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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '92880.a', 'ambient_counter': 1, 'ambient_order': 92880, 'ambient_tex': 'C_{215}\\times D_{216}', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': False, 'core_order': 1032, 'counter': 60, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '92880.a.90._.A', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '90.A', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '90.1', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': None, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 90, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_5\\times D_9', 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '1032.6', 'subgroup_hash': None, 'subgroup_order': 1032, 'subgroup_tex': 'C_{1032}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '92880.a', 'aut_centralizer_order': None, 'aut_label': None, 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': None, 'complements': None, 'conjugacy_class_count': 1, 'contained_in': None, 'contains': None, 'core': None, 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [23220, 46440, 2160, 61920, 11610], 'label': '92880.a.90._.A', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': None, 'normal_contained_in': None, 'normal_contains': None, 'normalizer': None, 'old_label': '90.A', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '90._.A', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '1032.6', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 42, 'aut_gen_orders': [2, 2, 2, 42], 'aut_gens': [[1], [259], [517], [689], [433]], 'aut_group': '336.228', 'aut_hash': 228, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 336, 'aut_permdeg': 48, 'aut_perms': [10091931383442369960742999986450796027504316391780441757879680, 9316061658908865418814106937435637395952569130064207477962880, 4732679730338884481652528321590894592000, 11500183496481238404928372873442436158600531269021504619520], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [6, 1, 2, 1], [8, 1, 4, 1], [12, 1, 4, 1], [24, 1, 8, 1], [43, 1, 42, 1], [86, 1, 42, 1], [129, 1, 84, 1], [172, 1, 84, 1], [258, 1, 84, 1], [344, 1, 168, 1], [516, 1, 168, 1], [1032, 1, 336, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^3\\times C_{42}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 42, 'autcent_group': '336.228', 'autcent_hash': 228, 'autcent_nilpotent': True, 'autcent_order': 336, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\times C_{42}', '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], [3, 1, 2], [4, 1, 2], [6, 1, 2], [8, 1, 4], [12, 1, 4], [24, 1, 8], [43, 1, 42], [86, 1, 42], [129, 1, 84], [172, 1, 84], [258, 1, 84], [344, 1, 168], [516, 1, 168], [1032, 1, 336]], 'center_label': '1032.6', 'center_order': 1032, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '43.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 6, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['43.1', 1], ['8.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [6, 1, 2, 1], [8, 1, 4, 1], [12, 1, 4, 1], [24, 1, 8, 1], [43, 1, 42, 1], [86, 1, 42, 1], [129, 1, 84, 1], [172, 1, 84, 1], [258, 1, 84, 1], [344, 1, 168, 1], [516, 1, 168, 1], [1032, 1, 336, 1]], 'element_repr_type': 'PC', 'elementary': 258, 'eulerian_function': 1, 'exponent': 1032, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 43], 'faithful_reps': [[1, 0, 336]], 'familial': True, 'frattini_label': '4.1', 'frattini_quotient': '258.6', 'hash': 6, 'hyperelementary': 258, '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': 336, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 1032]], 'label': '1032.6', 'linC_count': None, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C1032', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, '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': 16, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 1032, 'number_divisions': 16, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 16, 'number_subgroups': 16, 'old_label': None, 'order': 1032, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 