-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1568.264', 'ambient_counter': 264, 'ambient_order': 1568, 'ambient_tex': 'C_{14}^2.D_4', 'central': False, 'central_factor': False, 'centralizer_order': 28, 'characteristic': False, 'core_order': 392, 'counter': 7, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1568.264.4.b1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '4.b1.b1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '4.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '392.23', 'subgroup_hash': 23, 'subgroup_order': 392, 'subgroup_tex': 'C_7^2:Q_8', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '1568.264', 'aut_centralizer_order': 4, 'aut_label': '4.b1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '2016.m', 'aut_weyl_index': 8, 'centralizer': '56.a1.a1', 'complements': ['392.d1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['8.b1.a1', '8.e1.b1', '28.c1.b1', '28.i1.b1'], 'core': '4.b1.b1', 'coset_action_label': None, 'count': 1, 'diagramx': [5196, 2734, 5653, 2267, 4024, 9542, 4630, 9545], 'generators': [785, 224, 56, 812, 16], 'label': '1568.264.4.b1.b1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '4.b1.b1', 'normal_contained_in': ['2.a1.a1'], 'normal_contains': ['8.b1.a1', '28.c1.b1'], 'normalizer': '1.a1.a1', 'old_label': '4.b1.b1', 'projective_image': '112.13', 'quotient_action_image': '2.1', 'quotient_action_kernel': '2.1', 'quotient_action_kernel_order': 2, 'quotient_fusion': None, 'short_label': '4.b1.b1', 'subgroup_fusion': None, 'weyl_group': '56.5'}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '28.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [6, 2, 6, 28], 'aut_gens': [[1, 14], [1, 266], [13, 14], [9, 210], [43, 14]], 'aut_group': '2016.m', 'aut_hash': 5723319100097859086, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2016, 'aut_permdeg': 34, 'aut_perms': [130753247177941128966070519651690831149, 2376719306881848238599083666613369600, 184734970254990532564576819913791125723, 122067805514612699330858407970036251083], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 14, 2, 1], [7, 1, 6, 1], [7, 2, 3, 1], [7, 2, 18, 1], [14, 1, 6, 1], [14, 2, 3, 1], [14, 2, 18, 1], [28, 2, 6, 2], [28, 2, 36, 1], [28, 14, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_{28}:C_6^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '24.15', 'autcent_hash': 15, 'autcent_nilpotent': True, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '84.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 84, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 2, 1], [4, 14, 2], [7, 1, 6], [7, 2, 21], [14, 1, 6], [14, 2, 21], [28, 2, 48], [28, 14, 12]], 'center_label': '14.2', 'center_order': 14, 'central_product': True, 'central_quotient': '28.3', 'commutator_count': 1, 'commutator_label': '14.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '7.1', '7.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 23, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['56.3', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 1], [4, 14, 1, 2], [7, 1, 6, 1], [7, 2, 3, 1], [7, 2, 6, 3], [14, 1, 6, 1], [14, 2, 3, 1], [14, 2, 6, 3], [28, 2, 6, 2], [28, 2, 12, 3], [28, 14, 6, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 28, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[2, 0, 36]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '196.10', 'hash': 23, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 14], 'inner_gens': [[1, 378], [29, 14]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 28, 'inner_split': True, 'inner_tex': 'D_{14}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 4, 'irrep_stats': [[1, 28], [2, 91]], 'label': '392.23', 'linC_count': 36, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 14, 'linQ_dim': 16, 'linQ_dim_count': 14, 'linR_count': 18, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C7^2:Q8', 'ngens': 5, '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': 14, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 119, 'number_divisions': 22, 'number_normal_subgroups': 18, 'number_subgroup_autclasses': 23, 'number_subgroup_classes': 33, 'number_subgroups': 78, 'old_label': None, 'order': 392, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 1], [4, 30], [7, 48], [14, 48], [28, 264]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6, 6], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 210], [5, 