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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1760.46', 'ambient_counter': 46, 'ambient_order': 1760, 'ambient_tex': '(C_2\\times C_{44}).C_{20}', 'central': False, 'central_factor': False, 'centralizer_order': 440, 'characteristic': False, 'core_order': 8, 'counter': 62, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1760.46.220.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '220.b1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '220.1', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 220, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{11}:C_{20}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '8.2', 'subgroup_hash': 2, 'subgroup_order': 8, 'subgroup_tex': 'C_2\\times C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1760.46', 'aut_centralizer_order': 880, 'aut_label': '220.b1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '8.3', 'aut_weyl_index': 1760, 'centralizer': '4.b1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['20.b1.a1', '44.b1.a1', '110.a1.a1'], 'contains': ['440.a1.a1', '440.c1.a1'], 'core': '220.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3991, 2808, 6397, 1121, 4481, 9041, 7619, 7475], 'generators': [20, 440], 'label': '1760.46.220.b1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '220.b1.a1', 'normal_contained_in': ['20.b1.a1', '110.a1.a1'], 'normal_contains': ['440.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '220.b1.a1', 'projective_image': '880.22', 'quotient_action_image': '4.1', 'quotient_action_kernel': '55.1', 'quotient_action_kernel_order': 55, 'quotient_fusion': None, 'short_label': '220.b1.a1', 'subgroup_fusion': None, 'weyl_group': '4.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 3], [5, 3]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [1, 22], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.3', 'autcent_hash': 3, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_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, 3], [4, 1, 4]], 'center_label': '8.2', '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': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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': '8.2', 'linC_count': 12, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 4, 'linQ_dim': 3, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C4', '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': 8, 'number_divisions': 6, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 4]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 3], [5, 3]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [1, 22], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2]], 'representations': {'PC': {'code': 34, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16426]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [251, 504]}, 'Perm': {'d': 6, 'gens': [22, 120, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_4', 'transitive_degree': 8, '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': '40.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 440, 'aut_gen_orders': [10, 20, 20, 20, 10], 'aut_gens': [[1, 20, 40], [891, 20, 630], [1221, 900, 790], [1431, 20, 1500], [31, 20, 710], [301, 20, 1400]], 'aut_group': None, 'aut_hash': 7302058934348597158, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 14080, 'aut_permdeg': 176, 'aut_perms': [144848133071376041531403230065050049398964783826210653878199010901127893423122108599424442827740198097076564155966039835462769789994178299346498741919722732734935638445057274452904963569596175763861823713352977085331151356306462830032037392088747640887195520336736857260767311764420411183853480909151357432734188975726203, 4267013877253555137020728930569090536043132521969936538673300554733309486827334385585991287406340563568761026151093510402712553008529669432457564775946857304883728008868622290973803178396418110719691845230341342940808806684600250678447340883423860023026124299286384737251560700159797800225740816166704429494045297013343, 126974394740389127844715257837098916347324446159178141304061874406679315965086580808742767437671361955833487704372707064868745773749159351185156699605913927270918797109420698546228954509054608872688359972044494388460889289563091057054382935107602387416740826834489938248737211108688196672856837387550485979927062896310475, 63309574215553323350289253332136963531632414967563256535771778595143067537257610299534235452944205560836269935901127408798779391495015219532518963284948289497093315141085924637572591677899627332986458164693999929485006346534179883910052276260442679588616562414243351896824696002388776045294929176172887344718593390969667, 99492926065018010510059313035417894750124418120531156633659843914346806647120050918223625207590651087797097854694028007158916608141728523740461995125397560393295298679869042337353864945058199506737983596773035904376408839936667392036168244810805720113426068789724296560892783065263640901468744603569939783572629370345017], 'aut_phi_ratio': 22.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [4, 2, 2, 1], [4, 4, 2, 1], [5, 11, 1, 4], [8, 44, 4, 1], [10, 11, 1, 4], [10, 22, 1, 4], [11, 10, 1, 1], [20, 22, 2, 4], [20, 44, 2, 4], [22, 10, 1, 1], [22, 10, 2, 1], [40, 44, 4, 4], [44, 20, 2, 1], [44, 20, 4, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_{22}\\times D_4).C_5.C_2^4', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 220, 'autcentquo_group': '3520.b', 'autcentquo_hash': 1968696860554166433, 'autcentquo_nilpotent': False, 'autcentquo_order': 3520, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\wr C_2\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [4, 2, 2], [4, 4, 2], [5, 11, 4], [8, 44, 4], [10, 11, 4], [10, 22, 4], [11, 10, 1], [20, 22, 8], [20, 44, 8], [22, 10, 3], [40, 44, 16], [44, 20, 6]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '880.22', 'commutator_count': 1, 'commutator_label': '44.