2], [6, 2], [8, 4], [12, 4], [24, 8], [43, 42], [86, 42], [129, 84], [172, 84], [258, 84], [344, 168], [516, 168], [1032, 336]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 42, 'outer_gen_orders': [2, 2, 2, 42], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[259], [517], [689], [433]], 'outer_group': '336.228', 'outer_hash': 228, 'outer_nilpotent': True, 'outer_order': 336, 'outer_permdeg': 48, 'outer_perms': [10091931383442369960742999986450796027504316391780441757879680, 9316061658908865418814106937435637395952569130064207477962880, 4732679730338884481652528321590894592000, 11500183496481238404928372873442436158600531269021504619520], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_{42}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 54, 'pgroup': 0, 'primary_abelian_invariants': [8, 3, 43], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 2], [2, 3], [4, 2], [8, 1], [42, 2], [84, 3], [168, 2], [336, 1]], 'representations': {'PC': {'code': 2451798891734250346931, 'gens': [1], 'pres': [5, -2, -2, -2, -3, -43, 10, 26, 42, 78]}, 'GLFp': {'d': 2, 'p': 173, 'gens': [232754763]}, 'Perm': {'d': 54, 'gens': [30414458170831798604149131517362761621603774100659339526144000000000000, 12985968052195387737912190492362074541654491115683674576650240000000000, 4276435011353800222356967049300996629350395915608859722383360000000000, 239244441730960389123926322991315430128767467520000000000, 59010256945620955738811989462269485937328128000000000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [1032], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{1032}', 'transitive_degree': 1032, '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': '860.11', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': None, 'aut_cyclic': None, 'aut_derived_length': None, 'aut_exponent': None, 'aut_gen_orders': None, 'aut_gens': [[1, 2], [51601, 2], [1, 10322], [17201, 2], [1, 34402], [46441, 34402], [1, 80842], [69661, 34402], [1, 11182], [1, 30082], [1, 52114], [11611, 33538], [61921, 2], [1, 30962]], 'aut_group': None, 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 2612736, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': 108.0, 'aut_solvable': None, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 108, 2, 1], [3, 2, 1, 1], [4, 2, 1, 1], [5, 1, 4, 1], [6, 2, 1, 1], [8, 2, 2, 1], [9, 2, 3, 1], [10, 1, 4, 1], [10, 108, 8, 1], [12, 2, 2, 1], [15, 2, 4, 1], [18, 2, 3, 1], [20, 2, 4, 1], [24, 2, 4, 1], [27, 2, 9, 1], [30, 2, 4, 1], [36, 2, 6, 1], [40, 2, 8, 1], [43, 1, 42, 1], [45, 2, 12, 1], [54, 2, 9, 1], [60, 2, 8, 1], [72, 2, 12, 1], [86, 1, 42, 1], [86, 108, 84, 1], [90, 2, 12, 1], [108, 2, 18, 1], [120, 2, 16, 1], [129, 2, 42, 1], [135, 2, 36, 1], [172, 2, 42, 1], [180, 2, 24, 1], [215, 1, 168, 1], [216, 2, 36, 1], [258, 2, 42, 1], [270, 2, 36, 1], [344, 2, 84, 1], [360, 2, 48, 1], [387, 2, 126, 1], [430, 1, 168, 1], [430, 108, 336, 1], [516, 2, 84, 1], [540, 2, 72, 1], [645, 2, 168, 1], [774, 2, 126, 1], [860, 2, 168, 1], [1032, 2, 168, 1], [1080, 2, 144, 1], [1161, 2, 378, 1], [1290, 2, 168, 1], [1548, 2, 252, 1], [1720, 2, 336, 1], [1935, 2, 504, 1], [2322, 2, 378, 1], [2580, 2, 336, 1], [3096, 2, 504, 1], [3870, 2, 504, 1], [4644, 2, 756, 1], [5160, 2, 672, 1], [5805, 2, 1512, 1], [7740, 2, 1008, 1], [9288, 2, 1512, 1], [11610, 2, 1512, 1], [15480, 2, 2016, 1], [23220, 2, 3024, 1], [46440, 2, 6048, 1]], 'aut_supersolvable': None, 'aut_tex': None, 'autcent_abelian': None, 'autcent_cyclic': None, 'autcent_exponent': None, 'autcent_group': None, 'autcent_hash': None, 'autcent_nilpotent': None, 'autcent_order': None, 'autcent_solvable': None, 'autcent_split': None, 'autcent_supersolvable': None, 'autcent_tex': None, 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': None, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 108, 2], [3, 2, 1], [4, 2, 1], [5, 1, 4], [6, 2, 1], [8, 2, 2], [9, 2, 3], [10, 1, 4], [10, 108, 8], [12, 2, 2], [15, 2, 4], [18, 2, 3], [20, 2, 4], [24, 2, 4], [27, 2, 9], [30, 2, 4], [36, 2, 6], [40, 2, 8], [43, 1, 42], [45, 2, 12], [54, 2, 9], [60, 2, 