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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '7680.fs', 'ambient_counter': 149, 'ambient_order': 7680, 'ambient_tex': 'C_2\\wr S_3\\times F_5', 'central': False, 'central_factor': False, 'centralizer_order': 1280, 'characteristic': True, 'core_order': 64, 'counter': 560, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '7680.fs.120.A', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '120.a1.N', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '120.36', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 36, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 120, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'S_3\\times F_5', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '64.267', 'subgroup_hash': 267, 'subgroup_order': 64, 'subgroup_tex': 'C_2^6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '7680.fs', 'aut_centralizer_order': None, 'aut_label': '120.A', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '6.A', 'complements': ['64.A'], 'conjugacy_class_count': 1, 'contained_in': ['24.B', '40.A'], 'contains': ['240.A'], 'core': '120.A', 'coset_action_label': None, 'count': 1, 'diagramx': [-1, 3262, -1, 4745], 'generators': [11536, 5167, 362880, 40279680, 16, 7], 'label': '7680.fs.120.A', 'mobius_quo': 0, 'mobius_sub': None, 'normal_closure': '120.A', 'normal_contained_in': ['24.B', '40.A'], 'normal_contains': ['240.A', '480.A'], 'normalizer': '1.a1', 'old_label': '120.a1.N', 'projective_image': '3840.fl', 'quotient_action_image': '6.1', 'quotient_action_kernel': '20.3', 'quotient_action_kernel_order': 20, 'quotient_fusion': None, 'short_label': '120.A', 'subgroup_fusion': None, 'weyl_group': '6.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '64.267', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 78120, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 2, 4, 8, 16, 32], [13, 2, 4, 8, 16, 32], [44, 57, 28, 55, 38, 54]], 'aut_group': '20158709760.a', 'aut_hash': 1127810123506825908, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20158709760, 'aut_permdeg': 63, 'aut_perms': [1088780861340731732174280893943159511512640016119869733192292622602376850177766021539680, 193386930810790965654717163452553945898231780614988004231894671695879290786822421539277], 'aut_phi_ratio': 629959680.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 63, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(6,2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 78120, 'autcent_group': '20158709760.a', 'autcent_hash': 1127810123506825908, 'autcent_nilpotent': False, 'autcent_order': 20158709760, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(6,2)', '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, 63]], 'center_label': '64.267', 'center_order': 64, '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', '2.1', '2.1', '2.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 267, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 6]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 63]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [6], 'factors_of_aut_order': [2, 3, 5, 7, 31], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '64.267', 'hash': 267, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1, 1, 1, 1], 'inner_gens': [[1, 2, 4, 8, 16, 32], [1, 2, 4, 8, 16, 32], [1, 2, 4, 8, 16, 32], [1, 2, 4, 8, 16, 32], [1, 2, 4, 8, 16, 32], [1, 2, 4, 8, 16, 32]], '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, 64]], 'label': '64.267', 'linC_count': None, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^6', 'ngens': 6, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 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': 64, 'number_divisions': 64, 'number_normal_subgroups': 2825, 'number_subgroup_autclasses': 7, 'number_subgroup_classes': 2825, 'number_subgroups': 2825, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 63]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 78120, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[13, 2, 4, 8, 16, 32], [44, 57, 28, 55, 38, 54]], 'outer_group': '20158709760.a', 'outer_hash': 1127810123506825908, 'outer_nilpotent': False, 'outer_order': 20158709760, 'outer_permdeg': 63, 