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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1672.31', 'ambient_counter': 31, 'ambient_order': 1672, 'ambient_tex': 'C_{22}\\times D_{38}', 'central': False, 'central_factor': True, 'centralizer_order': 44, 'characteristic': False, 'core_order': 836, 'counter': 5, 'cyclic': False, 'direct': True, 'hall': 0, 'label': '1672.31.2.a1.d1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '2.a1.d1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '2.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 2, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '836.6', 'subgroup_hash': 6, 'subgroup_order': 836, 'subgroup_tex': 'C_{11}\\times D_{38}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1672.31', 'aut_centralizer_order': 2, 'aut_label': '2.a1', 'aut_quo_index': 1, 'aut_stab_index': 6, 'aut_weyl_group': None, 'aut_weyl_index': 12, 'centralizer': '38.a1.a1', 'complements': ['836.a1.a1', '836.b1.a1', '836.a1.c1', '836.b1.c1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.b1', '4.b1.b1', '4.b1.d1', '22.a1.d1', '38.b1.d1'], 'core': '2.a1.d1', 'coset_action_label': None, 'count': 1, 'diagramx': [4367, 4190, 6239, 5778, 2830, 2285, 4806, 4320], 'generators': [1, 46, 88, 1064], 'label': '1672.31.2.a1.d1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.a1.d1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['4.a1.b1', '4.b1.b1', '4.b1.d1', '22.a1.d1'], 'normalizer': '1.a1.a1', 'old_label': '2.a1.d1', 'projective_image': '76.3', 'quotient_action_image': '1.1', 'quotient_action_kernel': '2.1', 'quotient_action_kernel_order': 2, 'quotient_fusion': None, 'short_label': '2.a1.d1', 'subgroup_fusion': None, 'weyl_group': '38.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '44.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 1710, 'aut_gen_orders': [10, 2, 18, 19], 'aut_gens': [[1, 2], [1, 302], [419, 834], [1, 178], [45, 2]], 'aut_group': None, 'aut_hash': 1833356378742793410, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6840, 'aut_permdeg': 48, 'aut_perms': [2075169291006304263371377854806207709047453104761148884183393, 1315490940785934276800146421585094761380472044415980191277367, 8283302024492390132344993668817604359337095039962142740760593, 528496412035958192620708651764713819826048307461984546756705], 'aut_phi_ratio': 19.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 19, 2, 1], [11, 1, 10, 1], [19, 2, 9, 1], [22, 1, 10, 1], [22, 19, 20, 1], [38, 2, 9, 1], [209, 2, 90, 1], [418, 2, 90, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{19}:(C_2^2\\times C_{90})', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 10, 'autcent_group': '20.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 20, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{10}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 342, 'autcentquo_group': '342.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 342, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{19}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 19, 2], [11, 1, 10], [19, 2, 9], [22, 1, 10], [22, 19, 20], [38, 2, 9], [209, 2, 90], [418, 2, 90]], 'center_label': '22.2', 'center_order': 22, 'central_product': True, 'central_quotient': '38.1', 'commutator_count': 1, 'commutator_label': '19.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '11.1', '19.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 6, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['11.1', 1], ['2.1', 1], ['38.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 19, 1, 2], [11, 1, 10, 1], [19, 2, 9, 1], [22, 1, 10, 1], [22, 19, 10, 2], [38, 2, 9, 1], [209, 2, 90, 1], [418, 2, 90, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 36, 'exponent': 418, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3, 5, 19], 'factors_of_order': [2, 11, 19], 'faithful_reps': [[2, 0, 90]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '836.6', 'hash': 6, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 38, 'inner_gen_orders': [2, 19], 'inner_gens': [[1, 530], [309, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 38, 'inner_split': True, 'inner_tex': 'D_{19}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 180, 'irrQ_dim': 180, 'irrR_degree': 4, 'irrep_stats': [[1, 44], [2, 198]], 'label': '836.6', 'linC_count': 90, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 28, 'linQ_degree_count': 6, 'linQ_dim': 28, 'linQ_dim_count': 6, 'linR_count': 315, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C11*D38', '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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 242, 'number_divisions': 12, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 20, 'number_subgroups': 128, 'old_label': None, 'order': 836, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 39], [11, 10], [19, 18], [22, 390], [38, 18], [209, 180], [418, 180]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 90, 'outer_gen_orders': [2, 90], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 834], [419, 126]], 'outer_group': '180.12', 'outer_hash': 12, 'outer_nilpotent': True, 'outer_order': 180, 'outer_permdeg': 18, 'outer_perms': [1307674368000, 357001852440993], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{90}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 32, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 11], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [10, 4], [18, 2], [180, 2]], 'representations': {'PC': {'code': 446464791312732926099, 'gens': [1, 2], 'pres': [4, -2, -2, -11, -19, 4241, 21, 2690, 142, 12675]}, 'GLFp': {'d': 2, 'p': 419, 'gens': [22950738780, 175980, 20081896363]}, 'Perm': {'d': 32, 'gens': [265558152069838740888854465587200, 1, 522956310, 8753954958317600299503184632883200]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 22], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{11}\\times D_{38}', 'transitive_degree': 418, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '88.12', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 3420, 'aut_gen_orders': [30, 90, 18, 180], 'aut_gens': [[1, 2, 4], [836, 1322, 717], [1, 1190, 932], [837, 443, 313], [1, 662, 1469]], 'aut_group': None, 'aut_hash': 2572722440972311456, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 82080, 'aut_permdeg': 86, 'aut_perms': [20394387967446299261935870539420429334747192225893978744721171879382001207192981093270961852860148198829309123553770367610201816321, 1605614178168910873388710550497782713615779011404814292703804518966232004151343163636814440845763635291338652058435122471931829641, 154412898300147286380467348969862281480365069927108641874782679148838539917250193731013803762599647355115916527393276071138181760, 19371177777045817087234393811412912378883793960116936101660039338861511652747848296846936067480604517473439761209628831665154424733], 'aut_phi_ratio': 114.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 19, 4, 1], [11, 1, 10, 1], [19, 2, 9, 1], [22, 1, 30, 1], [22, 19, 40, 1], [38, 2, 27, 1], [209, 2, 90, 1], [418, 2, 270, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_{19}\\times A_4).C_{90}.C_2^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 60, 'autcent_group': '240.196', 'autcent_hash': 196, 'autcent_nilpotent': False, 'autcent_order': 240, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_{10}\\times S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 342, 'autcentquo_group': '342.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 342, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{19}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 19, 4], [11, 1, 10], [19, 2, 9], [22, 1, 30], [22, 19, 40], [38, 2, 27], [209, 2, 90], [418, 2, 270]], 'center_label': '44.4', 'center_order': 44, 'central_product': True, 'central_quotient': '38.1', 'commutator_count': 1, 'commutator_label': '19.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '11.1', '19.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 31, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['11.1', 1], ['2.1', 2], ['38.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 19, 1, 4], [11, 1, 10, 1], [19, 2, 9, 1], [22, 1, 10, 3], [22, 19, 10, 4], [38, 2, 9, 3], [209, 2, 90, 1], [418, 2, 90, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 18620, 'exponent': 418, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 5, 19], 'factors_of_order': [2, 11, 19], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1672.31', 'hash': 31, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 38, 'inner_gen_orders': [1, 2, 19], 'inner_gens': [[1, 2, 4], [1, 2, 1060], [1, 618, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 38, 'inner_split': True, 'inner_tex': 'D_{19}', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 88], [2, 396]], 'label': '1672.31', 'linC_count': 12960, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 28, 'linQ_degree_count': 12, 'linQ_dim': 28, 'linQ_dim_count': 12, 'linR_count': 540, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C22*D38', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 484, 'number_divisions': 24, 'number_normal_subgroups': 42, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 64, 'number_subgroups': 460, 'old_label': None, 'order': 1672, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 79], [11, 10], [19, 18], [22, 790], [38, 54], [209, 180], [418, 540]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 180, 'outer_gen_orders': [90, 45, 90], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[837, 2, 900], [836, 3, 477], [837, 2, 817]], 'outer_group': None, 'outer_hash': 1840319424225423585, 'outer_nilpotent': False, 'outer_order': 2160, 'outer_permdeg': 20, 'outer_perms': [256096282903211553, 512546938502860896, 257203903035340833], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'A_4.C_{90}.C_2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 34, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 11], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [10, 8], [18, 4], [180, 4]], 'representations': {'PC': {'code': 4113892862747775420718643, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -11, -19, 7957, 42, 4488, 238, 19809]}, 'Perm': {'d': 34, 'gens': [1126433986787154089647, 2350183339898634240000, 3476616969684418560000, 8954945705218201060967611351695360000, 7116376445267293]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 22], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{22}\\times D_{38}', 'transitive_degree': 836, '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': '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}
