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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '165888.eb', 'ambient_counter': 106, 'ambient_order': 165888, 'ambient_tex': 'C_4\\times A_4^3:S_4', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': True, 'core_order': 165888, 'counter': 1, 'cyclic': False, 'direct': None, 'hall': 6, 'label': '165888.eb.1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '1.a1', 'outer_equivalence': True, 'perfect': False, 'proper': False, 'quotient': '1.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 1, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_1', 'simple': False, 'solvable': True, 'special_labels': ['R', 'L0', 'D0', 'C0'], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '165888.eb', 'subgroup_hash': 813647509787346299, 'subgroup_order': 165888, 'subgroup_tex': 'C_4\\times A_4^3:S_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '165888.eb', 'aut_centralizer_order': None, 'aut_label': '1.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '41472.A', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': None, 'contains': None, 'core': '1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [26, 2969, 6, 373], 'generators': [791394514560, 10867651201560, 95880, 241003462, 19006560, 270655227360, 7, 387132496560, 1093680, 8033837137200, 15307625906040, 355659096960, 355477656960, 1960645680, 474904782142], 'label': '165888.eb.1.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '1.a1', 'normal_contained_in': [], 'normal_contains': [], 'normalizer': '1.a1', 'old_label': '1.a1', 'projective_image': '41472.j', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '1.a1', 'subgroup_fusion': None, 'weyl_group': '41472.j'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 72, 'aut_gen_orders': [8, 6, 12, 6], 'aut_gens': [[1408754672767, 1500846370697], [6722052392887, 4577362968257], [15042388965967, 13868106749417], [1361106810720, 7245948302777], [13078881573840, 10063361070742]], 'aut_group': '331776.gh', 'aut_hash': 9100794241782841809, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 331776, 'aut_permdeg': 866, 'aut_perms': 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'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [2, 27, 1, 4], [2, 108, 1, 2], [2, 324, 1, 2], [2, 432, 2, 1], [3, 24, 1, 1], [3, 192, 1, 1], [3, 512, 1, 1], [3, 1152, 1, 1], [4, 1, 2, 1], [4, 9, 2, 1], [4, 27, 2, 2], [4, 108, 1, 2], [4, 108, 2, 2], [4, 216, 1, 2], [4, 216, 2, 1], [4, 324, 1, 2], [4, 324, 2, 2], [4, 432, 2, 5], [4, 648, 1, 2], [4, 648, 2, 1], [4, 1296, 2, 6], [6, 24, 1, 1], [6, 144, 1, 2], [6, 192, 1, 1], [6, 216, 1, 2], [6, 512, 1, 1], [6, 576, 1, 2], [6, 864, 1, 2], [6, 1152, 1, 1], [6, 3456, 1, 2], [6, 3456, 2, 1], [8, 432, 2, 2], [8, 1296, 2, 2], [9, 4608, 2, 1], [12, 24, 2, 1], [12, 144, 2, 1], [12, 192, 2, 1], [12, 216, 2, 1], [12, 512, 2, 1], [12, 576, 2, 1], [12, 864, 1, 2], [12, 864, 2, 2], [12, 1152, 2, 1], [12, 1728, 1, 2], [12, 1728, 2, 1], [12, 1728, 4, 2], [12, 3456, 2, 4], [18, 4608, 2, 1], [24, 1728, 4, 2], [36, 4608, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_2\\times S_4^3).D_6', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 72, 'autcentquo_group': '82944.d', 'autcentquo_hash': 6931629392945487332, 'autcentquo_nilpotent': False, 'autcentquo_order': 82944, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'A_4^3.