'outer_perms': [1088780861340731732174280893943159511512640016119869733192292622602376850177766021539680, 193386930810790965654717163452553945898231780614988004231894671695879290786822421539277], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\GL(6,2)', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 12, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 6, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 64]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3, 4, 5, 6], 'pres': [6, -2, 2, 2, 2, 2, 2]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101750005088214, 125101750005092586, 25038659807093174, 125101729084386182, 24992906222183252, 125101729074820244]}, 'GLFp': {'d': 5, 'p': 2, 'gens': [5226561, 7292993, 5128257, 20225089, 19896385, 6210625]}, 'Perm': {'d': 12, 'gens': [39916800, 362880, 5040, 120, 6, 1]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 2, 2], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^6', 'transitive_degree': 64, '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': '16.10', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [12, 12, 12, 6, 4, 12, 6], 'aut_gens': [[1313941673647, 1313901763216, 1313901405496, 23544365644800, 1313988504338, 4097419966080, 1313941685183, 1313901400327, 1313988508662, 1313941317136], [1313941685167, 1313941317143, 1313901400336, 66784798080000, 1314028789789, 4097419966080, 1313941673656, 1313901405503, 1314028794110, 1313901763223], [64250440910767, 64250440536376, 64250400619583, 66784798080000, 64250487729337, 84998833920000, 64250440905623, 64250400631096, 64250487730781, 64250400982456], [64250440899256, 64250400988823, 64250400625927, 66784798080000, 64250487730781, 84998874199680, 64250440910767, 64250400619583, 64250487729938, 64250440542743], [45774870197183, 45774869822767, 45774829912327, 42019936358400, 45774917021514, 22411047859200, 45774870192016, 45774829905983, 45774917031597, 45774830268847], [87794806555576, 87794806187527, 87794766264367, 66784798080000, 87794893664510, 62861774976000, 87794806544063, 87794766270736, 87794893658742, 87794766633607], [45774870197167, 45774830268856, 45774829912336, 86480864870400, 45774957300342, 22411088138880, 45774870192023, 45774829905967, 45774957301194, 45774869822776], [45774870197176, 45774830275207, 45774829912343, 42019936358400, 45774957296861, 22411047859200, 45774870185663, 45774829917487, 45774957300342, 45774869829127]], 'aut_group': '10000.cb', 'aut_hash': 6508615515983140359, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 92160, 'aut_permdeg': 36, 'aut_perms': [186128879731455088386397247113876848293543, 121274920549065587544524335286964710914891, 55601215417986925979403520875673289654887, 121971118589304380161459977495306628926500, 277109440219773494795393360367282357573514, 146320496780632514487440865885824122999829, 122059645940425172979972520030793564087230], 'aut_phi_ratio': 45.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 3, 3, 2], [2, 5, 1, 2], [2, 6, 1, 2], [2, 6, 2, 1], [2, 6, 3, 1], [2, 10, 1, 1], [2, 15, 3, 2], [2, 24, 1, 1], [2, 30, 1, 2], [2, 30, 2, 1], [2, 30, 3, 1], [2, 120, 1, 1], [3, 32, 1, 1], [4, 5, 2, 2], [4, 10, 1, 2], [4, 15, 6, 2], [4, 24, 1, 1], [4, 24, 3, 2], [4, 30, 2, 4], [4, 30, 3, 2], [4, 120, 1, 5], [4, 120, 3, 6], [5, 4, 1, 1], [6, 32, 1, 1], [6, 32, 2, 1], [6, 160, 1, 2], [6, 160, 2, 1], [10, 4, 1, 1], [10, 8, 1, 1], [10, 12, 3, 2], [10, 24, 1, 2], [10, 24, 2, 1], [10, 24, 3, 1], [10, 96, 1, 1], [12, 160, 2, 4], [15, 128, 1, 1], [20, 96, 1, 1], [20, 96, 3, 2], [30, 128, 1, 1], [30, 128, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_5^4.\\OD_{16}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '11520.ck', 'autcentquo_hash': 5484343431313454187, 'autcentquo_nilpotent': False, 'autcentquo_order': 11520, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\times \\POPlus(4,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 3, 6], [2, 5, 2], [2, 6, 7], [2, 10, 1], [2, 15, 6], [2, 24, 1], [2, 30, 7], [2, 120, 1], [3, 32, 1], [4, 5, 4], [4, 10, 2], [4, 15, 12], [4, 24, 7], [4, 30, 14], [4, 120, 23], [5, 4, 1], [6, 32, 3], [6, 160, 4], [10, 4, 1], [10, 8, 1], [10, 12, 6], [10, 24, 7], [10, 96, 1], [12, 160, 8], [15, 128, 1], [20, 96, 7], [30, 128, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '3840.fl', 'commutator_count': 2, 'commutator_label': '480.1209', 'complements_known': True, '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', '3.1', '5.