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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '576.6608', 'ambient_counter': 6608, 'ambient_order': 576, 'ambient_tex': 'D_8:S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 96, 'counter': 17, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '576.6608.3.a1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '3.a1.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': 3, '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': '192.1331', 'subgroup_hash': 1331, 'subgroup_order': 192, 'subgroup_tex': 'D_8:D_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '576.6608', 'aut_centralizer_order': 2, 'aut_label': '3.a1', 'aut_quo_index': None, 'aut_stab_index': 6, 'aut_weyl_group': '384.20133', 'aut_weyl_index': 12, 'centralizer': '288.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['6.a1.b1', '6.b1.a1', '6.c1.b1', '6.e1.a1', '6.f1.b1', '6.g1.a1', '6.h1.a1', '6.i1.a1', '6.j1.b1', '6.k1.a1', '6.l1.b1', '6.m1.a1', '6.n1.b1', '6.o1.a1', '6.p1.a1', '9.a1.a1'], 'core': '6.a1.b1', 'coset_action_label': None, 'count': 3, 'diagramx': [7944, -1, 8422, -1, 7742, -1, 3953, -1], 'generators': [1, 288, 432, 2, 8, 12, 72], 'label': '576.6608.3.a1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '3.a1.a1', 'projective_image': '288.958', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '3.a1.a1', 'subgroup_fusion': None, 'weyl_group': '96.209'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 6, 4, 2, 4, 2], 'aut_gens': [[1, 2, 4, 8], [1, 98, 4, 136], [1, 2, 132, 105], [97, 50, 4, 57], [97, 2, 4, 8], [1, 2, 4, 153], [1, 2, 100, 8]], 'aut_group': '768.1076474', 'aut_hash': 1076474, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 768, 'aut_permdeg': 13, 'aut_perms': [6, 1037927568, 1037967840, 3628801, 1516838401, 1], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 3, 2, 1], [2, 4, 1, 1], [2, 4, 2, 1], [2, 6, 1, 1], [2, 12, 1, 1], [2, 12, 2, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 4, 1, 1], [4, 6, 1, 2], [4, 12, 1, 1], [6, 2, 1, 1], [6, 4, 1, 1], [6, 8, 1, 1], [6, 8, 2, 1], [8, 4, 2, 1], [8, 12, 2, 1], [12, 4, 1, 2], [12, 8, 1, 1], [24, 8, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6\\times D_4^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '48.51', 'autcentquo_hash': 51, 'autcentquo_nilpotent': False, 'autcentquo_order': 48, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times D_6', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 3, 2], [2, 4, 3], [2, 6, 1], [2, 12, 3], [3, 2, 1], [4, 2, 2], [4, 4, 1], [4, 6, 2], [4, 12, 1], [6, 2, 1], [6, 4, 1], [6, 8, 3], [8, 4, 2], [8, 12, 2], [12, 4, 2], [12, 8, 1], [24, 8, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '96.209', 'commutator_count': 1, 'commutator_label': '12.2', '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'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1331, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['32.43', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 3, 1, 2], [2, 4, 1, 3], [2, 6, 1, 1], [2, 12, 1, 3], [3, 2, 1, 1], [4, 2, 1, 2], [4, 4, 1, 1], [4, 6, 1, 2], [4, 12, 1, 1], [6, 2, 1, 1], [6, 4, 1, 1], [6, 8, 1, 3], [8, 4, 1, 2], [8, 12, 1, 2], [12, 4, 1, 2], [12, 8, 1, 1], [24, 8, 1, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 524160, 'exponent': 24, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 1, 1]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '48.51', 'hash': 1331, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 2, 12], 'inner_gens': [[1, 98, 4, 104], [97, 2, 4, 56], [1, 2, 4, 136], [97, 146, 68, 8]], 'inner_hash': 209, 'inner_nilpotent': False, 'inner_order': 96, 'inner_split': True, 'inner_tex': 'D_4\\times D_6', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 12], [4, 4], [8, 1]], 'label': '192.1331', 'linC_count': 16, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 16, 'linQ_dim': 6, 'linQ_dim_count': 16, 'linR_count': 16, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D8:D6', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 26, 'number_characteristic_subgroups': 51, 'number_conjugacy_classes': 33, 'number_divisions': 33, 'number_normal_subgroups': 101, 'number_subgroup_autclasses': 214, 'number_subgroup_classes': 298, 'number_subgroups': 944, 'old_label': None, 'order': 192, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 63], [3, 2], [4, 32], [6, 30], [8, 32], [12, 16], [24, 16]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 2, 4, 105], [1, 2, 100, 8], [97, 2, 4, 8]], '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': 4, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 4], [8, 1]], 'representations': {'PC': {'code': 1181801189995173760068150233185281, 