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 46, '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], [4, 2, 1, 2], [4, 4, 1, 2], [5, 11, 4, 1], [8, 44, 2, 2], [10, 11, 4, 1], [10, 22, 4, 1], [11, 10, 1, 1], [20, 22, 4, 2], [20, 44, 4, 2], [22, 10, 1, 1], [22, 10, 2, 1], [40, 44, 8, 2], [44, 20, 1, 2], [44, 20, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72, 'exponent': 440, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [[20, 0, 2]], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '220.7', 'hash': 46, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 220, 'inner_gen_orders': [20, 2, 22], 'inner_gens': [[1, 900, 700], [881, 20, 40], [1141, 20, 40]], 'inner_hash': 22, 'inner_nilpotent': False, 'inner_order': 880, 'inner_split': True, 'inner_tex': 'C_2^3.F_{11}', 'inner_used': [1, 2, 3], 'irrC_degree': 20, 'irrQ_degree': 40, 'irrQ_dim': 40, 'irrR_degree': 40, 'irrep_stats': [[1, 40], [2, 10], [4, 5], [10, 8], [20, 2]], 'label': '1760.46', 'linC_count': 40, 'linC_degree': 14, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 4, 'linQ_dim': 18, 'linQ_dim_count': 2, 'linR_count': 6, 'linR_degree': 18, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': '(C2*C44).C20', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 35, 'number_characteristic_subgroups': 17, 'number_conjugacy_classes': 65, 'number_divisions': 25, 'number_normal_subgroups': 33, 'number_subgroup_autclasses': 52, 'number_subgroup_classes': 76, 'number_subgroups': 434, 'old_label': None, 'order': 1760, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 3], [4, 12], [5, 44], [8, 176], [10, 132], [11, 10], [20, 528], [22, 30], [40, 704], [44, 120]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [20, 0, 0, 0], 'outer_gens': [[441, 20, 840], [11, 900, 930], [891, 900, 40], [11, 900, 840]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [288, 121, 1, 126], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 5], [8, 4], [10, 4], [16, 1], [20, 2], [40, 1]], 'representations': {'PC': {'code': 29390069696332825863924223737834786188915380886037993510030009, 'gens': [1, 4, 5], 'pres': [7, -2, -2, -5, -2, 2, -2, -11, 14, 36, 4636, 25203, 24504, 2111, 3168, 102, 57125, 5052, 7579, 124, 47046, 11773, 17660]}, 'Perm': {'d': 27, 'gens': [15566290945153314513496892, 2986867059894, 32962608360045418070016000, 442648564714, 4491470470252, 5720734548052, 449879257393666699382784000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 20], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_2\\times C_{44}).C_{20}', 'transitive_degree': 176, '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': '20.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 110, 'aut_gen_orders': [10, 22], 'aut_gens': [[1, 20], [1, 40], [131, 20]], 'aut_group': '220.7', 'aut_hash': 7, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 220, 'aut_permdeg': 13, 'aut_perms': [141765265, 980890615], 'aut_phi_ratio': 2.75, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 11, 2, 1], [5, 11, 1, 4], [10, 11, 1, 4], [11, 10, 1, 1], [20, 11, 2, 4], [22, 10, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times F_{11}', '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': 110, 'autcentquo_group': '110.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 110, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 11, 2], [5, 11, 4], [10, 11, 4], [11, 10, 1], [20, 11, 8], [22, 10, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '110.1', 'commutator_count': 1, 'commutator_label': '11.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1', '11.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 11, 2, 1], [5, 11, 4, 1], [10, 11, 4, 1], [11, 10, 1, 1], [20, 11, 8, 1], [22, 10, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 144, 'exponent': 220, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 5, 11], 'faithful_reps': [[10, -1, 1]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '110.1', 'hash': 1, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 110, 'inner_gen_orders': [10, 11], 'inner_gens': [[1, 40], [201, 20]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 110, 'inner_split': True, 'inner_tex': 'F_{11}', 'inner_used': [1, 2], 'irrC_degree': 10, 'irrQ_degree': 10, 'irrQ_dim': 20, 'irrR_degree': 20, 'irrep_stats': [[1, 20], [10, 2]], 'label': '220.1', 'linC_count': 1, 'linC_degree': 10, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 1, 'linQ_dim': 12, 'linQ_dim_count': 1, 'linR_count': 5, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C11:C20', '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': 17, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 22, 'number_divisions': 8, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 12, 'number_subgroups': 52, 'old_label': None, 'order': 220, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 1], [4, 22], [5, 44], [10, 44], [11, 10], [20, 88], [22, 10]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[11, 200]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [4, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 2], [8, 1], [10, 2]], 'representations': {'PC': {'code': 48139442564190039, 'gens': [1, 4], 'pres': [4, -2, -2, -5, -11, 8, 21, 643, 647, 411]}, 'GLFq': {'d': 2, 'q': 121, 'gens': [15680518, 67421834, 148067751, 17715620]}, 'Perm': {'d': 15, 'gens': [6267312009, 13499992920, 16, 105871456080]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [20], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{11}:C_{20}', 'transitive_degree': 44, 'wreath_data': None, 'wreath_product': False}
