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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '256.4554', 'ambient_counter': 4554, 'ambient_order': 256, 'ambient_tex': '(C_2^3\\times C_4).D_4', 'central': True, 'central_factor': False, 'centralizer_order': 256, 'characteristic': True, 'core_order': 2, 'counter': 238, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '256.4554.128.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '128.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '128.859', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 859, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 128, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^4.D_4', 'simple': True, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': True, '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': '256.4554', 'aut_centralizer_order': None, 'aut_label': '128.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['64.a1.a1', '64.d1.a1', '64.e1.a1', '64.f1.a1', '64.g1.a1', '64.h1.a1', '64.i1.a1', '64.j1.a1', '64.k1.a1'], 'contains': ['256.a1.a1'], 'core': '128.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [128], 'label': '256.4554.128.a1.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '128.a1.a1', 'normal_contained_in': ['64.a1.a1'], 'normal_contains': ['256.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '128.a1.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '128.a1.a1', 'subgroup_fusion': None, 'weyl_group': None}
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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': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4, 2, 4, 4, 4, 4, 4], 'aut_gens': [[1, 4, 16, 64], [43, 188, 16, 224], [145, 148, 184, 64], [137, 36, 16, 192], [195, 132, 24, 96], [115, 20, 144, 96], [203, 148, 144, 64], [17, 148, 184, 96], [195, 12, 56, 192]], 'aut_group': None, 'aut_hash': 5074040165636225497, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 4096, 'aut_permdeg': 80, 'aut_perms': [20444710710774564985177209771245365628953936662367577509314946798973025673346537949293445977549995656213875089613662317, 15621448355732972785184071404033980872675412072013521620103143478506193553584039161566735130243308625816295428485968316, 15557964103119327606452876504751542284443498312555989439022259838243664529358983540469068476987786018860820034548041431, 53436211184657174653671928950334601873093234823996145185533715805890933955373174666337465653084886902267832618322034420, 58803801850952988408948710144338742566167051245040221064898603344581623747980283233147593752453659460621383262451372996, 18637806501072822349079874927140032144530705265965711569658382768323396031786593515827644639833564139373691649471848960, 68684004578791068557496212656390474754962614561508770779030837482771449504193841190500633244739949822476283526147732479, 58731609698623016812631376293460253501992425220329276416761473927027496154521329610046694865330409168179869033595056729], 'aut_phi_ratio': 32.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 4, 2, 1], [2, 8, 1, 1], [2, 16, 1, 1], [4, 2, 2, 1], [4, 4, 1, 1], [4, 4, 2, 3], [4, 8, 1, 1], [4, 8, 2, 2], [4, 16, 1, 1], [8, 8, 8, 1], [8, 16, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^8.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '64.202', 'autcentquo_hash': 202, 'autcentquo_nilpotent': True, 'autcentquo_order': 64, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^3:D_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 4, 2], [2, 8, 1], [2, 16, 1], [4, 2, 2], [4, 4, 7], [4, 8, 5], [4, 16, 1], [8, 8, 8], [8, 16, 4]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '64.90', 'commutator_count': 1, 'commutator_label': '16.10', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 4554, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 4, 1, 2], [2, 8, 1, 1], [2, 16, 1, 1], [4, 2, 2, 1], [4, 4, 1, 3], [4, 4, 2, 2], [4, 8, 1, 3], [4, 8, 2, 1], [4, 16, 1, 1], [8, 8, 4, 2], [8, 16, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1344, 'exponent': 8, 'exponents_of_order': [8], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '32.23', 'frattini_quotient': '8.5', 'hash': 4554, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [4, 4, 2, 2], 'inner_gens': [[1, 156, 152, 64], [185, 4, 16, 224], [137, 4, 16, 64], [1, 164, 16, 64]], 'inner_hash': 90, 'inner_nilpotent': True, 'inner_order': 64, 'inner_split': None, 'inner_tex': 'C_2^4:C_4', 'inner_used': [1, 2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 12], [4, 8], [8, 1]], 'label': '256.4554', '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': '(C2^3*C4).D4', 'ngens': 3, 'nilpotency_class': 4, 'nilpotent': True, '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': 20, 'number_characteristic_subgroups': 39, 'number_conjugacy_classes': 37, 'number_divisions': 25, 'number_normal_subgroups': 51, 