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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '256.12517', 'ambient_counter': 12517, 'ambient_order': 256, 'ambient_tex': 'C_4^2.C_2^4', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 64, 'counter': 28, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '256.12517.4.k1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '4.k1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '4.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^2', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '64.12', 'subgroup_hash': 12, 'subgroup_order': 64, 'subgroup_tex': 'C_4.D_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '256.12517', 'aut_centralizer_order': None, 'aut_label': '4.k1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '64.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['2.c1.a1', '2.h1.a1', '2.i1.b1'], 'contains': ['8.d1.a1', '8.k1.a1', '8.l1.b1'], 'core': '4.k1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5114, 4333, 4832, 4224, 5071, 4346, 4813, 4216], 'generators': [35, 4], 'label': '256.12517.4.k1.a1', 'mobius_quo': 0, 'mobius_sub': 2, 'normal_closure': '4.k1.a1', 'normal_contained_in': ['2.c1.a1', '2.h1.a1', '2.i1.b1'], 'normal_contains': ['8.d1.a1', '8.k1.a1', '8.l1.b1'], 'normalizer': '1.a1.a1', 'old_label': '4.k1.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.k1.a1', 'subgroup_fusion': None, 'weyl_group': None}
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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': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 8, 'aut_gen_orders': [2, 2, 2, 2, 2, 2, 2, 2, 2], 'aut_gens': [[1, 2, 16], [1, 3, 4], [1, 6, 16], [17, 34, 48], [9, 34, 16], [1, 22, 48], [21, 38, 48], [41, 10, 16], [33, 34, 16], [41, 34, 16]], 'aut_group': '512.418967', 'aut_hash': 4972023617141994759, 'aut_nilpotency_class': 4, 'aut_nilpotent': True, 'aut_order': 512, 'aut_permdeg': 12, 'aut_perms': [41505125, 11290327, 167160, 98155463, 295108696, 172200, 16, 83629560, 98155440], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 8, 2, 1], [4, 2, 2, 2], [4, 4, 1, 1], [8, 4, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4^2:D_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': 4, 'autcentquo_group': '32.27', 'autcentquo_hash': 27, 'autcentquo_nilpotent': True, 'autcentquo_order': 32, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\wr C_2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 8, 2], [4, 2, 4], [4, 4, 1], [8, 4, 8]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '16.3', 'commutator_count': 1, 'commutator_label': '8.2', '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': 12, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 8, 1, 2], [4, 2, 1, 4], [4, 4, 1, 1], [8, 4, 4, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 8, 'exponents_of_order': [6], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.2', 'frattini_quotient': '4.2', 'hash': 12, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4, 2], 'inner_gens': [[1, 62, 48], [21, 2, 48], [33, 34, 16]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 16, 'inner_split': True, 'inner_tex': 'C_2^2:C_4', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 10], [4, 1]], 'label': '64.12', 'linC_count': 16, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 8, 'linQ_dim': 8, 'linQ_dim_count': 8, 'linR_count': 4, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4.D8', 'ngens': 2, 'nilpotency_class': 3, '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': 8, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 19, 'number_divisions': 13, 'number_normal_subgroups': 19, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 42, 'number_subgroups': 105, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 19], [4, 12], [8, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [4, 4, 0, 20], 'outer_gens': [[37, 2, 16], [5, 2, 16], [33, 6, 16], [5, 7, 4]], 'outer_group': '32.27', 'outer_hash': 27, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [36273, 10905, 38579, 6322], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\wr C_2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 5]], 'representations': {'PC': {'code': 73257990429348214, 'gens': [1, 2, 5], 'pres': [6, -2, 2, -2, -2, 2, -2, 745, 31, 218, 50, 1444, 730, 88]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [34876415, 29555524, 37559389, 28816400, 28233362, 33722916]}, 'Perm': {'d': 16, 'gens': [287408224654, 1340370222031, 2890861761247, 18058, 4296764102400, 4296764127293]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4.D_8', 'transitive_degree': 32, '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': 