8], [72, 2, 12], [86, 1, 42], [86, 108, 84], [90, 2, 12], [108, 2, 18], [120, 2, 16], [129, 2, 42], [135, 2, 36], [172, 2, 42], [180, 2, 24], [215, 1, 168], [216, 2, 36], [258, 2, 42], [270, 2, 36], [344, 2, 84], [360, 2, 48], [387, 2, 126], [430, 1, 168], [430, 108, 336], [516, 2, 84], [540, 2, 72], [645, 2, 168], [774, 2, 126], [860, 2, 168], [1032, 2, 168], [1080, 2, 144], [1161, 2, 378], [1290, 2, 168], [1548, 2, 252], [1720, 2, 336], [1935, 2, 504], [2322, 2, 378], [2580, 2, 336], [3096, 2, 504], [3870, 2, 504], [4644, 2, 756], [5160, 2, 672], [5805, 2, 1512], [7740, 2, 1008], [9288, 2, 1512], [11610, 2, 1512], [15480, 2, 2016], [23220, 2, 3024], [46440, 2, 6048]], 'center_label': '430.4', 'center_order': 430, 'central_product': None, 'central_quotient': '216.6', 'commutator_count': None, 'commutator_label': '108.2', 'complements_known': False, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '5.1', '43.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': None, 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 108, 1, 2], [3, 2, 1, 1], [4, 2, 1, 1], [5, 1, 4, 1], [6, 2, 1, 1], [8, 2, 2, 1], [9, 2, 3, 1], [10, 1, 4, 1], [10, 108, 4, 2], [12, 2, 2, 1], [15, 2, 4, 1], [18, 2, 3, 1], [20, 2, 4, 1], [24, 2, 4, 1], [27, 2, 9, 1], [30, 2, 4, 1], [36, 2, 6, 1], [40, 2, 8, 1], [43, 1, 42, 1], [45, 2, 12, 1], [54, 2, 9, 1], [60, 2, 8, 1], [72, 2, 12, 1], [86, 1, 42, 1], [86, 108, 42, 2], [90, 2, 12, 1], [108, 2, 18, 1], [120, 2, 16, 1], [129, 2, 42, 1], [135, 2, 36, 1], [172, 2, 42, 1], [180, 2, 24, 1], [215, 1, 168, 1], [216, 2, 36, 1], [258, 2, 42, 1], [270, 2, 36, 1], [344, 2, 84, 1], [360, 2, 48, 1], [387, 2, 126, 1], [430, 1, 168, 1], [430, 108, 168, 2], [516, 2, 84, 1], [540, 2, 72, 1], [645, 2, 168, 1], [774, 2, 126, 1], [860, 2, 168, 1], [1032, 2, 168, 1], [1080, 2, 144, 1], [1161, 2, 378, 1], [1290, 2, 168, 1], [1548, 2, 252, 1], [1720, 2, 336, 1], [1935, 2, 504, 1], [2322, 2, 378, 1], [2580, 2, 336, 1], [3096, 2, 504, 1], [3870, 2, 504, 1], [4644, 2, 756, 1], [5160, 2, 672, 1], [5805, 2, 1512, 1], [7740, 2, 1008, 1], [9288, 2, 1512, 1], [11610, 2, 1512, 1], [15480, 2, 2016, 1], [23220, 2, 3024, 1], [46440, 2, 6048, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': None, 'exponent': 46440, 'exponents_of_order': [4, 3, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 5, 43], 'faithful_reps': None, 'familial': False, 'frattini_label': '36.2', 'frattini_quotient': '2580.a', 'hash': 8252518514231004068, 'hyperelementary': 2, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': [2, 108], 'inner_gens': [[1, 862], [92021, 2]], 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': 216, 'inner_split': None, 'inner_tex': 'D_{108}', 'inner_used': None, 'irrC_degree': None, 'irrQ_degree': None, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': None, 'label': '92880.a', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C215*D216', 'ngens': 9, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 3, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 3, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 3, 0, 1, 1, 1, 1, 0, 3, 1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 68, 'number_characteristic_subgroups': 68, 'number_conjugacy_classes': 23865, 'number_divisions': 72, 'number_normal_subgroups': 76, 'number_subgroup_autclasses': None, 'number_subgroup_classes': 176, 'number_subgroups': 2464, 'old_label': None, 'order': 92880, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 217], [3, 2], [4, 2], [5, 4], [6, 2], [8, 4], [9, 6], [10, 868], [12, 4], [15, 8], [18, 6], [20, 8], [24, 8], [27, 18], [30, 8], [36, 12], [40, 16], [43, 42], [45, 24], [54, 18], [60, 16], [72, 24], [86, 9114], [90, 24], [108, 36], [120, 32], [129, 84], [135, 72], [172, 84], [180, 48], [215, 168], [216, 72], [258, 84], [270, 72], [344, 168], [360, 96], [387, 252], [430, 36456], [516, 168], [540, 144], [645, 336], [774, 252], [860, 336], [1032, 336], [1080, 288], [1161, 756], [1290, 336], [1548, 504], [1720, 672], [1935, 1008], [2322, 756], [2580, 672], [3096, 1008], [3870, 1008], [4644, 1512], [5160, 1344], [5805, 