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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1070.2', 'ambient_counter': 2, 'ambient_order': 1070, 'ambient_tex': 'C_5\\times D_{107}', 'central': False, 'central_factor': False, 'centralizer_order': 535, 'characteristic': True, 'core_order': 107, 'counter': 4, 'cyclic': True, 'direct': False, 'hall': 107, 'label': '1070.2.10.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '10.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '10.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 10, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{10}', 'simple': True, 'solvable': True, 'special_labels': ['D', 'L1', 'D1', 'C2'], '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': '1070.2', 'aut_centralizer_order': 428, 'aut_label': '10.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '106.2', 'aut_weyl_index': 428, 'centralizer': '2.a1.a1', 'complements': ['107.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1', '5.a1.a1'], 'contains': ['1070.a1.a1'], 'core': '10.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [1748, 4401, 1748, 4401, 2128, 2986, 2128, 2986], 'generators': [10], 'label': '1070.2.10.a1.a1', 'mobius_quo': -1, 'mobius_sub': 1, 'normal_closure': '10.a1.a1', 'normal_contained_in': ['2.a1.a1', '5.a1.a1'], 'normal_contains': ['1070.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '10.a1.a1', 'projective_image': '1070.2', 'quotient_action_image': '2.1', 'quotient_action_kernel': '5.1', 'quotient_action_kernel_order': 5, 'quotient_fusion': None, 'short_label': '10.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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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': True, 'abelian': False, 'abelian_quotient': '10.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 22684, 'aut_gen_orders': [4, 106, 107], 'aut_gens': [[1, 2], [1, 426], [1, 472], [11, 2]], 'aut_group': None, 'aut_hash': 192398982627897821, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 45368, 'aut_permdeg': 111, 'aut_perms': [778447532888636816105473830590029824991544744646823252399834053771399569785691016380625508559334971823807470547266335662508495102647056026107545094208879099301020376364220415999999, 1461470156931703002457822352487925598439703509329408904041929435382418343517107822569884852725300524802428367352322612106912662190606528446463277342554893653082616725163975388401435, 80140891865725221357861724167061900619734602130549482692033947042393685753250777318366889751388040832702942783350234796063957860691239681798564712428640516851415369291943470820930], 'aut_phi_ratio': 107.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 107, 1, 1], [5, 1, 4, 1], [10, 107, 4, 1], [107, 2, 53, 1], [535, 2, 212, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{107}:(C_2\\times C_{212})', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 11342, 'autcentquo_group': '11342.a', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 11342, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{107}', 'cc_stats': [[1, 1, 1], [2, 107, 1], [5, 1, 4], [10, 107, 4], [107, 2, 53], [535, 2, 212]], 'center_label': '5.1', 'center_order': 5, 'central_product': True, 'central_quotient': '214.1', 'commutator_count': 1, 'commutator_label': '107.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '5.1', '107.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['214.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 107, 1, 1], [5, 1, 4, 1], [10, 107, 4, 1], [107, 2, 53, 1], [535, 2, 212, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 18, 'exponent': 1070, 'exponents_of_order': [1, 1, 1], 'factors_of_aut_order': [2, 53, 107], 'factors_of_order': [2, 5, 107], 'faithful_reps': [[2, 0, 212]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1070.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 214, 'inner_gen_orders': [2, 107], 'inner_gens': [[1, 212], [861, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 214, 'inner_split': True, 'inner_tex': 'D_{107}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 424, 'irrQ_dim': 424, 'irrR_degree': 4, 'irrep_stats': [[1, 10], [2, 265]], 'label': '1070.2', 'linC_count': 212, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 110, 'linQ_degree_count': 2, 'linQ_dim': 110, 'linQ_dim_count': 2, 'linR_count': 318, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5*D107', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, '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': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 275, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 220, 'old_label': None, 'order': 1070, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 107], [5, 4], [10, 428], [107, 106], [535, 424]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 212, 'outer_gen_orders': [212], 'outer_gen_pows': [0], 'outer_gens': [[1, 446]], 'outer_group': '212.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 212, 'outer_permdeg': 57, 'outer_perms': [723926903980137676150974511088689270272945160474611502323653083958673792000], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{212}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 112, 'pgroup': 0, 'primary_abelian_invariants': [2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 2], [106, 1], [424, 1]], 'representations': {'PC': {'code': 4582533016755799, 'gens': [1, 2], 'pres': [3, -2, -5, -107, 1273, 34, 9542]}, 'GLFp': {'d': 2, 'p': 641, 'gens': [168559821441, 89299304506]}, 'Perm': {'d': 112, 'gens': [48806427963310445734127381919780547023926658175409759816344851670953030415534263482025576963860688793943375768749035053147355989627311916026479083772570209845090562543386774521600, 33, 1843672250229380505816155757023669047895481251969262142449749950077627788985931157208006518735054806532873874317277484009982985512507436174238080053969596659543180259023797072104320]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [10], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times D_{107}', 'transitive_degree': 535, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '10.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [4], 'aut_gens': [[1], [7]], 'aut_group': '4.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [5, 1, 4, 1], [10, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4', '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], [5, 1, 4], [10, 1, 4]], 'center_label': '10.2', 'center_order': 10, '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', '5.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [5, 1, 4, 1], [10, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 10, 'eulerian_function': 1, 'exponent': 10, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 5], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '10.2', 'hash': 2, 'hyperelementary': 10, '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': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 10]], 'label': '10.2', 'linC_count': 4, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C10', 'ngens': 2, '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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 10, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 10, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [5, 4], [10, 4]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [0], 'outer_gens': [[7]], 'outer_group': '4.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 2]], 'representations': {'PC': {'code': 83, 'gens': [1], 'pres': [2, -2, -5, 4]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16717348]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 504]}, 'Perm': {'d': 7, 'gens': [720, 96]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [10], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}', 'transitive_degree': 10, 'wreath_data': None, 'wreath_product': False}