2, 'aut_exponent': 4, 'aut_gen_orders': [4, 4, 4, 4, 4, 4, 4, 2], 'aut_gens': [[1, 2, 4, 32], [17, 219, 133, 232], [17, 91, 133, 184], [129, 83, 205, 40], [17, 10, 196, 48], [129, 195, 77, 104], [129, 67, 213, 104], [17, 138, 20, 224], [17, 18, 20, 224]], 'aut_group': None, 'aut_hash': 5881906593575344611, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8192, 'aut_permdeg': 48, 'aut_perms': [331531126110071326122108115072544529124175742622693827038471, 452544950607759194533951770684697806450324570406744711777380, 4799337277827740948803793915664648617152766401277392880854173, 5089036730186500649183709236022578498608850729405280645733690, 8385507020187956728234266232582110311425813152551025602135029, 11867431507758223627801782413731791516251177127696200652156958, 4971377159603862334635765740644258759889185051791078986620962, 3628008674850554482830459189246747085267443974241439424315560], 'aut_phi_ratio': 64.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [2, 16, 1, 1], [4, 2, 2, 2], [4, 4, 1, 4], [4, 8, 2, 2], [4, 16, 1, 3], [8, 4, 4, 1], [8, 8, 2, 1], [8, 8, 4, 2], [8, 16, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^9.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '256.56092', 'autcent_hash': 56092, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8', 'autcentquo_abelian': True, 'autcentquo_cyclic': False, 'autcentquo_exponent': 2, 'autcentquo_group': '32.51', 'autcentquo_hash': 51, 'autcentquo_nilpotent': True, 'autcentquo_order': 32, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 4, 1], [2, 16, 1], [4, 2, 4], [4, 4, 4], [4, 8, 4], [4, 16, 3], [8, 4, 4], [8, 8, 10], [8, 16, 2]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '64.203', 'commutator_count': 1, 'commutator_label': '16.2', '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'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 12517, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [2, 16, 1, 1], [4, 2, 1, 4], [4, 4, 1, 4], [4, 8, 1, 4], [4, 16, 1, 3], [8, 4, 2, 2], [8, 8, 1, 2], [8, 8, 2, 4], [8, 16, 1, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 161280, 'exponent': 8, 'exponents_of_order': [8], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.2', 'frattini_quotient': '16.14', 'hash': 12517, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 2, 4, 4], 'inner_gens': [[1, 130, 132, 48], [129, 2, 156, 32], [129, 138, 4, 96], [17, 2, 196, 32]], 'inner_hash': 203, 'inner_nilpotent': True, 'inner_order': 64, 'inner_split': True, 'inner_tex': 'C_2^3:D_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 12], [4, 8], [8, 1]], 'label': '256.12517', 'linC_count': 16, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 8, 'linQ_dim': 16, 'linQ_dim_count': 2, 'linR_count': 14, 'linR_degree': 16, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4^2.C2^4', 'ngens': 4, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 22, 'number_characteristic_subgroups': 47, 'number_conjugacy_classes': 37, 'number_divisions': 31, 'number_normal_subgroups': 105, 'number_subgroup_autclasses': 203, 'number_subgroup_classes': 285, 'number_subgroups': 695, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 23], [4, 104], [8, 128]], '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': [[1, 2, 20, 240], [17, 130, 20, 176], [129, 2, 4, 240], [1, 130, 4, 32], [1, 2, 20, 32], [1, 67, 5, 56]], 'outer_group': '128.1009', 'outer_hash': 1009, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 14, 'outer_perms': [375930, 997933050, 14519, 17764, 13412044800, 6745945381], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5:C_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 32, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 8], [4, 2], [8, 5]], 'representations': {'PC': {'code': 10718214655679770747861489650499635694, 'gens': [1, 2, 3, 6], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 2081, 137, 3170, 1882, 66, 395, 91, 2309, 1173, 141, 2710, 166]}, 'Perm': {'d': 32, 'gens': [1397935648789478997973159291739784, 8798828843109387210295494581235039, 18074785779454785849230938051749155, 1936978301547452806823226840678906, 26066734767839994115710534451714795, 35084145068101308346937119494144000, 43051791433894663320640725198711314, 43051791433894663320632115191808000]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4^2.C_2^4', 'transitive_degree': 128, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', '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, 3]], 'center_label': '4.2', 'center_order': 4, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', '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]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
