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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '576.2132', 'ambient_counter': 2132, 'ambient_order': 576, 'ambient_tex': 'C_4^2.S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 288, 'characteristic': True, 'core_order': 6, 'counter': 199, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '576.2132.96.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '96.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '96.152', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 152, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 96, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3:C_4\\times Q_8', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '6.2', 'subgroup_hash': 2, 'subgroup_order': 6, 'subgroup_tex': 'C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '576.2132', 'aut_centralizer_order': 4608, 'aut_label': '96.a1', 'aut_quo_index': 12, 'aut_stab_index': 1, 'aut_weyl_group': '2.1', 'aut_weyl_index': 4608, 'centralizer': '2.b1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['32.a1.a1', '48.a1.a1', '48.c1.a1', '48.d1.a1'], 'contains': ['192.a1.a1', '288.a1.a1'], 'core': '96.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5501, 4057, 4939, 3182, 5780, 3332, 6046, 6797], 'generators': [312, 192], 'label': '576.2132.96.a1.a1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '96.a1.a1', 'normal_contained_in': ['32.a1.a1', '48.a1.a1', '48.d1.a1', '48.c1.a1'], 'normal_contains': ['192.a1.a1', '288.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '96.a1.a1', 'projective_image': '288.491', 'quotient_action_image': '2.1', 'quotient_action_kernel': '48.13', 'quotient_action_kernel_order': 48, 'quotient_fusion': None, 'short_label': '96.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [5]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_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, 1], [3, 1, 2], [6, 1, 2]], 'center_label': '6.2', 'center_order': 6, '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', '3.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 6, 'eulerian_function': 1, 'exponent': 6, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '6.2', 'hash': 2, 'hyperelementary': 6, '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 6]], 'label': '6.2', 'linC_count': 2, 'linC_degree': 1, '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': 'C6', 'ngens': 2, 'nilpotency_class': 1, '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': 6, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 6, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [3, 2], [6, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[5]], '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': 1, 'perfect': False, 'permutation_degree': 5, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2]], 'representations': {'PC': {'code': 21, 'gens': [1], 'pres': [2, -2, -3, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [73]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [31, 56]}, 'Perm': {'d': 5, 'gens': [24, 4]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6', 'transitive_degree': 6, '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': '32.36', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [4, 6, 6, 6, 6, 12, 6, 2, 4], 'aut_gens': [[1, 4, 48], [155, 452, 360], [329, 212, 360], [443, 460, 336], [465, 284, 336], [331, 220, 240], [185, 436, 72], [27, 500, 336], [3, 140, 528], [443, 476, 48]], 'aut_group': None, 'aut_hash': 1609739320405297462, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 9216, 'aut_permdeg': 72, 'aut_perms': [24724805921113586222024596191312426510679559898937294596624835922701730571625208788870174883445530271488, 5223024514767853899704787056351667791324550171018167673212285017470009866658891457284758323803676828660, 25252653296887664586029157417445434144196329638792097023148420212229292828842951937294006610791475167016, 37870521287056057790367227697732965941118084028804070507423299294435268889697833170815573029633675635206, 58842023021206000440308100463062481875000704360494142423160420402786243064131359226190280660244206622642, 35151181205594122679059389635926973737605546482077528019097816193773851410247384144063443177981089523911, 5185704859922526854165836621863100386285251183553843381443402603787839270339644703314371207833266137717, 15356189333506491933188203438086694136572433276121111457833285255838917368029002464095453708197818739584, 50777794181403380090203586328515887399750033406093584284050202818017382916036873088264062429307265857568], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 2], [3, 4, 1, 1], [4, 1, 2, 2], [4, 2, 2, 2], [4, 6, 4, 2], [6, 2, 1, 6], [6, 4, 1, 3], [8, 6, 8, 1], [8, 9, 8, 1], [8, 18, 4, 1], [12, 2, 2, 4], [12, 4, 2, 6], [12, 4, 4, 2], [12, 12, 4, 2], [24, 12, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_6^2.C_2^6.C_2^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '128.2328', 'autcent_hash': 2328, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^7', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '72.46', 'autcentquo_hash': 46, 'autcentquo_nilpotent': False, 'autcentquo_order': 72, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times D_6', 'cc_stats': [[1, 1, 1], [2, 1, 3], [3, 2, 2], [3, 4, 1], [4, 1, 4], [4, 2, 4], [4, 6, 8], [6, 2, 6], [6, 4, 3], [8, 6, 8], [8, 9, 8], [8, 18, 4], [12, 2, 8], [12, 4, 20], [12, 12, 8], [24, 12, 8]], 'center_label': '8.2', 'center_order': 8, 'central_product': False, 'central_quotient': '72.46', 'commutator_count': 1, 'commutator_label': '18.5', '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': 2132, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 2], [3, 4, 1, 1], [4, 1, 2, 