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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '7200.eb', 'ambient_counter': 106, 'ambient_order': 7200, 'ambient_tex': 'A_6.D_{10}', 'central': False, 'central_factor': False, 'centralizer_order': 5, 'characteristic': True, 'core_order': 3600, 'counter': 4, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '7200.eb.2.c1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '2.c1.a1', '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': False, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '3600.dv', 'subgroup_hash': 9218817735889571177, 'subgroup_order': 3600, 'subgroup_tex': 'A_6.C_{10}', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '7200.eb', 'aut_centralizer_order': None, 'aut_label': '2.c1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1440.a1.a1', 'complements': ['3600.c1.a1', '3600.b1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1', '10.a1.a1', '20.c1.a1', '72.b1.a1', '90.g1.a1'], 'core': '2.c1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5456, 8594, 4481, 8594, 5675, 8112, 5703, 8112], 'generators': [194035930998, 94363448751, 94363315200], 'label': '7200.eb.2.c1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.c1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['4.a1.a1', '10.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2.c1.a1', 'projective_image': '7200.eb', 'quotient_action_image': '2.1', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '2.c1.a1', 'subgroup_fusion': None, 'weyl_group': '1440.5841'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '10.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 120, 'aut_gen_orders': [20, 4, 8], 'aut_gens': [[261578315121, 194038414576, 94363315200], [194036829962, 94366603135, 367434144000], [367434421050, 261575988722, 194035564800], [367436172004, 261574979436, 194035564800]], 'aut_group': '5760.db', 'aut_hash': 4554243923608446656, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5760, 'aut_permdeg': 184, 'aut_perms': [167453651961307078932111922499322224033518370800028818305729144786782313723381557154200994072721792866559482167480962900658381624436989957223296202553903952981597780019862943453231074371350439837135108382483063760014292981908073313561444545700165700924107365954515617068951374790485283692159251668046774423431943870097577666725749517255357, 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'C_4\\times S_6:C_2', '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': 120, 'autcentquo_group': '1440.5841', 'autcentquo_hash': 5841, 'autcentquo_nilpotent': False, 'autcentquo_order': 1440, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_6:C_2', 'cc_stats': [[1, 1, 1], [2, 45, 1], [3, 80, 1], [4, 90, 1], [4, 180, 1], [5, 1, 4], [5, 144, 5], [8, 90, 2], [10, 45, 4], [15, 80, 4], [20, 90, 4], [20, 180, 4], [40, 90, 8]], 'center_label': '5.1', 'center_order': 5, 'central_product': True, 'central_quotient': '720.765', 'commutator_count': 1, 'commutator_label': '360.118', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '5.1', '360.118'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 100, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['5.1', 1], ['720.765', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 45, 1, 1], [3, 80, 1, 1], [4, 90, 1, 1], [4, 180, 1, 1], [5, 1, 4, 1], [5, 144, 1, 1], [5, 144, 4, 1], [8, 90, 2, 1], [10, 45, 4, 1], [15, 80, 4, 1], [20, 90, 4, 1], [20, 180, 4, 1], [40, 90, 8, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 1404, 'exponent': 120, 'exponents_of_order': [4, 2, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[9, 0, 8], [10, 0, 12], [16, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '3600.dv', 'hash': 9218817735889571177, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 120, 'inner_gen_orders': [2, 4, 1], 'inner_gens': [[261578315121, 194037065182, 94363315200], [261575120685, 194038414576, 94363315200], [261578315121, 194038414576, 94363315200]], 'inner_hash': 765, 'inner_nilpotent': False, 'inner_order': 720, 'inner_split': True, 'inner_tex': 'A_6.C_2', 'inner_used': [1, 2], 'irrC_degree': 9, 'irrQ_degree': 36, 'irrQ_dim': 36, 'irrR_degree': 18, 'irrep_stats': [[1, 10], [9, 10], [10, 15], [16, 5]], 'label': '3600.dv', 'linC_count': 8, 'linC_degree': 9, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 13, 'linQ_degree_count': 4, 'linQ_dim': 13, 'linQ_dim_count': 4, 'linR_count': 8, 'linR_degree': 11, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'A6.C10', 'ngens': 3, '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, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 14, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 40, 'number_divisions': 14, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 49, 'number_subgroup_classes': 51, 'number_subgroups': 1730, 'old_label': None, 'order': 