(C_2\\times S_4)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 9, 2], [2, 27, 4], [2, 108, 2], [2, 324, 2], [2, 432, 2], [3, 24, 1], [3, 192, 1], [3, 512, 1], [3, 1152, 1], [4, 1, 2], [4, 9, 2], [4, 27, 4], [4, 108, 6], [4, 216, 4], [4, 324, 6], [4, 432, 10], [4, 648, 4], [4, 1296, 12], [6, 24, 1], [6, 144, 2], [6, 192, 1], [6, 216, 2], [6, 512, 1], [6, 576, 2], [6, 864, 2], [6, 1152, 1], [6, 3456, 4], [8, 432, 4], [8, 1296, 4], [9, 4608, 2], [12, 24, 2], [12, 144, 2], [12, 192, 2], [12, 216, 2], [12, 512, 2], [12, 576, 2], [12, 864, 6], [12, 1152, 2], [12, 1728, 12], [12, 3456, 8], [18, 4608, 2], [24, 1728, 8], [36, 4608, 4]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '41472.j', 'commutator_count': 1, 'commutator_label': '20736.r', 'complements_known': False, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 15, 'conjugacy_classes_known': True, 'counter': 106, 'cyclic': False, 'derived_length': 5, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['41472.j', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [2, 27, 1, 4], [2, 108, 1, 2], [2, 324, 1, 2], [2, 432, 1, 2], [3, 24, 1, 1], [3, 192, 1, 1], [3, 512, 1, 1], [3, 1152, 1, 1], [4, 1, 2, 1], [4, 9, 2, 1], [4, 27, 2, 2], [4, 108, 1, 2], [4, 108, 2, 2], [4, 216, 1, 2], [4, 216, 2, 1], [4, 324, 1, 2], [4, 324, 2, 2], [4, 432, 1, 4], [4, 432, 2, 3], [4, 648, 1, 2], [4, 648, 2, 1], [4, 1296, 1, 6], [4, 1296, 2, 3], [6, 24, 1, 1], [6, 144, 1, 2], [6, 192, 1, 1], [6, 216, 1, 2], [6, 512, 1, 1], [6, 576, 1, 2], [6, 864, 1, 2], [6, 1152, 1, 1], [6, 3456, 1, 4], [8, 432, 1, 2], [8, 432, 2, 1], [8, 1296, 1, 2], [8, 1296, 2, 1], [9, 4608, 1, 2], [12, 24, 2, 1], [12, 144, 2, 1], [12, 192, 2, 1], [12, 216, 2, 1], [12, 512, 2, 1], [12, 576, 2, 1], [12, 864, 1, 2], [12, 864, 2, 2], [12, 1152, 2, 1], [12, 1728, 1, 2], [12, 1728, 2, 3], [12, 1728, 4, 1], [12, 3456, 1, 2], [12, 3456, 2, 3], [18, 4608, 1, 2], [24, 1728, 2, 2], [24, 1728, 4, 1], [36, 4608, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 72, 'exponents_of_order': [11, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[9, 0, 8], [18, 0, 2], [27, 0, 12], [36, 0, 8], [54, 0, 12], [81, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '82944.o', 'hash': 813647509787346299, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 72, 'inner_gen_orders': [4, 9], 'inner_gens': [[1408754672767, 11976219900737], [2790750792367, 1500846370697]], 'inner_hash': 4961977302514827477, 'inner_nilpotent': False, 'inner_order': 41472, 'inner_split': True, 'inner_tex': 'A_4^3:S_4', 'inner_used': [1, 2], 'irrC_degree': 9, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 18, 'irrep_stats': [[1, 8], [2, 4], [3, 8], [6, 16], [8, 12], [9, 16], [12, 8], [18, 4], [27, 24], [36, 16], [54, 24], [81, 8]], 'label': '165888.eb', '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': False, 'metacyclic': False, 'monomial': None, 'name': 'C4*A4^3:S4', 'ngens': 2, '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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 90, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 148, 'number_divisions': 105, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 14195, 'number_subgroup_classes': 15920, 'number_subgroups': 4765433, 'old_label': None, 'order': 165888, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 1855], [3, 1880], [4, 26048], [6, 19304], [8, 6912], [9, 9216], [12, 59200], [18, 9216], [24, 13824], [36, 18432]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1408754672760, 1500846370702], [1408754672760, 1500846370697], [5426624811007, 13360009622537]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 10, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [3, 