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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '4032.dt', 'ambient_counter': 98, 'ambient_order': 4032, 'ambient_tex': 'D_{12}\\times \\PSL(2,7)', 'central': False, 'central_factor': False, 'centralizer_order': 12, 'characteristic': False, 'core_order': 4, 'counter': 104, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '4032.dt.48.d1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '48.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': 48, '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': '84.2', 'subgroup_hash': 2, 'subgroup_order': 84, 'subgroup_tex': 'C_7:C_{12}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '4032.dt', 'aut_centralizer_order': 24, 'aut_label': '48.d1', 'aut_quo_index': None, 'aut_stab_index': 8, 'aut_weyl_group': '84.7', 'aut_weyl_index': 192, 'centralizer': '336.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['6.a1.a1', '16.b1.a1', '24.b1.a1'], 'contains': ['96.b1.a1', '144.b1.a1', '336.j1.a1'], 'core': '1008.a1.a1', 'coset_action_label': None, 'count': 8, 'diagramx': [6365, -1, 211, -1, 2556, -1, 1415, -1], 'generators': [4032186, 10890600, 4032000, 7983360], 'label': '4032.dt.48.d1.a1', 'mobius_quo': None, 'mobius_sub': -3, 'normal_closure': '6.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '8.b1.a1', 'old_label': '48.d1.a1', 'projective_image': '2016.dd', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '48.d1.a1', 'subgroup_fusion': None, 'weyl_group': '42.2'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '12.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 42, 'aut_gen_orders': [6, 14], 'aut_gens': [[1, 3], [1, 51], [73, 45]], 'aut_group': '84.7', 'aut_hash': 7, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 84, 'aut_permdeg': 9, 'aut_perms': [6055, 203047], 'aut_phi_ratio': 3.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 7, 1, 2], [4, 1, 2, 1], [6, 7, 1, 2], [7, 3, 2, 1], [12, 7, 2, 2], [14, 3, 2, 1], [28, 3, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times F_7', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '42.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 42, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 7, 2], [4, 1, 2], [6, 7, 2], [7, 3, 2], [12, 7, 4], [14, 3, 2], [28, 3, 4]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '21.1', 'commutator_count': 1, 'commutator_label': '7.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '7.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['21.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 7, 2, 1], [4, 1, 2, 1], [6, 7, 2, 1], [7, 3, 2, 1], [12, 7, 4, 1], [14, 3, 2, 1], [28, 3, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 48, 'exponent': 84, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[3, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '42.2', 'hash': 2, 'hyperelementary': 3, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 21, 'inner_gen_orders': [3, 7], 'inner_gens': [[1, 27], [61, 3]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 21, 'inner_split': True, 'inner_tex': 'C_7:C_3', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 6, 'irrep_stats': [[1, 12], [3, 8]], 'label': '84.2', 'linC_count': 4, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 2, 'linQ_dim': 8, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C7:C12', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, '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': 12, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 20, 'number_divisions': 9, 'number_normal_subgroups': 9, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 12, 'number_subgroups': 30, 'old_label': None, 'order': 84, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 1], [3, 14], [4, 2], [6, 14], [7, 6], [12, 28], [14, 6], [28, 12]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [1, 0], 'outer_gens': [[1, 51], [1, 45]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [4, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 1], [6, 2], [12, 1]], 'representations': {'PC': {'code': 34618937875133, 'gens': [1, 2], 'pres': [4, -3, -2, -2, -7, 217, 21, 650, 34, 387]}, 'GLFq': {'d': 2, 'q': 49, 'gens': [5411900, 1410219, 822992, 705900]}, 'Perm': {'d': 11, 'gens': [9, 404640, 16, 4083240]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [12], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_7:C_{12}', 'transitive_degree': 28, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 168, 'aut_gen_orders': [2, 6, 2, 2, 6, 3, 2], 'aut_gens': [[4032000, 12454041600, 12458077034, 479772167, 7983360, 6230689920], [4032000, 12454041600, 12458077034, 486626567, 7983360, 6238673280], [10886400, 12454041600, 12464928595, 12940667221, 7983360, 482670720], [4032000, 6706022400, 6710057834, 12933813767, 7983360, 6230689920], [10886400, 12454041600, 12464931434, 482675207, 7983360, 6227786880], [10886400, 12454041600, 12464931001, 6238674080, 7983360, 12940663680], [4032000, 12454041600, 12458077034, 12933813767, 7983360, 482670720], [10886400, 12454041600, 12464931434, 479772167, 7983360, 6238673280]], 'aut_group': '16128.ca', 'aut_hash': 585455335392977831, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 16128, 'aut_permdeg': 33, 'aut_perms': [1910452983301690629701448784475574, 3212763573964545642903718857611078753, 265252860432639460725322525747200, 3563862481978281287181230870204166, 78489352199828413007999272210889803, 5835879113541820973345696591763606, 1910452983301690628213228175400320], 'aut_phi_ratio': 14.