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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1344.9851', 'ambient_counter': 9851, 'ambient_order': 1344, 'ambient_tex': 'C_{84}.C_2^4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 24, 'counter': 254, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1344.9851.28.l1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '28.l1.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': 28, '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': '48.10', 'subgroup_hash': 10, 'subgroup_order': 48, 'subgroup_tex': 'C_3:\\OD_{16}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.9851', 'aut_centralizer_order': 48, 'aut_label': '28.l1', 'aut_quo_index': None, 'aut_stab_index': 7, 'aut_weyl_group': '96.209', 'aut_weyl_index': 336, 'centralizer': '168.o1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.g1.a1', '14.c1.a1', '14.e1.a1', '14.f1.a1'], 'contains': ['56.a1.a1', '56.bd1.a1', '56.bd1.b1', '84.j1.a1'], 'core': '56.a1.a1', 'coset_action_label': None, 'count': 7, 'diagramx': [7782, -1, 8514, -1, 8167, -1, 2285, -1], 'generators': [5, 448, 336, 8, 672], 'label': '1344.9851.28.l1.a1', 'mobius_quo': None, 'mobius_sub': -2, 'normal_closure': '4.g1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '7.a1.a1', 'old_label': '28.l1.a1', 'projective_image': '672.1175', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '28.l1.a1', 'subgroup_fusion': None, 'weyl_group': '24.14'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 2, 2, 3], 'aut_gens': [[1, 4], [1, 20], [3, 4], [3, 28], [39, 28], [25, 4], [17, 4]], 'aut_group': '96.209', 'aut_hash': 209, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96, 'aut_permdeg': 9, 'aut_perms': [5160, 120, 1, 143, 7, 45360], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [3, 2, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [6, 2, 1, 1], [6, 2, 2, 1], [8, 6, 4, 1], [12, 2, 2, 2]], 'aut_supersolvable': True, 'aut_tex': 'D_4\\times D_6', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '12.4', 'autcentquo_hash': 4, 'autcentquo_nilpotent': False, 'autcentquo_order': 12, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [3, 2, 1], [4, 1, 2], [4, 2, 1], [6, 2, 3], [8, 6, 4], [12, 2, 4]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, 'central_quotient': '12.4', 'commutator_count': 1, 'commutator_label': '6.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 10, '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, 1], [3, 2, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [6, 2, 1, 1], [6, 2, 2, 1], [8, 6, 2, 2], [12, 2, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 24, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 4]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '12.4', 'hash': 10, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6], 'inner_gens': [[1, 44], [9, 4]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'D_6', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 10]], 'label': '48.10', 'linC_count': 4, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 4, 'linQ_dim': 6, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3:OD16', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 11, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 18, 'number_divisions': 12, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 20, 'number_subgroups': 28, 'old_label': None, 'order': 48, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 4], [6, 6], [8, 24], [12, 8]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [12, 0, 0], 'outer_gens': [[13, 4], [3, 4], [1, 28]], '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': 2, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 1], [8, 1]], 'representations': {'PC': {'code': 6693510545677793, 'gens': [1, 3], 'pres': [5, -2, -2, -2, -2, -3, 10, 126, 662, 42, 803, 58, 804]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [41715827558031495, 58415889939761677]}, 'GLFp': {'d': 2, 'p': 13, 'gens': [26365, 273, 15381]}, 'Perm': {'d': 11, 'gens': [4124305, 1134264, 8807280, 12136560, 3]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3:\\OD_{16}', 'transitive_degree': 24, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [12, 12, 6, 6, 6, 6, 4, 2], 'aut_gens': [[1, 2, 4, 16], [385, 10, 228, 208], [385, 10, 228, 880], [193, 674, 12, 368], [289, 10, 908, 208], [289, 2, 1012, 1264], [1057, 2, 684, 80], [865, 2, 1020, 208], [1153, 10, 12, 208]], 'aut_group': None, 'aut_hash': 9161810123478319507, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 32256, 'aut_permdeg': 42, 'aut_perms': [725113179591997391725753474808974338221457436377093, 709609911308210471413803773629975919971531298637063, 500540458054023812290399810510206398406343375470080, 651394535397641431670575679931988351280679195053040, 962577903757016129873014821271226332442034441286878, 1270280166319439181676085640645112941378728739297368, 1218081736839141388262810893090825455268255233185323, 480051274647955995613324713775586841265333366979280], 'aut_phi_ratio': 84.