4], [4, 1], [6, 6], [8, 6], [9, 8], [12, 8], [16, 3], [18, 6], [24, 3], [27, 12], [36, 9], [54, 14], [72, 4], [81, 4], [108, 6], [162, 2], [216, 1]], 'representations': {'PC': {'code': '63160286675392184216458933786033274469142055384192784219940781148552140595833172523061819287019647846324092945916504621013713792491538246959302826962517560635103073208132070856591173453567519266123256076710060887444807762286904196069195934557619298556762002292852193748099419610173854588843062428489038936285217570040608198414526352744989904488919368345429898770353279361845139389883338179564355663756535989573852126094595620038419825609557296707747502038063033817801139349034075392336', 'gens': [1, 2, 5, 6, 8, 10, 12, 13, 14, 15], 'pres': [15, 2, 2, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 2, 803160, 4866781, 76, 1877942, 122, 4763043, 1085808, 10558804, 308719, 1457134, 982624, 154864, 4598645, 1119980, 1481255, 388040, 83045, 260, 13366086, 1300341, 1179396, 1549851, 906, 17959687, 3841942, 2796517, 1893652, 277987, 79282, 13417, 352, 5145128, 3803783, 4445318, 411533, 42188, 178283, 23716809, 8179224, 691239, 2547054, 540069, 29784, 20349, 82014, 28929, 444, 7413130, 7945, 1021720, 233695, 47590, 23845, 25855, 18940, 15863051, 7931546, 1866281, 2099576, 42206, 1721, 1010907, 4296282, 273837, 252807, 42237, 14172, 4827, 3672, 1179388, 4898923, 22738, 544393, 15238, 22813, 7708, 1423, 22161614, 16912829, 2381444, 3207659, 923474, 465839, 220829, 4169, 38609]}, 'Perm': {'d': 16, 'gens': [1408754672767, 1500846370697]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 288, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_4\\times A_4^3:S_4', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
-
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 72, 'aut_gen_orders': [8, 6, 12, 6], 'aut_gens': [[1408754672767, 1500846370697], [6722052392887, 4577362968257], [15042388965967, 13868106749417], [1361106810720, 7245948302777], [13078881573840, 10063361070742]], 'aut_group': '331776.gh', 'aut_hash': 9100794241782841809, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 331776, 'aut_permdeg': 866, 'aut_perms': 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'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [2, 27, 1, 4], [2, 108, 1, 2], [2, 324, 1, 2], [2, 432, 2, 1], [3, 24, 1, 1], [3, 192, 1, 1], [3, 512, 1, 1], [3, 1152, 1, 1], [4, 1, 2, 1], [4, 9, 2, 1], [4, 27, 2, 2], [4, 108, 1, 2], [4, 108, 2, 2], [4, 216, 1, 2], [4, 216, 2, 1], [4, 324, 1, 2], [4, 324, 2, 2], [4, 432, 2, 5], [4, 648, 1, 2], [4, 648, 2, 1], [4, 1296, 2, 6], [6, 24, 1, 1], [6, 144, 1, 2], [6, 192, 1, 1], [6, 216, 1, 2], [6, 512, 1, 1], [6, 576, 1, 2], [6, 864, 1, 2], [6, 1152, 1, 1], [6, 3456, 1, 2], [6, 3456, 2, 1], [8, 432, 2, 2], [8, 1296, 2, 2], [9, 4608, 2, 1], [12, 24, 2, 1], [12, 144, 2, 1], [12, 192, 2, 1], [12, 216, 2, 1], [12, 512, 2, 1], [12, 576, 2, 1], [12, 864, 1, 2], [12, 864, 2, 2], [12, 1152, 2, 1], [12, 1728, 1, 2], [12, 1728, 2, 1], [12, 1728, 4, 2], [12, 3456, 2, 4], [18, 4608, 2, 1], [24, 1728, 4, 2], [36, 4608, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_2\\times S_4^3).D_6', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 72, 'autcentquo_group': '82944.d', 'autcentquo_hash': 6931629392945487332, 'autcentquo_nilpotent': False, 'autcentquo_order': 82944, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'A_4^3.