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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '960.414', 'ambient_counter': 414, 'ambient_order': 960, 'ambient_tex': 'C_{15}:\\SD_{64}', 'central': False, 'central_factor': False, 'centralizer_order': 30, 'characteristic': False, 'core_order': 4, 'counter': 53, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '960.414.120.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '120.b1.a1', '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': 120, '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': '8.3', 'subgroup_hash': 3, 'subgroup_order': 8, 'subgroup_tex': 'D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '960.414', 'aut_centralizer_order': 192, 'aut_label': '120.b1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '8.3', 'aut_weyl_index': 768, 'centralizer': '32.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['24.b1.a1', '40.b1.a1', '60.b1.a1'], 'contains': ['240.a1.a1', '240.b1.a1'], 'core': '240.a1.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [3465, -1, 2840, -1, 3439, -1, 3439, -1], 'generators': [1, 48], 'label': '960.414.120.b1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '30.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.b1.a1', 'old_label': '120.b1.a1', 'projective_image': '480.145', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '120.b1.a1', 'subgroup_fusion': None, 'weyl_group': '8.3'}
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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': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 6], [3, 2]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [5, 9], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [4, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', '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': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [4, 2, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [4, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 6], [5, 2]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 1]], 'label': '8.3', 'linC_count': 1, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D4', 'ngens': 2, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 5, 'number_divisions': 5, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 10, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 5], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [2], 'outer_gens': [[3, 2]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1]], 'representations': {'PC': {'code': 294, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 37, 16]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [29, 56, 24], 'family': 'COPlus'}, {'d': 1, 'q': 4, 'gens': [7, 16, 1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [12, 55, 56]}, 'Perm': {'d': 4, 'gens': [6, 16, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4', 'transitive_degree': 4, 'wreath_data': ['C_2', 'C_2', '2T1'], 'wreath_product': True}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 48, 'aut_gen_orders': [6, 12, 8, 12, 16, 8], 'aut_gens': [[1, 2, 64], [25, 350, 256], [53, 366, 64], [25, 678, 704], [45, 382, 832], [21, 10, 64], [57, 690, 128]], 'aut_group': None, 'aut_hash': 6950560539663341654, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6144, 'aut_permdeg': 68, 'aut_perms': [2461780145879062200730792622449939043551955063681635858637599709449468411784282345376169275012474, 1014792707258886901549527036773705070028667494378949901602854452484321887321276942372250287482494, 1549972735519110186835926131041640578426084462831704262078638580788456188760199712054293706079575, 1916982968239194572648954821156502651934227945607364819368652182351650046027396293078305119893995, 221280448575936090142466024486558205174523782380114951677095945069073012347942129759940644562530, 1435301145371438351489557142525859388431071726080906858613440863209014863986458531306482401281267], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 16, 1, 1], [3, 2, 1, 1], [4, 2, 1, 1], [4, 48, 1, 1], [5, 1, 4, 1], [6, 2, 1, 1], [6, 16, 2, 1], [8, 2, 2, 1], [10, 1, 4, 1], [10, 16, 4, 1], [12, 4, 1, 1], [15, 2, 4, 1], [16, 2, 4, 1], [20, 2, 4, 1], [20, 48, 4, 1], [24, 4, 2, 1], [30, 