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 149, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['20.3', 1], ['384.20100', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 3, 1, 6], [2, 5, 1, 2], [2, 6, 1, 7], [2, 10, 1, 1], [2, 15, 1, 6], [2, 24, 1, 1], [2, 30, 1, 7], [2, 120, 1, 1], [3, 32, 1, 1], [4, 5, 2, 2], [4, 10, 2, 1], [4, 15, 2, 6], [4, 24, 1, 7], [4, 30, 2, 7], [4, 120, 1, 7], [4, 120, 2, 8], [5, 4, 1, 1], [6, 32, 1, 1], [6, 32, 2, 1], [6, 160, 1, 2], [6, 160, 2, 1], [10, 4, 1, 1], [10, 8, 1, 1], [10, 12, 1, 6], [10, 24, 1, 7], [10, 96, 1, 1], [12, 160, 2, 2], [12, 160, 4, 1], [15, 128, 1, 1], [20, 96, 1, 7], [30, 128, 1, 1], [30, 128, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 60, 'exponents_of_order': [9, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[24, 1, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '3840.fl', 'hash': 7850125273742898706, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 2, 2, 5, 2, 4, 2, 2, 2, 2], 'inner_gens': [[1313941673647, 1313901763216, 1313901405496, 42019936358400, 1313988519597, 4097419966080, 1313941685183, 1313901400327, 1313988508662, 1313941317136], [1313941673647, 1313901763216, 1313901405496, 42019936358400, 1314028799277, 4097419966080, 1313941685183, 1313901400327, 1314028783417, 1313941317136], [1313941673647, 1313901763216, 1313901405496, 42019936358400, 1313988510109, 4097419966080, 1313941685183, 1313901400327, 1313988508662, 1313941317136], [64250440899247, 64250400988816, 64250400631096, 23544365644800, 64250487729938, 84998874199680, 64250440910783, 64250400625927, 64250487734262, 64250440542736], [1313941680016, 1313941310767, 1313901393983, 42019936358400, 1313988504338, 4097419966080, 1313941685183, 1313901400327, 1313988514430, 1313901756847], [1313941673647, 1313901763216, 1313901405496, 86480864870400, 1313988504338, 4097419966080, 1313941685183, 1313901400327, 1313988508662, 1313941317136], [1313941673647, 1313901763216, 1313901405496, 42019936358400, 1313988504338, 4097419966080, 1313941685183, 1313901400327, 1313988503737, 1313941317136], [1313941673647, 1313901763216, 1313901405496, 42019936358400, 1313988504338, 4097419966080, 1313941685183, 1313901400327, 1313988520193, 1313941317136], [1313941673647, 1313941322303, 1313901405496, 42019936358400, 1313988514430, 4097419966080, 1313941680016, 1313901393983, 1313988508662, 1313901768383], [1313941673647, 1313901763216, 1313901405496, 42019936358400, 1314028799277, 4097419966080, 1313941685183, 1313901400327, 1314028783417, 1313941317136]], 'inner_hash': 2006955795942604059, 'inner_nilpotent': False, 'inner_order': 3840, 'inner_split': True, 'inner_tex': 'C_2^3:S_4\\times F_5', 'inner_used': [1, 2, 3, 4, 5, 6, 9], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 16], [2, 20], [3, 48], [4, 4], [6, 28], [8, 5], [12, 12], [24, 7]], 'label': '7680.fs', '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': 'C2wrS3*F5', 'ngens': 10, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 72, 'number_characteristic_subgroups': 64, 'number_conjugacy_classes': 140, 'number_divisions': 108, 'number_normal_subgroups': 118, 'number_subgroup_autclasses': 5074, 'number_subgroup_classes': 26488, 'number_subgroups': 455764, 'old_label': None, 'order': 7680, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 527], [3, 32], [4, 3568], [5, 4], [6, 736], [10, 348], [12, 1280], [15, 128], [20, 672], [30, 384]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1313941680016, 1313901756847, 1313901393983, 23544365644800, 1313988504338, 4097419966080, 1313941685183, 1313901400327, 1313988514430, 1313941310767], [1313941680023, 1313901768367, 1313901405496, 23544365644800, 1313988504338, 4097419966080, 1313941673656, 1313901400327, 1313988520193, 1313941322287], [1313941680023, 1313901768367, 1313901393967, 23544365644800, 1313988504338, 4097379686400, 1313941673656, 1313901405503, 1313988520193, 1313941322287]], 'outer_group': '24.14', 'outer_hash': 14, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 7, 'outer_perms': [23, 127, 847], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_6', 'pc_rank': 7, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [3, 24], [4, 9], [6, 26], [8, 4], [12, 19], [16, 1], [24, 7]], 'representations': {'PC': {'code': '864813536507956599260082900613784069878335311816391975937847916763159466510065151463932366567181620990005855359621003849093758120783562717249144000', 'gens': [1, 2, 4, 7, 9, 10, 11], 'pres': [11, 2, 2, 2, 2, 2, 3, 2, 5, 2, 2, 2, 133, 56, 127075, 64254, 37073, 124, 185684, 15415, 6846, 158, 192197, 97168, 55995, 66534, 114593, 30531, 15758, 7222, 226, 67591, 6376, 4275, 53501, 32722, 13428, 52842, 39653, 13264, 348501, 87163, 14574, 21845]}, 'Perm': {'d': 17, 'gens': [1313941673647, 1313901763216, 1313901405496, 23544365644800, 1313988504338, 4097419966080, 1313941685183, 1313901400327, 1313988508662, 1313941317136]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 40, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\wr S_3\\times F_5', 'transitive_degree': None, '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': 2, 'aut_exponent': 60, 'aut_gen_orders': [2, 4, 15], 'aut_gens': [[1, 2, 8], [1, 2, 88], [1, 2, 104], [81, 98, 8]], 'aut_group': '120.36', 'aut_hash': 36, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 120, 'aut_permdeg': 8, 'aut_perms': [1, 2160, 15147], 'aut_phi_ratio': 3.75, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 5, 1, 1], [2, 15, 1, 1], [3, 2, 1, 1], [4, 5, 1, 2], [4, 15, 1, 2], [5, 4, 1, 1], [6, 10, 1, 1], [10, 12, 1, 1], [12, 10, 1, 2], [15, 8, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3\\times F_5', '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': 60, 'autcentquo_group': '120.36', 'autcentquo_hash': 36, 'autcentquo_nilpotent': False, 'autcentquo_order': 120, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_5', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 5, 1], [2, 15, 1], [3, 2, 1], [4, 5, 2], [4, 15, 2], [5, 4, 1], [6, 10, 1], [10, 12, 1], [12, 10, 2], [15, 8, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '120.36', 'commutator_count': 1, 'commutator_label': '15.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 36, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['20.3', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 5, 1, 1], [2, 15, 1, 1], [3, 2, 1, 1], [4, 5, 2, 1], [4, 15, 2, 1], [5, 4, 1, 1], [6, 10, 1, 1], [10, 12, 1, 1], [12, 10, 2, 1], [15, 8, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 60, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[8, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '120.36', 'hash': 36, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 4, 15], 'inner_gens': [[1, 2, 88], [1, 2, 104], [41, 26, 8]], 'inner_hash': 36, 'inner_nilpotent': False, 'inner_order': 120, 'inner_split': True, 'inner_tex': 'S_3\\times F_5', 'inner_used': [1, 2, 3], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 4], [4, 2], [8, 1]], 'label': '120.36', 'linC_count': 8, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 4, 'linQ_dim': 6, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'S3*F5', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, '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': 15, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 15, 'number_divisions': 12, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 32, 'number_subgroup_classes': 32, 'number_subgroups': 132, 'old_label': None, 'order': 120, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 23], [3, 2], [4, 40], [5, 4], [6, 10], [10, 12], [12, 20], [15, 8]], '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': 3, 'perfect': False, 'permutation_degree': 8, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 3], [8, 1]], 'representations': {'PC': {'code': 150714042188947859463, 'gens': [1, 2, 4], 'pres': [5, -2, -2, -2, -3, -5, 26, 1763, 1048, 173, 78, 909, 614]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [108470310248616366, 91793119286428885]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [105573001540, 122216099379, 105504904686, 71239560474, 146113600095]}, 'Perm': {'d': 8, 'gens': [864, 1, 1680, 3, 6000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times F_5', 'transitive_degree': 15, 'wreath_data': None, 'wreath_product': False}