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 1], [2, 7, 2, 1], [2, 14, 1, 1], [2, 28, 1, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 4, 1, 1], [4, 12, 2, 1], [4, 14, 1, 2], [4, 28, 1, 1], [4, 84, 2, 1], [6, 2, 1, 1], [6, 4, 1, 1], [6, 4, 2, 1], [6, 14, 2, 1], [6, 28, 1, 1], [6, 28, 2, 1], [7, 2, 3, 1], [8, 12, 2, 1], [8, 84, 2, 1], [12, 2, 2, 1], [12, 4, 1, 1], [12, 4, 2, 1], [12, 14, 2, 1], [12, 28, 1, 1], [12, 28, 2, 1], [14, 2, 3, 1], [14, 4, 3, 1], [14, 8, 3, 1], [21, 4, 3, 1], [28, 4, 3, 2], [28, 8, 3, 1], [28, 24, 6, 1], [42, 4, 3, 1], [42, 8, 3, 1], [42, 8, 6, 1], [56, 24, 6, 1], [84, 4, 6, 1], [84, 8, 3, 1], [84, 8, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{42}.(C_2^4\\times C_6).C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': None, 'autcentquo_hash': 5451351369529179689, 'autcentquo_nilpotent': False, 'autcentquo_order': 2016, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^3\\times S_3\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 4, 1], [2, 7, 2], [2, 14, 1], [2, 28, 1], [3, 2, 1], [4, 2, 2], [4, 4, 1], [4, 12, 2], [4, 14, 2], [4, 28, 1], [4, 84, 2], [6, 2, 1], [6, 4, 3], [6, 14, 2], [6, 28, 3], [7, 2, 3], [8, 12, 2], [8, 84, 2], [12, 2, 2], [12, 4, 3], [12, 14, 2], [12, 28, 3], [14, 2, 3], [14, 4, 3], [14, 8, 3], [21, 4, 3], [28, 4, 6], [28, 8, 3], [28, 24, 6], [42, 4, 3], [42, 8, 9], [56, 24, 6], [84, 4, 6], [84, 8, 9]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '672.1175', 'commutator_count': 1, 'commutator_label': '84.6', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 9851, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['14.1', 1], ['96.158', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 1], [2, 7, 1, 2], [2, 14, 1, 1], [2, 28, 1, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 4, 1, 1], [4, 12, 1, 2], [4, 14, 1, 2], [4, 28, 1, 1], [4, 84, 1, 2], [6, 2, 1, 1], [6, 4, 1, 1], [6, 4, 2, 1], [6, 14, 1, 2], [6, 28, 1, 1], [6, 28, 2, 1], [7, 2, 3, 1], [8, 12, 1, 2], [8, 84, 1, 2], [12, 2, 2, 1], [12, 4, 1, 1], [12, 4, 2, 1], [12, 14, 2, 1], [12, 28, 1, 1], [12, 28, 2, 1], [14, 2, 3, 1], [14, 4, 3, 1], [14, 8, 3, 1], [21, 4, 3, 1], [28, 4, 3, 2], [28, 8, 3, 1], [28, 24, 3, 2], [42, 4, 3, 1], [42, 8, 3, 1], [42, 8, 6, 1], [56, 24, 3, 2], [84, 4, 6, 1], [84, 8, 3, 1], [84, 8, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 29877120, 'exponent': 168, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[8, -1, 6]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '336.219', 'hash': 9851, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 2, 4, 42], 'inner_gens': [[1, 2, 4, 208], [1, 2, 12, 16], [1, 682, 4, 1136], [1153, 2, 228, 16]], 'inner_hash': 1175, 'inner_nilpotent': False, 'inner_order': 672, 'inner_split': True, 'inner_tex': 'D_{42}:C_2^3', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 48, 'irrQ_dim': 96, 'irrR_degree': 16, 'irrep_stats': [[1, 16], [2, 44], [4, 36], [8, 9]], 'label': '1344.9851', 'linC_count': 96, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 128, 'linQ_dim': 16, 'linQ_dim_count': 128, 'linR_count': 96, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C84.C2^4', 'ngens': 8, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 46, 'number_characteristic_subgroups': 88, 'number_conjugacy_classes': 105, 'number_divisions': 54, 'number_normal_subgroups': 140, 'number_subgroup_autclasses': 400, 'number_subgroup_classes': 516, 'number_subgroups': 3684, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 63], [3, 2], [4, 256], [6, 126], [7, 6], [8, 192], [12, 128], [14, 42], [21, 12], [28, 192], [42, 84], [56, 144], [84, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 6], 'outer_gen_pows': [0, 0, 0, 1008], 'outer_gens': [[1, 2, 4, 464], [673, 2, 4, 16], [1, 2, 4, 688], [1, 2, 340, 1168]], 'outer_group': '48.52', 'outer_hash': 52, 'outer_nilpotent': True, 'outer_order': 48, 'outer_permdeg': 11, 'outer_perms': [24, 3628800, 40320, 723], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 6], [6, 8], [8, 2], [12, 6], [24, 3], [48, 1]], 'representations': {'PC': {'code': 24285089751156425609012034167528813687793829954642631851019, 'gens': [1, 2, 3, 5], 'pres': [8, -2, -2, -2, -2, -2, -2, -3, -7, 154, 66, 10891, 2715, 8324, 11380, 116, 19973, 11157, 141, 46598, 7190, 222, 73735]}, 'Perm': {'d': 26, 'gens': [621574835363025223680000, 357298057206213, 44141565448, 2968616036117, 4410546813871, 5802181616106, 6758061133824000, 16754357281155028254720000]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{84}.C_2^4', 'transitive_degree': 336, 'wreath_data': None, 'wreath_product': False}
