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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1056.85', 'ambient_counter': 85, 'ambient_order': 1056, 'ambient_tex': 'D_{44}:C_{12}', 'central': False, 'central_factor': False, 'centralizer_order': 24, 'characteristic': False, 'core_order': 1, 'counter': 98, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '1056.85.528.d1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '528.d1.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': 528, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': True, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1056.85', 'aut_centralizer_order': 160, 'aut_label': '528.d1', 'aut_quo_index': None, 'aut_stab_index': 88, 'aut_weyl_group': '1.1', 'aut_weyl_index': 14080, 'centralizer': '44.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['48.d1.a1', '176.d1.a1', '264.d1.a1', '264.e1.a1', '264.f1.a1'], 'contains': ['1056.a1.a1'], 'core': '1056.a1.a1', 'coset_action_label': None, 'count': 44, 'diagramx': [7484, -1, 6136, -1, 2161, -1, 2039, -1], 'generators': [5], 'label': '1056.85.528.d1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '12.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '44.b1.a1', 'old_label': '528.d1.a1', 'projective_image': '1056.85', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '528.d1.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': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], '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': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, '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': '24.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 220, 'aut_gen_orders': [44, 10, 20, 22, 20, 20, 10, 4], 'aut_gens': [[1, 2, 8], [653, 794, 712], [817, 266, 56], [889, 266, 1000], [889, 2, 536], [125, 794, 296], [413, 798, 424], [365, 2, 152], [557, 794, 520]], 'aut_group': None, 'aut_hash': 4279695281879324969, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 14080, 'aut_permdeg': 98, 'aut_perms': [7433627951489627561067983085278834637308924234002751208137286147972171309313951087698711302961133347182473220763916571569355459950028899496656956103908403, 6893351698344636491392519897299797722130104914841135313242459404096848680851930617178321108547586897372731941063343577937110346864842091156052206406045044, 7521762564959631770549705357931557737750287201985882485003008982759736157663601183645336665357410841472342556637565395304255321353031456153768985979630118, 962543594517973384823492143431872364900070712408690076348918674628103383693464884488902435636260589889545965168759916673413509947605991665418468761620016, 6368221523527801265379560590945205998980280392998821863087362869063272618579577369692634779366379346190569657253688810736747030222974475965900787115829975, 4490871076715793315656703672597020859742881609856123950333225957810791094572147269712146008594377452337478088161770643394707411983755503507058318885331807, 5916277197442842216166102912004867764804552517126850829244944635725758240118915043539242044563565465094784592691236413576741023063950850822954451821710912, 5219733263011781438619848206040432611884732840762409481006813563159486908496052931618484015440436701636429642602081287981941707068655118203089994295023203], 'aut_phi_ratio': 44.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 44, 2, 1], [3, 1, 2, 1], [4, 2, 1, 2], [4, 4, 2, 1], [6, 1, 2, 3], [6, 44, 4, 1], [8, 22, 4, 1], [11, 2, 5, 1], [12, 2, 2, 2], [12, 4, 4, 1], [22, 2, 5, 3], [24, 22, 8, 1], [33, 2, 10, 1], [44, 4, 5, 2], [44, 4, 20, 1], [66, 2, 10, 3], [132, 4, 10, 2], [132, 4, 40, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_4\\times C_{11}:C_5).C_2^6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '32.51', 'autcent_hash': 51, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^5', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 110, 'autcentquo_group': '440.42', 'autcentquo_hash': 42, 'autcentquo_nilpotent': False, 'autcentquo_order': 440, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 44, 2], [3, 1, 2], [4, 2, 2], [4, 4, 2], [6, 1, 6], [6, 44, 4], [8, 22, 4], [11, 2, 5], [12, 2, 4], [12, 4, 4], [22, 2, 15], [24, 22, 8], [33, 2, 10], [44, 4, 30], [66, 2, 30], [132, 4, 60]], 'center_label': '12.5', 'center_order': 12, 'central_product': True, 'central_quotient': '88.7', 'commutator_count': 1, 'commutator_label': '44.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 85, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['352.15', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 44, 1, 2], [3, 1, 2, 1], [4, 2, 1, 2], [4, 4, 2, 1], [6, 1, 2, 3], [6, 44, 2, 2], [8, 22, 4, 1], [11, 2, 5, 1], [12, 2, 2, 2], [12, 4, 4, 1], [22, 2, 5, 3], [24, 22, 8, 1], [33, 2, 10, 1], [44, 4, 5, 2], [44, 4, 20, 1], [66, 2, 10, 3], [132, 4, 10, 2], [132, 4, 40, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 264, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 3, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '132.7', 'hash': 85, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 44, 'inner_gen_orders': [2, 2, 22], 'inner_gens': [[1, 794, 344], [265, 2, 536], [721, 530, 8]], 'inner_hash': 7, 'inner_nilpotent': False, 'inner_order': 88, 'inner_split': True, 'inner_tex': 'C_{11}:D_4', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 24], [2, 138], [4, 30]], 'label': '1056.85', 'linC_count': 640, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 60, 'linQ_dim': 18, 'linQ_dim_count': 60, 'linR_count': 360, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D44:C12', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 32, 'number_characteristic_subgroups': 42, 'number_conjugacy_classes': 192, 'number_divisions': 34, 'number_normal_subgroups': 46, 'number_subgroup_autclasses': 88, 'number_subgroup_classes': 100, 'number_subgroups': 804, 'old_label': None, 'order': 1056, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 91], [3, 2], [4, 12], [6, 182], [8, 88], [11, 10], [12, 24], [22, 30], [24, 176], [33, 20], [44, 120], [66, 60], [132, 240]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 10, 'outer_gen_orders': [2, 2, 2, 2, 10], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[529, 2, 872], [5, 2, 872], [1, 6, 872], [1, 2, 520], [1, 2, 584]], 'outer_group': '160.238', 'outer_hash': 238, 'outer_nilpotent': True, 'outer_order': 160, 'outer_permdeg': 15, 'outer_perms': [87178291200, 479001600, 3628800, 40320, 753], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4\\times C_{10}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 6], [8, 2], [10, 2], [20, 7], [40, 5]], 'representations': {'PC': {'code': 107412984140890247622528942514655845060000019, 'gens': [1, 2, 4], 'pres': [7, -2, -2, -2, -2, -2, -3, -11, 11117, 36, 9635, 7514, 80, 24084, 102, 13445, 166, 47046]}, 'Perm': {'d': 26, 'gens': [621574841786321730667687, 44800577, 93405312000, 94484887, 132180480, 94484880, 16754359726882706632704000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 12], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{44}:C_{12}', 'transitive_degree': 528, 'wreath_data': None, 'wreath_product': False}