3024], [7740, 2016], [9288, 3024], [11610, 3024], [15480, 4032], [23220, 6048], [46440, 12096]], 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': False, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': None, 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 12096, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': None, 'pc_rank': 2, 'perfect': False, 'permutation_degree': 83, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5, 43], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': None, 'representations': {'PC': {'code': '1096951812441675889561543958029022806192386800727960228303521620920099157509397257716289539', 'gens': [1, 2], 'pres': [9, -2, -2, -2, -2, -3, -3, -3, -5, -43, 15517, 46, 46550, 74, 124131, 102, 310324, 175, 1117157, 212, 3910038, 249, 430]}, 'GLFp': {'d': 2, 'p': 431, 'gens': [33741641391, 34427086131]}, 'Perm': {'d': 83, 'gens': [5798020784168351923471795668096371402475786408871669769057970501287286461613705117439410510689797966827356160000000000000, 486960375269620878689081798325597689448764866845210205901594702668733794514823671976214876832195391921340087568920420940193]}}, 'schur_multiplier': [2], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [2, 430], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{215}\\times D_{216}', 'transitive_degree': 46440, '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': '10.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 36, 'aut_gen_orders': [6, 36], 'aut_gens': [[1, 2], [1, 22], [81, 56]], 'aut_group': '216.53', 'aut_hash': 53, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 216, 'aut_permdeg': 13, 'aut_perms': [91571880, 4115104569], 'aut_phi_ratio': 9.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 1, 1], [5, 1, 4, 1], [9, 2, 3, 1], [10, 9, 4, 1], [15, 2, 4, 1], [45, 2, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{36}:C_6', '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': 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], [5, 1, 4], [9, 2, 3], [10, 9, 4], [15, 2, 4], [45, 2, 12]], 'center_label': '5.1', 'center_order': 5, 'central_product': True, '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', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['18.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 2, 1, 1], [5, 1, 4, 1], [9, 2, 3, 1], [10, 9, 4, 1], [15, 2, 4, 1], [45, 2, 12, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 18, 'exponent': 90, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[2, 0, 12]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '30.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [2, 9], 'inner_gens': [[1, 52], [41, 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': 24, 'irrQ_dim': 24, 'irrR_degree': 4, 'irrep_stats': [[1, 10], [2, 20]], 'label': '90.1', 'linC_count': 12, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 2, 'linQ_dim': 10, 'linQ_dim_count': 2, 'linR_count': 18, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5*D9', '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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 30, 'number_divisions': 8, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 12, 'number_subgroups': 32, 'old_label': None, 'order': 90, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 9], [3, 2], [5, 4], [9, 6], [10, 36], [15, 8], [45, 24]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [12], 'outer_gen_pows': [0], 'outer_gens': [[1, 76]], 'outer_group': '12.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [867], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{12}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 2], [6, 1], [8, 1], [24, 1]], 'representations': {'PC': {'code': 793089809943, 'gens': [1, 2], 'pres': [4, -2, -3, -3, -5, 417, 29, 794, 46]}, 'GLFp': {'d': 2, 'p': 71, 'gens': [25053771, 24203612]}, 'Perm': {'d': 14, 'gens': [1118437200, 33, 7784552160, 14545077840]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [10], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times D_9', 'transitive_degree': 45, 'wreath_data': None, 'wreath_product': False}