2, 4, 1], [30, 16, 8, 1], [32, 6, 8, 1], [40, 2, 8, 1], [48, 4, 4, 1], [60, 4, 4, 1], [80, 2, 16, 1], [120, 4, 8, 1], [160, 6, 32, 1], [240, 4, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3:((C_2\\times C_4\\times C_8).C_2^5)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '16.10', 'autcent_hash': 10, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '384.12439', 'autcentquo_hash': 12439, 'autcentquo_nilpotent': False, 'autcentquo_order': 384, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_8:C_4\\times S_3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 16, 1], [3, 2, 1], [4, 2, 1], [4, 48, 1], [5, 1, 4], [6, 2, 1], [6, 16, 2], [8, 2, 2], [10, 1, 4], [10, 16, 4], [12, 4, 1], [15, 2, 4], [16, 2, 4], [20, 2, 4], [20, 48, 4], [24, 4, 2], [30, 2, 4], [30, 16, 8], [32, 6, 8], [40, 2, 8], [48, 4, 4], [60, 4, 4], [80, 2, 16], [120, 4, 8], [160, 6, 32], [240, 4, 16]], 'center_label': '10.2', 'center_order': 10, 'central_product': True, 'central_quotient': '96.33', 'commutator_count': 1, 'commutator_label': '48.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 414, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['192.79', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 16, 1, 1], [3, 2, 1, 1], [4, 2, 1, 1], [4, 48, 1, 1], [5, 1, 4, 1], [6, 2, 1, 1], [6, 16, 2, 1], [8, 2, 2, 1], [10, 1, 4, 1], [10, 16, 4, 1], [12, 4, 1, 1], [15, 2, 4, 1], [16, 2, 4, 1], [20, 2, 4, 1], [20, 48, 4, 1], [24, 4, 2, 1], [30, 2, 4, 1], [30, 16, 8, 1], [32, 6, 8, 1], [40, 2, 8, 1], [48, 4, 4, 1], [60, 4, 4, 1], [80, 2, 16, 1], [120, 4, 8, 1], [160, 6, 32, 1], [240, 4, 16, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 36, 'exponent': 480, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[4, 0, 16]], 'familial': False, 'frattini_label': '16.1', 'frattini_quotient': '60.11', 'hash': 414, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 48, 'inner_gen_orders': [2, 16, 3], 'inner_gens': [[1, 30, 64], [37, 2, 704], [1, 322, 64]], 'inner_hash': 33, 'inner_nilpotent': False, 'inner_order': 96, 'inner_split': True, 'inner_tex': 'C_3:D_{16}', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 64, 'irrQ_dim': 128, 'irrR_degree': 8, 'irrep_stats': [[1, 20], [2, 95], [4, 35]], 'label': '960.414', 'linC_count': 784, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 20, 'linQ_degree_count': 4, 'linQ_dim': 22, 'linQ_dim_count': 8, 'linR_count': 32, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C15:SD64', 'ngens': 8, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 28, 'number_characteristic_subgroups': 30, 'number_conjugacy_classes': 150, 'number_divisions': 28, 'number_normal_subgroups': 30, 'number_subgroup_autclasses': 64, 'number_subgroup_classes': 64, 'number_subgroups': 280, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 17], [3, 2], [4, 50], [5, 4], [6, 34], [8, 4], [10, 68], [12, 4], [15, 8], [16, 8], [20, 200], [24, 8], [30, 136], [32, 48], [40, 16], [48, 16], [60, 16], [80, 32], [120, 32], [160, 192], [240, 64]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 8, 'outer_gen_orders': [2, 4, 8], 'outer_gen_pows': [60, 0, 0], 'outer_gens': [[29, 2, 64], [1, 2, 448], [1, 10, 64]], 'outer_group': '64.83', 'outer_hash': 83, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 14, 'outer_perms': [6227020800, 43908480, 18551], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4\\times C_8', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 40, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 3], [4, 7], [8, 5], [16, 5], [32, 2], [64, 2]], 'representations': {'PC': {'code': 12500026086317505159869450276054114573877255, 'gens': [1, 2, 7], 'pres': [8, -2, -2, -2, -2, -2, -2, -3, -5, 481, 41, 1442, 66, 1795, 91, 1924, 116, 19726, 222]}, 'Perm': {'d': 40, 'gens': [22549638304972078078917952527688210704718885200, 2742459984884017090663705661449868715685996800, 33, 44074981004299847962731929876229936143673523200, 64983610912345163270973308885219756218504702080, 85930875187807345334493624000829403491086737280, 106867121372654151338721916856919182376937514880, 5760]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{15}:\\SD_{64}', 'transitive_degree': 480, 'wreath_data': None, 'wreath_product': False}