3600, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 45], [3, 80], [4, 270], [5, 724], [8, 180], [10, 180], [15, 320], [20, 1080], [40, 720]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [194036085462, 0], 'outer_gens': [[94366629779, 367435012836, 261574790400], [194039089521, 94366164976, 367434144000]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 2], [9, 2], [10, 1], [16, 1], [20, 1], [36, 2], [40, 1], [64, 1], [80, 1]], 'representations': {'Perm': {'d': 15, 'gens': [261578315121, 194038414576, 94363315200]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [10], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'A_6.C_{10}', 'transitive_degree': 50, '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': 120, 'aut_gen_orders': [8, 2, 20, 2, 4, 4, 5, 4, 8], 'aut_gens': [[361687258823, 87659010888, 94363315200], [87659158445, 361687085871, 261574790400], [361689243518, 87659063151, 94363315200], [361686337730, 281176352274, 194035564800], [361687258823, 87658116663, 94363315200], [361686337730, 87659190424, 94363315200], [361686225768, 87658711926, 94363315200], [361686225768, 87658253871, 94363315200], [13412076712, 281174482143, 367434144000], [361686127392, 87659201207, 94363315200]], 'aut_group': '28800.ds', 'aut_hash': 856545384635260607, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 28800, 'aut_permdeg': 330, 'aut_perms': 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1], [15, 160, 2, 1], [20, 180, 2, 1], [20, 360, 2, 1], [40, 180, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_5\\times S_6:C_2', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': '28800.ds', 'autcentquo_hash': 856545384635260607, 'autcentquo_nilpotent': False, 'autcentquo_order': 28800, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\times S_6:C_2', 'cc_stats': [[1, 1, 1], [2, 45, 1], [2, 150, 1], [2, 180, 1], [3, 80, 1], [4, 90, 1], [4, 180, 1], [4, 450, 1], [5, 2, 2], [5, 144, 1], [5, 288, 2], [6, 1200, 1], [8, 180, 1], [8, 900, 1], [10, 90, 2], [10, 720, 1], [15, 160, 2], [20, 180, 2], [20, 360, 2], [40, 180, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '7200.eb', 'commutator_count': 1, 'commutator_label': '1800.555', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '5.1', '360.118'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 106, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 45, 1, 1], [2, 150, 1, 1], [2, 180, 1, 1], [3, 80, 1, 1], [4, 90, 1, 1], [4, 180, 1, 1], [4, 450, 1, 1], [5, 2, 2, 1], [5, 144, 1, 1], [5, 288, 2, 1], [6, 1200, 1, 1], [8, 180, 1, 1], [8, 900, 1, 1], [10, 90, 2, 1], [10, 720, 1, 1], [15, 160, 2, 1], [20, 180, 2, 1], [20, 360, 2, 1], [40, 180, 4, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 468, 'exponent': 120, 'exponents_of_order': [5, 2, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[18, 1, 4], [20, 1, 6], [32, 1, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '7200.eb', 'hash': 3417747234816056484, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 120, 'inner_gen_orders': [2, 10, 5], 'inner_gens': [[361687258823, 13414532536, 261574790400], [281176909882, 87659010888, 261574790400], [13413178823, 361687842888, 94363315200]], 'inner_hash': 3417747234816056484, 'inner_nilpotent': False, 'inner_order': 7200, 'inner_split': True, 'inner_tex': 'A_6.D_{10}', 'inner_used': [1, 2], 'irrC_degree': 18, 'irrQ_degree': 36, 'irrQ_dim': 36, 'irrR_degree': 18, 'irrep_stats': [[1, 4], [2, 4], [9, 4], [10, 2], [16, 2], [18, 4], [20, 7], [32, 2]], 'label': '7200.eb', 'linC_count': 16, 'linC_degree': 11, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 13, 'linQ_degree_count': 8, 'linQ_dim': 13, 'linQ_dim_count': 8, 'linR_count': 16, 'linR_degree': 11, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'A6.D10', 'ngens': 3, '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, 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': 20, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 29, 'number_divisions': 20, 'number_normal_subgroups': 9, 'number_subgroup_autclasses': 132, 'number_subgroup_classes': 132, 'number_subgroups': 13166, 'old_label': None, 'order': 7200, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 375], [3, 80], [4, 720], [5, 724], [6, 1200], [8, 1080], [10, 900], [15, 320], [20, 1080], [40, 720]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [0], 'outer_gens': [[361687258823, 180625238088, 367434144000]], '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': None, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [4, 2], [9, 4], [10, 2], [16, 2], [20, 1], [36, 2], [40, 1], [64, 1], [80, 1]], 'representations': {'Perm': {'d': 15, 'gens': [361687258823, 87659010888, 94363315200]}}, 'schur_multiplier': [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': 'A_6.D_{10}', 'transitive_degree': 50, '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}
