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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '648.295', 'ambient_counter': 295, 'ambient_order': 648, 'ambient_tex': 'C_2\\times C_9^2:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 1, 'counter': 38, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '648.295.36.c1.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '36.c1.c1', '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': 36, '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': '18.1', 'subgroup_hash': 1, 'subgroup_order': 18, 'subgroup_tex': 'D_9', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '648.295', 'aut_centralizer_order': 4, 'aut_label': '36.c1', 'aut_quo_index': None, 'aut_stab_index': 108, 'aut_weyl_group': '54.6', 'aut_weyl_index': 432, 'centralizer': '324.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['12.b1.a1', '18.a1.c1'], 'contains': ['72.b1.c1', '108.b1.a1'], 'core': '648.a1.a1', 'coset_action_label': '36T1035', 'count': 18, 'diagramx': [3303, -1, 8816, -1, 4727, -1, 3761, -1], 'generators': [2, 84, 216], 'label': '648.295.36.c1.c1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '18.a1.c1', 'old_label': '36.c1.c1', 'projective_image': '648.295', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '36.c1.c1', 'subgroup_fusion': None, 'weyl_group': '18.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 18, 'aut_gen_orders': [6, 9], 'aut_gens': [[1, 2], [1, 4], [15, 2]], 'aut_group': '54.6', 'aut_hash': 6, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 54, 'aut_permdeg': 9, 'aut_perms': [100930, 59573], 'aut_phi_ratio': 9.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 1, 1], [9, 2, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_9:C_6', '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': 18, 'autcentquo_group': '54.6', 'autcentquo_hash': 6, 'autcentquo_nilpotent': False, 'autcentquo_order': 54, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_9:C_6', 'cc_stats': [[1, 1, 1], [2, 9, 1], [3, 2, 1], [9, 2, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '18.1', 'commutator_count': 1, 'commutator_label': '9.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 1, 1], [9, 2, 3, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 18, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 1, 3]], 'familial': True, 'frattini_label': '3.1', 'frattini_quotient': '6.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [2, 9], 'inner_gens': [[1, 16], [5, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 18, 'inner_split': True, 'inner_tex': 'D_9', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 2, 'irrep_stats': [[1, 2], [2, 4]], 'label': '18.1', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D9', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 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': 6, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 16, 'old_label': None, 'order': 18, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 9], [3, 2], [9, 6]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 3, 'outer_gen_orders': [3], 'outer_gen_pows': [1], 'outer_gens': [[1, 4]], 'outer_group': '3.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 3, 'outer_permdeg': 3, 'outer_perms': [3], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [6, 1]], 'representations': {'PC': {'code': 106355, 'gens': [1, 2], 'pres': [3, -2, -3, -3, 97, 22, 110]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [58301504795026264, 69483174513730069]}, 'Lie': [{'d': 2, 'q': 8, 'gens': [577, 613], 'family': 'CSOMinus'}], 'GLFp': {'d': 2, 'p': 17, 'gens': [38240, 78609]}, 'Perm': {'d': 9, 'gens': [5759, 357120, 80884]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_9', 'transitive_degree': 9, '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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 72, 'aut_gen_orders': [18, 12, 12, 4], 'aut_gens': [[1, 4, 36], [451, 4, 612], [391, 296, 260], [407, 508, 644], [223, 592, 196]], 'aut_group': None, 'aut_hash': 7935501427042882995, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 23328, 'aut_permdeg': 324, 'aut_perms': 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155371710701012845664043747635513138717390229465834291504879574517818036100860869095444771890777017108955011462934672607594790523545107869950740707967791437404760244384864458265707838054165954431153573570510029663152966253788776651808841941098168967527510868544296775156631721438270084651895847870697207447860241908711812375031177158037317327352871696699460565379280553393641494805770469871603354726877694852413515474802220703001330853207376160437604755217086370327642112817623597552675592019821999794371132978902884035901133014158112073811609113937343278716591781092521629217844262858394339833639842864735585969174876204801657823412343586607048266429111509891433128514594611], 'aut_phi_ratio': 108.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 81, 1, 2], [3, 4, 2, 1], [4, 81, 4, 1], [6, 4, 2, 1], [9, 4, 18, 1], [18, 4, 18, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_9^2.C_{12}.C_6.C_2', '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': 72, 'autcentquo_group': '11664.lk', 'autcentquo_hash': 4509669642411733234, 'autcentquo_nilpotent': False, 'autcentquo_order': 11664, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_9^2:C_{24}:C_6', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 81, 2], [3, 4, 2], [4, 81, 4], [6, 4, 2], [9, 4, 18], [18, 4, 18]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '324.35', 'commutator_count': 1, 'commutator_label': '81.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 295, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['324.35', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 81, 1, 2], [3, 4, 1, 2], [4, 81, 2, 2], [6, 4, 1, 2], [9, 4, 3, 6], [18, 4, 3, 6]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 36, 'exponents_of_order': [4, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 1, 18]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '72.45', 'hash': 295, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [4, 9, 9], 'inner_gens': [[1, 72, 340], [581, 4, 36], [381, 4, 36]], 'inner_hash': 35, 'inner_nilpotent': False, 'inner_order': 324, 'inner_split': False, 'inner_tex': 'C_9^2:C_4', 'inner_used': [1, 2], 'irrC_degree': 4, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [4, 40]], 'label': '648.295', 'linC_count': 18, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 6, 'linQ_dim': 12, 'linQ_dim_count': 6, 'linR_count': 18, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*C9^2:C4', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 9, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 48, 'number_divisions': 22, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 36, 'number_subgroup_classes': 74, 'number_subgroups': 1276, 'old_label': None, 'order': 648, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 163], [3, 8], [4, 324], [6, 8], [9, 72], [18, 72]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6, 6], 'outer_gen_pows': [1, 2, 1], 'outer_gens': [[1, 172, 284], [3, 76, 632], [325, 292, 64]], 'outer_group': '72.48', 'outer_hash': 48, 'outer_nilpotent': False, 'outer_order': 72, 'outer_permdeg': 10, 'outer_perms': [720, 40350, 725761], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6\\times D_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [4, 4], [12, 12]], 'representations': {'PC': {'code': 61495217765367759772513961991141978309998402203, 'gens': [1, 3, 5], 'pres': [7, 2, 2, 3, 3, 2, 3, 3, 14, 1514, 345, 79, 6051, 346, 11904, 10721, 102, 1349, 12108, 166, 1182, 10597]}, 'Perm': {'d': 20, 'gens': [123600953843512129, 1, 2335037506744344, 265140936561700206, 393592528344800160, 399987211639843080, 32392094382437184]}}, 'schur_multiplier': [18], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_9^2:C_4', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
