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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '856.12', 'ambient_counter': 12, 'ambient_order': 856, 'ambient_tex': 'C_2^2\\times C_{214}', 'central': True, 'central_factor': False, 'centralizer_order': 856, 'characteristic': True, 'core_order': 107, 'counter': 16, 'cyclic': True, 'direct': True, 'hall': 107, 'label': '856.12.8.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '8.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '8.5', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 5, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^3', 'simple': True, 'solvable': True, 'special_labels': ['C3'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '107.1', 'subgroup_hash': 1, 'subgroup_order': 107, 'subgroup_tex': 'C_{107}', 'supersolvable': True, 'sylow': 107}
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gps_subgroup_data • Show schema
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{'ambient': '856.12', 'aut_centralizer_order': 168, 'aut_label': '8.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '106.2', 'aut_weyl_index': 168, 'centralizer': '1.a1.a1', 'complements': ['107.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '4.a1.b1', '4.a1.c1', '4.a1.d1', '4.a1.e1', '4.a1.f1', '4.a1.g1'], 'contains': ['856.a1.a1'], 'core': '8.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [8598, 8598, 4123, 4123, 6737, 6737, 4110, 4110], 'generators': [8], 'label': '856.12.8.a1.a1', 'mobius_quo': -1, 'mobius_sub': -8, 'normal_closure': '8.a1.a1', 'normal_contained_in': ['4.a1.e1', '4.a1.c1', '4.a1.a1', '4.a1.b1', '4.a1.f1', '4.a1.g1', '4.a1.d1'], 'normal_contains': ['856.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '8.a1.a1', 'projective_image': '8.5', 'quotient_action_image': '1.1', 'quotient_action_kernel': '8.5', 'quotient_action_kernel_order': 8, 'quotient_fusion': None, 'short_label': '8.a1.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': '107.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 106, 'aut_gen_orders': [106], 'aut_gens': [[1], [54]], 'aut_group': '106.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 106, 'aut_permdeg': 55, 'aut_perms': [235037922448130462157819095799560576965993618854733871579136000000000000], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [107, 1, 106, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{106}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 106, 'autcent_group': '106.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 106, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{106}', '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], [107, 1, 106]], 'center_label': '107.1', 'center_order': 107, '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': ['107.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [107, 1, 106, 1]], 'element_repr_type': 'PC', 'elementary': 107, 'eulerian_function': 1, 'exponent': 107, 'exponents_of_order': [1], 'factors_of_aut_order': [2, 53], 'factors_of_order': [107], 'faithful_reps': [[1, 0, 106]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '107.1', 'hash': 1, 'hyperelementary': 107, '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': 106, 'irrQ_dim': 106, 'irrR_degree': 2, 'irrep_stats': [[1, 107]], 'label': '107.1', 'linC_count': 106, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 106, 'linQ_degree_count': 1, 'linQ_dim': 106, 'linQ_dim_count': 1, 'linR_count': 53, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C107', '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': 107, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 107, 'order_factorization_type': 1, 'order_stats': [[1, 1], [107, 106]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 106, 'outer_gen_orders': [106], 'outer_gen_pows': [0], 'outer_gens': [[54]], 'outer_group': '106.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 106, 'outer_permdeg': 55, 'outer_perms': [235037922448130462157819095799560576965993618854733871579136000000000000], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{106}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 107, 'pgroup': 107, 'primary_abelian_invariants': [107], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [106, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -107]}, 'Lie': [{'d': 1, 'q': 107, 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 107, 'gens': [1225151]}, 'Perm': {'d': 107, 'gens': [12150573975587908558064082271972503944261179666958805379263968688536773312539694302781998252415217388457727618432222125237219155992191776065669038080000000000000000000000000]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [107], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{107}', 'transitive_degree': 107, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '856.12', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 4452, 'aut_gen_orders': [212, 742], 'aut_gens': [[1, 2, 4], [431, 3, 333], [3, 431, 234]], 'aut_group': None, 'aut_hash': 2849052560199589475, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 17808, 'aut_permdeg': 113, 'aut_perms': [990811388720128251821007621149842439624670880146600837597729740680012821297324379325235198137634751306571690451769966032221365580839741687118828910232728856571128844162068689715200000, 199245404520971152620726722781377002745567922600342716201234430969408000627375556244307313751396181369833023153671789907826767879925166918048503200052196440845089958173396172800000000], 'aut_phi_ratio': 42.