'number_subgroup_autclasses': 199, 'number_subgroup_classes': 247, 'number_subgroups': 879, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 39], [4, 88], [8, 128]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4], 'outer_gen_pows': [0, 154, 154, 18, 0], 'outer_gens': [[43, 188, 16, 224], [25, 180, 184, 192], [185, 180, 184, 224], [17, 148, 184, 96], [219, 28, 176, 64]], 'outer_group': '64.261', 'outer_hash': 261, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 32, 'outer_perms': [8268628444230288865648587289804816, 18038196755204260493677924422300247, 112260373682779536584766187963204207, 52090320484070171814847344681975168, 146321174753995094309650109067370298], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_2^3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 24, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [4, 4], [8, 5]], 'representations': {'PC': {'code': 208351728608403483687705687424583511, 'gens': [1, 3, 5, 7], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 16, 1033, 3746, 1690, 66, 1283, 6084, 972, 116, 3158, 166]}, 'Perm': {'d': 24, 'gens': [30823692590920178845863, 58471055449100406970370, 85947181262698245651694, 111787520114829945659909, 138277539401031942925294, 10169658974225696175844, 165493671219134058908644, 165493671219134058890880]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_2^3\\times C_4).D_4', 'transitive_degree': 32, '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': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 8, 'aut_gen_orders': [2, 4, 8, 4, 2, 2, 2], 'aut_gens': [[1, 2, 8, 32], [1, 23, 88, 96], [17, 83, 72, 40], [65, 39, 72, 32], [17, 11, 24, 48], [17, 18, 24, 96], [1, 18, 8, 48], [1, 66, 8, 32]], 'aut_group': '1024.dnl', 'aut_hash': 454949056087899745, 'aut_nilpotency_class': 4, 'aut_nilpotent': True, 'aut_order': 1024, 'aut_permdeg': 32, 'aut_perms': [100414795240405806888590965839612792, 227725482748526830191907336617153578, 156424791988753458629266170407386412, 114373803356749651352900683219216781, 132062060875511454089921909158913812, 4253203942958765799836767412308383, 133586507436308610366929320990404538], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 2, 2, 1], [2, 4, 2, 2], [2, 8, 1, 1], [4, 4, 1, 2], [4, 8, 1, 3], [4, 8, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^4.D_4^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '128.1755', 'autcentquo_hash': 1755, 'autcentquo_nilpotent': True, 'autcentquo_order': 128, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^4:D_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 3], [2, 4, 4], [2, 8, 1], [4, 4, 2], [4, 8, 11]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '64.90', 'commutator_count': 1, 'commutator_label': '8.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 859, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 4, 1, 4], [2, 8, 1, 1], [4, 4, 1, 2], [4, 8, 1, 3], [4, 8, 2, 4]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 672, 'exponent': 4, 'exponents_of_order': [7], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[8, 1, 1]], 'familial': False, 'frattini_label': '16.11', 'frattini_quotient': '8.5', 'hash': 859, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4, 2, 4], 'inner_gens': [[1, 2, 8, 48], [1, 2, 88, 56], [1, 82, 8, 32], [17, 10, 8, 32]], 'inner_hash': 90, 'inner_nilpotent': True, 'inner_order': 64, 'inner_split': False, 'inner_tex': 'C_2^4:C_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 4], [4, 2], [8, 1]], 'label': '128.859', 'linC_count': 1, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 1, 'linQ_dim': 8, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^4.D4', 'ngens': 3, 'nilpotency_class': 4, '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': 13, 'number_characteristic_subgroups': 22, 'number_conjugacy_classes': 23, 'number_divisions': 19, 'number_normal_subgroups': 42, 'number_subgroup_autclasses': 87, 'number_subgroup_classes': 143, 'number_subgroups': 428, 'old_label': None, 'order': 128, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 31], [4, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [37, 0, 0, 0], 'outer_gens': [[65, 43, 88, 120], [1, 6, 88, 32], [1, 3, 24, 32], [17, 2, 8, 32]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [126, 55, 289, 288], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [4, 2], [8, 1]], 'representations': {'PC': {'code': 9976054716566501535228970, 'gens': [1, 2, 4, 6], 'pres': [7, 2, 2, 2, 2, 2, 2, 2, 36, 1242, 185, 80, 2021, 1188, 1027, 124, 1973]}, 'GLFp': {'d': 5, 'p': 2, 'gens': [19241027, 6210625, 19208257, 18750531, 30996566, 6670401, 6276161]}, 'Perm': {'d': 16, 'gens': [11219175095753, 11212821043200, 4103734089616, 5606275034856, 5606234766719, 2789792421136, 1313941673647]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^4.D_4', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