2], [4, 2, 1, 2], [4, 2, 2, 1], [4, 6, 1, 4], [4, 6, 2, 2], [6, 2, 1, 6], [6, 4, 1, 3], [8, 6, 4, 2], [8, 9, 4, 2], [8, 18, 4, 1], [12, 2, 2, 4], [12, 4, 1, 4], [12, 4, 2, 8], [12, 12, 1, 4], [12, 12, 2, 2], [24, 12, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 5376, 'exponent': 24, 'exponents_of_order': [6, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '72.46', 'hash': 2132, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6, 6], 'inner_gens': [[1, 20, 336], [33, 4, 528], [289, 100, 48]], 'inner_hash': 46, 'inner_nilpotent': False, 'inner_order': 72, 'inner_split': True, 'inner_tex': 'S_3\\times D_6', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 40], [4, 24]], 'label': '576.2132', 'linC_count': 128, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 40, 'linQ_dim': 10, 'linQ_dim_count': 48, 'linR_count': 36, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4^2.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': 40, 'number_characteristic_subgroups': 56, 'number_conjugacy_classes': 96, 'number_divisions': 56, 'number_normal_subgroups': 106, 'number_subgroup_autclasses': 154, 'number_subgroup_classes': 226, 'number_subgroups': 564, 'old_label': None, 'order': 576, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 3], [3, 8], [4, 60], [6, 24], [8, 192], [12, 192], [24, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0, 0, 0], 'outer_gens': [[289, 332, 48], [1, 20, 528], [27, 316, 336], [1, 316, 552], [3, 20, 240], [169, 164, 72]], 'outer_group': '128.2320', 'outer_hash': 2320, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 12, 'outer_perms': [16680, 87107880, 127387567, 40284841, 11520, 87096382], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_2^4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 8], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 18], [4, 19], [8, 11]], 'representations': {'PC': {'code': 11772575952342122912478170723414721005075001097, 'gens': [1, 3, 6], 'pres': [8, -2, -2, -2, -2, -3, -2, -2, -3, 16, 2505, 482, 66, 1283, 91, 1284, 16133, 6357, 141, 6742, 166, 6167]}, 'GLZN': {'d': 2, 'p': 30, 'gens': [189007, 40966, 640637, 297011, 630313, 626113, 513019, 32773]}, 'Perm': {'d': 22, 'gens': [58408856625034631545, 102438021137715221304, 163511332286129543544, 209376870451689434424, 209376870451676928000, 3719544, 6706022400, 3]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 8], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4^2.S_3^2', 'transitive_degree': 192, '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': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [4, 2, 12, 6], 'aut_gens': [[1, 2, 8], [77, 74, 40], [5, 54, 40], [29, 14, 60], [25, 46, 33]], 'aut_group': '2304.lo', 'aut_hash': 4355994722122854182, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2304, 'aut_permdeg': 15, 'aut_perms': [650168507640, 87660962047, 375700591560, 174367510583], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 1], [4, 2, 6, 1], [4, 3, 4, 1], [4, 6, 6, 1], [6, 2, 1, 3], [12, 4, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4:D_6^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '36.10', 'autcentquo_hash': 10, 'autcentquo_nilpotent': False, 'autcentquo_order': 36, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3^2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [3, 2, 1], [4, 2, 6], [4, 3, 4], [4, 6, 6], [6, 2, 3], [12, 4, 6]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '24.14', 'commutator_count': 1, 'commutator_label': '6.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 152, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.1', 1], ['8.4', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 1], [4, 2, 1, 6], [4, 3, 2, 2], [4, 6, 2, 3], [6, 2, 1, 3], [12, 4, 1, 6]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 112, 'exponent': 12, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '24.14', 'hash': 152, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 6], 'inner_gens': [[1, 2, 56], [1, 2, 88], [49, 18, 8]], 'inner_hash': 14, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': True, 'inner_tex': 'C_2\\times D_6', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 12], [4, 2]], 'label': '96.152', 'linC_count': 16, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 8, 'linQ_dim': 8, 'linQ_dim_count': 60, 'linR_count': 4, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3:C4*Q8', 'ngens': 6, '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': 12, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 30, 'number_divisions': 25, 'number_normal_subgroups': 51, 'number_subgroup_autclasses': 32, 'number_subgroup_classes': 70, 'number_subgroups': 114, 'old_label': None, 'order': 96, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 60], [6, 6], [12, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 6, 2, 2], 'outer_gen_pows': [0, 24, 0, 0, 0], 'outer_gens': [[1, 6, 8], [25, 26, 8], [24, 27, 9], [1, 2, 12], [5, 6, 8]], 'outer_group': '96.226', 'outer_hash': 226, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 8, 'outer_perms': [7, 143, 1456, 11520, 5160], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^2\\times S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 14], [4, 3]], 'representations': {'PC': {'code': 5150697430811021289153, 'gens': [1, 2, 4], 'pres': [6, -2, -2, -2, -2, -2, -3, 288, 31, 1347, 1065, 69, 1210, 88, 1163]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [122109387504, 27627749610, 116213468753, 75598583751, 136164604377, 124030584379]}, 'Perm': {'d': 15, 'gens': [6270929280, 87098239, 87103793, 87091200, 18619, 93405312000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3:C_4\\times Q_8', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