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1], [107, 1, 106, 1], [214, 1, 742, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{106}\\times \\PSL(2,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4452, 'autcent_group': None, 'autcent_hash': 2849052560199589475, 'autcent_nilpotent': False, 'autcent_order': 17808, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_{106}\\times \\PSL(2,7)', '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, 7], [107, 1, 106], [214, 1, 742]], 'center_label': '856.12', 'center_order': 856, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '107.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 12, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['107.1', 1], ['2.1', 3]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [107, 1, 106, 1], [214, 1, 106, 7]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 11557, 'exponent': 214, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3, 7, 53], 'factors_of_order': [2, 107], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '856.12', 'hash': 12, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], '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': None, 'irrep_stats': [[1, 856]], 'label': '856.12', 'linC_count': None, 'linC_degree': 3, '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*C214', '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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 856, 'number_divisions': 16, 'number_normal_subgroups': 32, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 32, 'number_subgroups': 32, 'old_label': None, 'order': 856, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 7], [107, 106], [214, 742]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4452, 'outer_gen_orders': [212, 318], 'outer_gen_pows': [0, 0], 'outer_gens': [[430, 428, 741], [429, 2, 313]], 'outer_group': None, 'outer_hash': 2849052560199589475, 'outer_nilpotent': False, 'outer_order': 17808, 'outer_permdeg': 113, 'outer_perms': [1191803717626209219439813759956056280166423970579069304182159635910950799786571543813322166567720788715280101538411833442467314377079903139423925690300788090339328000000000000000000000, 599404013093641330090554887018697542823975193243439870102568005381286889175767264941546613622242897579924791792680357987679684054574876007187743076036223149539328000000000000000000000], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_{106}\\times \\PSL(2,7)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 113, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 107], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [106, 8]], 'representations': {'PC': {'code': 65143811241, 'gens': [1, 2, 3], 'pres': [4, -2, -2, -2, -107, 34]}, 'Perm': {'d': 113, 'gens': [197450685722107402353682037275992488341277868034975337796656295094902858969771811440894224355027779366597957338237853638272334919686385621811850780464277094400000000000000000000000000, 15882455415227429404253703127090772871724410234473563207581748318444567162948183030959960131517678520479243672638179990208521148623422266876757623911219200000000000000000000000000, 1324641819451828974499891837121832599810209360673358065686551152497461815091591578895743130235002378688844343005686404521144382704205360039762937774080000000000000000000000000, 12150573975587908558064082271972503944261179666958805379263968688536773312539694302781998252415217388457727618432222125237219155992191776065669038080000000000000000000000000]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 214], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times C_{214}', 'transitive_degree': 856, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 84, 'aut_gen_orders': [3, 3], 'aut_gens': [[1, 2, 4], [4, 5, 3], [2, 4, 1]], 'aut_group': '168.42', 'aut_hash': 42, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 168, 'aut_permdeg': 7, 'aut_perms': [4361, 244], 'aut_phi_ratio': 42.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1]], 'aut_supersolvable': False, 'aut_tex': '\\PSL(2,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '168.42', 'autcent_hash': 42, 'autcent_nilpotent': False, 'autcent_order': 168, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\PSL(2,7)', '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, 7]], 'center_label': '8.5', 'center_order': 8, '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'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 3]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '8.5', 'hash': 5, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], '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, 8]], 'label': '8.5', 'linC_count': 28, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 28, 'linQ_dim': 3, 'linQ_dim_count': 28, 'linR_count': 28, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3', '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': 8, 'number_divisions': 8, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 16, 'number_subgroups': 16, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 84, 'outer_gen_orders': [3, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[4, 5, 3], [2, 4, 1]], 'outer_group': '168.42', 'outer_hash': 42, 'outer_nilpotent': False, 'outer_order': 168, 'outer_permdeg': 7, 'outer_perms': [4361, 244], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\PSL(2,7)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3], 'pres': [3, -2, 2, 2]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [16482, 16322, 3362]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [8156, 13286, 13933]}, 'Perm': {'d': 6, 'gens': [120, 6, 1]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}