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 2, 1], [2, 21, 1, 2], [2, 126, 2, 1], [3, 2, 1, 1], [3, 56, 1, 1], [3, 112, 1, 1], [4, 2, 1, 1], [4, 42, 1, 3], [4, 84, 1, 1], [4, 252, 2, 1], [6, 2, 1, 1], [6, 42, 1, 2], [6, 56, 1, 1], [6, 112, 1, 1], [6, 336, 2, 1], [7, 24, 2, 1], [12, 2, 2, 1], [12, 42, 2, 1], [12, 84, 1, 2], [12, 84, 2, 1], [12, 112, 1, 1], [12, 112, 2, 1], [14, 24, 2, 1], [14, 144, 4, 1], [21, 48, 2, 1], [28, 48, 2, 1], [42, 48, 2, 1], [84, 48, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times D_4\\times \\PGL(2,7)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 168, 'autcentquo_group': '4032.ds', 'autcentquo_hash': 5672968783997510635, 'autcentquo_nilpotent': False, 'autcentquo_order': 4032, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'D_6\\times \\PGL(2,7)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 6, 2], [2, 21, 2], [2, 126, 2], [3, 2, 1], [3, 56, 1], [3, 112, 1], [4, 2, 1], [4, 42, 3], [4, 84, 1], [4, 252, 2], [6, 2, 1], [6, 42, 2], [6, 56, 1], [6, 112, 1], [6, 336, 2], [7, 24, 2], [12, 2, 2], [12, 42, 2], [12, 84, 4], [12, 112, 3], [14, 24, 2], [14, 144, 4], [21, 48, 2], [28, 48, 2], [42, 48, 2], [84, 48, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '2016.dd', 'commutator_count': 1, 'commutator_label': '1008.884', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '168.42'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 98, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['168.42', 1], ['24.6', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 2], [2, 21, 1, 2], [2, 126, 1, 2], [3, 2, 1, 1], [3, 56, 1, 1], [3, 112, 1, 1], [4, 2, 1, 1], [4, 42, 1, 3], [4, 84, 1, 1], [4, 252, 1, 2], [6, 2, 1, 1], [6, 42, 1, 2], [6, 56, 1, 1], [6, 112, 1, 1], [6, 336, 1, 2], [7, 24, 2, 1], [12, 2, 2, 1], [12, 42, 2, 1], [12, 84, 1, 2], [12, 84, 2, 1], [12, 112, 1, 1], [12, 112, 2, 1], [14, 24, 2, 1], [14, 144, 2, 2], [21, 48, 2, 1], [28, 48, 2, 1], [42, 48, 2, 1], [84, 48, 4, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 171, 'exponent': 84, 'exponents_of_order': [6, 2, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[6, 0, 4], [12, 1, 2], [14, 1, 2], [16, 1, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '2016.dd', 'hash': 3169814912488974632, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 3, 12, 2, 1, 2], 'inner_gens': [[4032000, 12454041600, 12458077034, 486626567, 7983360, 6238673280], [4032000, 12454041600, 12458077034, 6227791367, 7983360, 12936712320], [4032000, 12454041600, 12458077034, 6234642005, 7983360, 12944695680], [10886400, 6706022400, 6716909113, 479772167, 7983360, 12944695680], [4032000, 12454041600, 12458077034, 479772167, 7983360, 6230689920], [10886400, 6706022400, 6716912234, 12940668167, 7983360, 6230689920]], 'inner_hash': 8333125712768441886, 'inner_nilpotent': False, 'inner_order': 2016, 'inner_split': True, 'inner_tex': 'D_6\\times \\GL(3,2)', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 6, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [2, 5], [3, 8], [6, 14], [7, 4], [8, 4], [12, 5], [14, 5], [16, 5]], 'label': '4032.dt', 'linC_count': 16, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 24, 'linQ_dim': 10, 'linQ_dim_count': 24, 'linR_count': 16, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'D12*PSL(2,7)', 'ngens': 6, '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, 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': 35, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 54, 'number_divisions': 40, 'number_normal_subgroups': 18, 'number_subgroup_autclasses': 273, 'number_subgroup_classes': 529, 'number_subgroups': 22010, 'old_label': None, 'order': 4032, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 307], [3, 170], [4, 716], [6, 926], [7, 48], [12, 760], [14, 624], [21, 96], [28, 96], [42, 96], [84, 192]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [313, 4032000, 0], 'outer_gens': [[4032000, 12454041600, 12458074962, 479769608, 7983360, 6230689920], [4032000, 12454041600, 12458077034, 482675207, 7983360, 6234641280], [4032000, 6706022400, 6710057834, 12933813767, 7983360, 6230689920]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': None, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 3], [4, 1], [6, 8], [7, 4], [8, 4], [12, 6], [14, 3], [16, 3], [24, 2], [28, 1], [32, 1]], 'representations': {'Perm': {'d': 14, 'gens': [4032000, 12454041600, 12458077034, 479772167, 7983360, 6230689920]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'D_{12}\\times \\PSL(2,7)', 'transitive_degree': 84, 'wreath_data': None, 'wreath_product': False}
