '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': '344.2', 'subgroup_hash': 2, 'subgroup_order': 344, 'subgroup_tex': 'C_{344}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1376.5', 'aut_centralizer_order': 4, 'aut_label': '4.e1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '168.57', 'aut_weyl_index': 16, 'centralizer': '2.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.b1.a1'], 'contains': ['8.c1.a1', '172.e1.a1'], 'core': '8.c1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [9390, -1, 570, -1, 271, -1, 7940, -1], 'generators': [2, 32, 8, 4], 'label': '1376.5.4.e1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.b1.a1', 'old_label': '4.e1.a1', 'projective_image': '8.3', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.e1.a1', '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': '344.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 42, 'aut_gen_orders': [2, 2, 42], 'aut_gens': [[1], [259], [173], [89]], 'aut_group': '168.57', 'aut_hash': 57, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 168, 'aut_permdeg': 46, 'aut_perms': [109050989124884473699447170889578704597969879389173989904, 87784816526576883555098164401461777476264286867995897104, 242091245115638712176566319733838055767028510100114067212], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [8, 1, 4, 1], [43, 1, 42, 1], [86, 1, 42, 1], [172, 1, 84, 1], [344, 1, 168, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times C_{42}', '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': 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], [4, 1, 2], [8, 1, 4], [43, 1, 42], [86, 1, 42], [172, 1, 84], [344, 1, 168]], 'center_label': '344.2', 'center_order': 344, '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', '2.1', '2.1', '43.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['43.1', 1], ['8.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [8, 1, 4, 1], [43, 1, 42, 1], [86, 1, 42, 1], [172, 1, 84, 1], [344, 1, 168, 1]], 'element_repr_type': 'PC', 'elementary': 86, 'eulerian_function': 1, 'exponent': 344, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 43], 'faithful_reps': [[1, 0, 168]], 'familial': True, 'frattini_label': '4.1', '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': 168, 'irrQ_dim': 168, 'irrR_degree': 2, 'irrep_stats': [[1, 344]], 'label': '344.2', 'linC_count': 168, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 46, 'linQ_degree_count': 2, 'linQ_dim': 46, 'linQ_dim_count': 2, 'linR_count': 84, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C344', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 344, 'number_divisions': 8, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 344, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 1], [4, 2], [8, 4], [43, 42], [86, 42], [172, 84], [344, 168]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 42, 'outer_gen_orders': [2, 2, 42], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[259], [173], [89]], 'outer_group': '168.57', 'outer_hash': 57, 'outer_nilpotent': True, 'outer_order': 168, 'outer_permdeg': 46, 'outer_perms': [109050989124884473699447170889578704597969879389173989904, 87784816526576883555098164401461777476264286867995897104, 242091245115638712176566319733838055767028510100114067212], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{42}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 51, 'pgroup': 0, 'primary_abelian_invariants': [8, 43], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 1], [42, 2], [84, 1], [168, 1]], 'representations': {'PC': {'code': 212235014905241, 'gens': [1], 'pres': [4, -2, -2, -2, -43, 8, 21, 34]}, 'GLFp': {'d': 2, 'p': 173, 'gens': [2210075]}, 'Perm': {'d': 51, 'gens': [216599033751535440633927791155991262120042770706947112960000000000, 59010256945620955738811989462269485937328128000000000, 92458860080319590419809567499679506678988139383362682880000000000, 30426512622586345503160869073461196830685671907286581248000000000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [344], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{344}', 'transitive_degree': 344, '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': '688.22', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [2, 42, 42, 84, 84, 6], 'aut_gens': [[1, 2, 16], [9, 10, 16], [1, 10, 1232], [689, 690, 144], [689, 11, 976], [697, 695, 528], [1, 698, 784]], 'aut_group': None, 'aut_hash': 7790671656276901314, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 2688, 'aut_permdeg': 58, 'aut_perms': [115504089006362473252940320088361393952303276335715627030949078578853910, 1318622727406602554750685513818429715076150552758109188161103263913932219570184, 1743855228536179521681154808030833466534286880683820932494041450878756066819097, 1137830720358858651179867253260826643215576368422280121086736297444668729250652, 2015404426528627413080809149942778541566047715915458404801335481010175553853927, 1409827516407082266538928987994271172740109515248265712271994009866465438517769], 'aut_phi_ratio': 4.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 2, 1], [4, 1, 4, 1], [4, 2, 2, 1], [8, 2, 8, 1], [43, 1, 42, 1], [86, 1, 42, 3], [86, 2, 84, 1], [172, 1, 168, 1], [172, 2, 84, 1], [344, 2, 336, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{21}:(C_2^4.C_2^3)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 42, 'autcent_group': '1344.11720', 'autcent_hash': 11720, 'autcent_nilpotent': True, 'autcent_order': 1344, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^5\\times C_{42}', '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, 2], [4, 1, 4], [4, 2, 2], [8, 2, 8], [43, 1, 42], [86, 1, 126], [86, 2, 84], [172, 1, 168], [172, 2, 84], [344, 2, 336]], 'center_label': '344.8', 'center_order': 344, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '43.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['32.5', 1], ['43.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [4, 1, 2, 2], [4, 2, 2, 1], [8, 2, 4, 2], [43, 1, 42, 1], [86, 1, 42, 3], [86, 2, 42, 2], [172, 1, 84, 2], [172, 2, 84, 1], [344, 2, 168, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 264, 'exponent': 344, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 43], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '172.4', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 1], 'inner_gens': [[1, 690, 16], [689, 2, 16], [1, 2, 16]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': None, 'irrep_stats': [[1, 688], [2, 172]], 'label': '1376.5', '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': 'C2^2:C344', 'ngens': 6, 'nilpotency_class': 2, 'nilpotent': True, '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': 16, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 860, 'number_divisions': 22, 'number_normal_subgroups': 32, 'number_subgroup_autclasses': 38, 'number_subgroup_classes': 50, 'number_subgroups': 68, 'old_label': None, 'order': 1376, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 7], [4, 8], [8, 16], [43, 42], [86, 294], [172, 336], [344, 672]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 84, 'outer_gen_orders': [2, 2, 2, 84], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 14, 16], [9, 2, 16], [1, 2, 1360], [9, 11, 208]], 'outer_group': '672.1249', 'outer_hash': 1249, 'outer_nilpotent': True, 'outer_order': 672, 'outer_permdeg': 18, 'outer_perms': [19678982400, 1313941305600, 6266937600, 1108927543453800], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:C_{42}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 55, 'pgroup': 0, 'primary_abelian_invariants': [2, 8, 43], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 3], [42, 4], [84, 4], [168, 3]], 'representations': {'PC': {'code': 959701495507207560614037929, 'gens': [1, 2, 5], 'pres': [6, -2, -2, -2, -2, -2, -43, 8281, 31, 50, 88]}, 'Perm': {'d': 55, 'gens': [235199270454485484889782128579939414630002677237461115834728448000000000, 230924418168220428849198408664174823935461500750528178133401600000000000, 470237161246602812072781512872833961465937816720349509713920000000000000, 470237254972199529314073681471640872687835973597618896518512640000000000, 62653808104498534005344290839085207646673279353723289600000000000, 59010256945620955738811989462269485937328128000000000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 344], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2:C_{344}', 'transitive_degree': 688, 'wreath_data': None, 'wreath_product': False}