'gens': [1, 2, 3, 4], 'pres': [7, -2, -2, -2, -2, -2, -2, -3, 1373, 2915, 794, 969, 80, 1971, 718, 102, 1699, 124, 1588]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101729095570500, 91793140190669372, 125101737214347233, 91678746541283614]}, 'GLZN': {'d': 2, 'p': 24, 'gens': [14017, 13969, 13831, 318173, 179719, 14113, 13853]}, 'Perm': {'d': 11, 'gens': [1, 1134264, 1625040, 3841464, 8807280, 12136560, 3]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_8:D_6', 'transitive_degree': 24, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [4, 2, 12, 4, 12, 4, 4], 'aut_gens': [[1, 2, 4, 24], [433, 2, 148, 24], [1, 2, 148, 552], [1, 106, 502, 512], [145, 482, 68, 408], [433, 306, 246, 232], [289, 482, 110, 224], [289, 10, 118, 512]], 'aut_group': None, 'aut_hash': 855117975400641664, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 4608, 'aut_permdeg': 20, 'aut_perms': [38100640314439359, 390190046418787113, 1795621573714778650, 22464718387480575, 34188073637943988, 92881772453629922, 1712011397134736234], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 1, 2], [2, 6, 2, 1], [2, 9, 2, 1], [2, 12, 2, 1], [2, 36, 1, 2], [3, 2, 2, 1], [3, 4, 1, 1], [4, 2, 1, 1], [4, 6, 2, 1], [4, 12, 2, 1], [4, 18, 1, 1], [6, 2, 2, 1], [6, 4, 1, 1], [6, 8, 2, 2], [6, 12, 2, 1], [6, 16, 1, 2], [6, 24, 2, 1], [8, 4, 1, 1], [8, 12, 2, 1], [8, 36, 1, 1], [12, 4, 2, 1], [12, 8, 1, 1], [12, 12, 2, 1], [12, 24, 2, 1], [24, 8, 2, 2], [24, 24, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times D_6^2.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '288.1031', 'autcentquo_hash': 1031, 'autcentquo_nilpotent': False, 'autcentquo_order': 288, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_6^2:D_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 4, 2], [2, 6, 2], [2, 9, 2], [2, 12, 2], [2, 36, 2], [3, 2, 2], [3, 4, 1], [4, 2, 1], [4, 6, 2], [4, 12, 2], [4, 18, 1], [6, 2, 2], [6, 4, 1], [6, 8, 4], [6, 12, 2], [6, 16, 2], [6, 24, 2], [8, 4, 1], [8, 12, 2], [8, 36, 1], [12, 4, 2], [12, 8, 1], [12, 12, 2], [12, 24, 2], [24, 8, 4], [24, 24, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '288.958', 'commutator_count': 1, 'commutator_label': '36.8', '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', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 6608, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 1, 2], [2, 6, 1, 2], [2, 9, 1, 2], [2, 12, 1, 2], [2, 36, 1, 2], [3, 2, 1, 2], [3, 4, 1, 1], [4, 2, 1, 1], [4, 6, 1, 2], [4, 12, 1, 2], [4, 18, 1, 1], [6, 2, 1, 2], [6, 4, 1, 1], [6, 8, 1, 4], [6, 12, 1, 2], [6, 16, 1, 2], [6, 24, 1, 2], [8, 4, 1, 1], [8, 12, 1, 2], [8, 36, 1, 1], [12, 4, 1, 2], [12, 8, 1, 1], [12, 12, 1, 2], [12, 24, 1, 2], [24, 8, 1, 2], [24, 8, 2, 1], [24, 24, 1, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6814080, 'exponent': 24, 'exponents_of_order': [6, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 1, 2]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '144.192', 'hash': 6608, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 6, 12], 'inner_gens': [[1, 2, 436, 168], [1, 2, 20, 408], [145, 10, 4, 264], [433, 194, 340, 24]], 'inner_hash': 958, 'inner_nilpotent': False, 'inner_order': 288, 'inner_split': True, 'inner_tex': 'D_4\\times S_3^2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 20], [4, 10], [8, 5]], 'label': '576.6608', 'linC_count': 138, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 136, 'linQ_dim': 8, 'linQ_dim_count': 136, 'linR_count': 138, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D8:S3^2', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 33, 'number_characteristic_subgroups': 40, 'number_conjugacy_classes': 51, 'number_divisions': 50, 'number_normal_subgroups': 128, 'number_subgroup_autclasses': 368, 'number_subgroup_classes': 634, 'number_subgroups': 3732, 'old_label': None, 'order': 576, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 135], [3, 8], [4, 56], [6, 144], [8, 64], [12, 88], [24, 80]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 2, 398, 232], [1, 290, 4, 408], [433, 2, 20, 168], [1, 2, 308, 408]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [6, 289, 1, 126], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 20], [4, 10], [8, 3], [16, 1]], 'representations': {'PC': {'code': 1295870016757700129045054198312846593094195145255647278097, 'gens': [1, 2, 3, 5], 'pres': [8, -2, -2, -2, -3, -2, -2, -2, -3, 10466, 250, 66, 267, 6724, 8172, 2660, 116, 16133, 5773, 6357, 141, 13454, 6742, 166, 12303, 6167]}, 'Perm': {'d': 14, 'gens': [6314114309, 6354393989, 3426, 5699, 13294, 17764, 13971242880, 20122058880]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_8:S_3^2', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