(C_2\\times S_4)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 9, 2], [2, 27, 4], [2, 108, 2], [2, 324, 2], [2, 432, 2], [3, 24, 1], [3, 192, 1], [3, 512, 1], [3, 1152, 1], [4, 1, 2], [4, 9, 2], [4, 27, 4], [4, 108, 6], [4, 216, 4], [4, 324, 6], [4, 432, 10], [4, 648, 4], [4, 1296, 12], [6, 24, 1], [6, 144, 2], [6, 192, 1], [6, 216, 2], [6, 512, 1], [6, 576, 2], [6, 864, 2], [6, 1152, 1], [6, 3456, 4], [8, 432, 4], [8, 1296, 4], [9, 4608, 2], [12, 24, 2], [12, 144, 2], [12, 192, 2], [12, 216, 2], [12, 512, 2], [12, 576, 2], [12, 864, 6], [12, 1152, 2], [12, 1728, 12], [12, 3456, 8], [18, 4608, 2], [24, 1728, 8], [36, 4608, 4]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '41472.j', 'commutator_count': 1, 'commutator_label': '20736.r', 'complements_known': False, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 15, 'conjugacy_classes_known': True, 'counter': 106, 'cyclic': False, 'derived_length': 5, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['41472.j', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [2, 27, 1, 4], [2, 108, 1, 2], [2, 324, 1, 2], [2, 432, 1, 2], [3, 24, 1, 1], [3, 192, 1, 1], [3, 512, 1, 1], [3, 1152, 1, 1], [4, 1, 2, 1], [4, 9, 2, 1], [4, 27, 2, 2], [4, 108, 1, 2], [4, 108, 2, 2], [4, 216, 1, 2], [4, 216, 2, 1], [4, 324, 1, 2], [4, 324, 2, 2], [4, 432, 1, 4], [4, 432, 2, 3], [4, 648, 1, 2], [4, 648, 2, 1], [4, 1296, 1, 6], [4, 1296, 2, 3], [6, 24, 1, 1], [6, 144, 1, 2], [6, 192, 1, 1], [6, 216, 1, 2], [6, 512, 1, 1], [6, 576, 1, 2], [6, 864, 1, 2], [6, 1152, 1, 1], [6, 3456, 1, 4], [8, 432, 1, 2], [8, 432, 2, 1], [8, 1296, 1, 2], [8, 1296, 2, 1], [9, 4608, 1, 2], [12, 24, 2, 1], [12, 144, 2, 1], [12, 192, 2, 1], [12, 216, 2, 1], [12, 512, 2, 1], [12, 576, 2, 1], [12, 864, 1, 2], [12, 864, 2, 2], [12, 1152, 2, 1], [12, 1728, 1, 2], [12, 1728, 2, 3], [12, 1728, 4, 1], [12, 3456, 1, 2], [12, 3456, 2, 3], [18, 4608, 1, 2], [24, 1728, 2, 2], [24, 1728, 4, 1], [36, 4608, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 72, 'exponents_of_order': [11, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[9, 0, 8], [18, 0, 2], [27, 0, 12], [36, 0, 8], [54, 0, 12], [81, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '82944.o', 'hash': 813647509787346299, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 72, 'inner_gen_orders': [4, 9], 'inner_gens': [[1408754672767, 11976219900737], [2790750792367, 1500846370697]], 'inner_hash': 4961977302514827477, 'inner_nilpotent': False, 'inner_order': 41472, 'inner_split': True, 'inner_tex': 'A_4^3:S_4', 'inner_used': [1, 2], 'irrC_degree': 9, 'irrQ_degree': 18, 'irrQ_dim': 18, 'irrR_degree': 18, 'irrep_stats': [[1, 8], [2, 4], [3, 8], [6, 16], [8, 12], [9, 16], [12, 8], [18, 4], [27, 24], [36, 16], [54, 24], [81, 8]], 'label': '165888.eb', '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': False, 'metacyclic': False, 'monomial': None, 'name': 'C4*A4^3:S4', 'ngens': 2, '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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 90, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 148, 'number_divisions': 105, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 14195, 'number_subgroup_classes': 15920, 'number_subgroups': 4765433, 'old_label': None, 'order': 165888, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 1855], [3, 1880], [4, 26048], [6, 19304], [8, 6912], [9, 9216], [12, 59200], [18, 9216], [24, 13824], [36, 18432]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1408754672760, 1500846370702], [1408754672760, 1500846370697], [5426624811007, 13360009622537]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 10, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [3, 4], [4, 1], [6, 6], [8, 6], [9, 8], [12, 8], [16, 3], [18, 6], [24, 3], [27, 12], [36, 9], [54, 14], [72, 4], [81, 4], [108, 6], [162, 2], [216, 1]], 'representations': {'PC': {'code': '63160286675392184216458933786033274469142055384192784219940781148552140595833172523061819287019647846324092945916504621013713792491538246959302826962517560635103073208132070856591173453567519266123256076710060887444807762286904196069195934557619298556762002292852193748099419610173854588843062428489038936285217570040608198414526352744989904488919368345429898770353279361845139389883338179564355663756535989573852126094595620038419825609557296707747502038063033817801139349034075392336', 'gens': [1, 2, 5, 6, 8, 10, 12, 13, 14, 15], 'pres': [15, 2, 2, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 2, 803160, 4866781, 76, 1877942, 122, 4763043, 1085808, 10558804, 308719, 1457134, 982624, 154864, 4598645, 1119980, 1481255, 388040, 83045, 260, 13366086, 1300341, 1179396, 1549851, 906, 17959687, 3841942, 2796517, 1893652, 277987, 79282, 13417, 352, 5145128, 3803783, 4445318, 411533, 42188, 178283, 23716809, 8179224, 691239, 2547054, 540069, 29784, 20349, 82014, 28929, 444, 7413130, 7945, 1021720, 233695, 47590, 23845, 25855, 18940, 15863051, 7931546, 1866281, 2099576, 42206, 1721, 1010907, 4296282, 273837, 252807, 42237, 14172, 4827, 3672, 1179388, 4898923, 22738, 544393, 15238, 22813, 7708, 1423, 22161614, 16912829, 2381444, 3207659, 923474, 465839, 220829, 4169, 38609]}, 'Perm': {'d': 16, 'gens': [1408754672767, 1500846370697]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 288, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_4\\times A_4^3:S_4', 'transitive_degree': 36, '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': '1.1', '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': [], '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]], '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]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': [], 'composition_length': 0, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1, 'exponent': 1, 'exponents_of_order': [], 'factors_of_aut_order': [], 'factors_of_order': [], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '1.1', 'hash': 1, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [], 'inner_gens': [], '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, 1]], 'label': '1.1', 'linC_count': 1, 'linC_degree': 0, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 0, 'linQ_degree_count': 1, 'linQ_dim': 0, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 0, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C1', 'ngens': 0, 'nilpotency_class': 0, 'nilpotent': True, 'normal_counts': [1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 1, 'number_characteristic_subgroups': 1, 'number_conjugacy_classes': 1, 'number_divisions': 1, 'number_normal_subgroups': 1, 'number_subgroup_autclasses': 1, 'number_subgroup_classes': 1, 'number_subgroups': 1, 'old_label': None, 'order': 1, 'order_factorization_type': 0, 'order_stats': [[1, 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': 0, 'perfect': True, 'permutation_degree': 1, 'pgroup': 1, 'primary_abelian_invariants': [], 'quasisimple': False, 'rank': 0, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 1]], 'representations': {'PC': {'code': 0, 'gens': [], 'pres': []}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_1', 'transitive_degree': 1, 'wreath_data': None, 'wreath_product': False}