14], [99, 42]], 'outer_group': '72.50', 'outer_hash': 50, 'outer_nilpotent': True, 'outer_order': 72, 'outer_permdeg': 12, 'outer_perms': [5040, 363024, 39916803], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 7], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [6, 6], [12, 8], [24, 3]], 'representations': {'PC': {'code': 2318448039646925158113245, 'gens': [1, 3], 'pres': [5, -2, -7, -2, -2, -7, 10, 986, 5672, 42, 7283, 58, 8404]}, 'GLFp': {'d': 2, 'p': 29, 'gens': [365837, 390225, 957]}, 'Perm': {'d': 22, 'gens': [2439325406344014720, 194163338880, 873, 288848891520, 55969571858264064000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 14], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_7^2:Q_8', 'transitive_degree': 56, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '56.8', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [42, 12, 12, 6, 2, 12, 6, 6], 'aut_gens': [[1, 2, 112], [809, 786, 336], [53, 850, 1456], [109, 90, 560], [97, 74, 1456], [1, 810, 112], [869, 794, 1456], [881, 78, 112], [785, 794, 336]], 'aut_group': None, 'aut_hash': 98983681409546200, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 16128, 'aut_permdeg': 70, 'aut_perms': [9470335266968974959858229052594742014287953015707225639854108835902284471511260682514450464316311614, 11285920279961906508370524444251226854881724245470611682829568758526502291655388438804093640582620730, 9767089154765091039236262632281059547004794059035677520978453544604352215055801368021630859019211673, 7733080900456212022049996569882133908306239780381329156560680621265924426057839583584967013311981413, 449785614039509010741697526557666197322405678688029060010571216460616612202471092932597053072695, 3863287798692865399981716577008302399839085287720992920677844064426335483316235541562384293036146958, 8738232526328008770455822939913005859128605213224768930932110290730122595825424809747280306262178074, 1688059237742208558747289271777894736201423153881962800821738796671430050280855703852950009873176153], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 2], [4, 28, 2, 2], [7, 1, 6, 1], [7, 2, 3, 1], [7, 2, 18, 1], [8, 2, 4, 1], [14, 1, 6, 3], [14, 2, 3, 3], [14, 2, 18, 3], [28, 2, 6, 4], [28, 2, 36, 2], [28, 28, 12, 2], [56, 2, 24, 2], [56, 2, 144, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_6\\times C_7:C_3).C_2^6.C_2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '96.231', 'autcent_hash': 231, 'autcent_nilpotent': True, 'autcent_order': 96, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4\\times C_6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '168.47', 'autcentquo_hash': 47, 'autcentquo_nilpotent': False, 'autcentquo_order': 168, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 2], [4, 28, 4], [7, 1, 6], [7, 2, 21], [8, 2, 4], [14, 1, 18], [14, 2, 63], [28, 2, 96], [28, 28, 24], [56, 2, 192]], 'center_label': '28.4', 'center_order': 28, 'central_product': True, 'central_quotient': '56.5', 'commutator_count': 1, 'commutator_label': '28.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '7.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 264, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['224.22', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 2], [4, 28, 1, 2], [4, 28, 2, 1], [7, 1, 6, 1], [7, 2, 3, 1], [7, 2, 6, 3], [8, 2, 4, 1], [14, 1, 6, 3], [14, 2, 3, 3], [14, 2, 6, 9], [28, 2, 6, 4], [28, 2, 12, 6], [28, 28, 6, 2], [28, 28, 12, 1], [56, 2, 24, 8]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 48, 'exponent': 56, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '196.10', 'hash': 264, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 28, 'inner_gen_orders': [2, 28, 1], 'inner_gens': [[1, 838, 112], [845, 2, 112], [1, 2, 112]], 'inner_hash': 5, 'inner_nilpotent': False, 'inner_order': 56, 'inner_split': True, 'inner_tex': 'D_{28}', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 56], [2, 378]], 'label': '1568.264', 'linC_count': 4608, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 12, 'linQ_dim': 16, 'linQ_dim_count': 6, 'linR_count': 1296, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C14^2.D4', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 32, 'number_characteristic_subgroups': 42, 'number_conjugacy_classes': 434, 'number_divisions': 51, 'number_normal_subgroups': 46, 'number_subgroup_autclasses': 86, 'number_subgroup_classes': 117, 'number_subgroups': 414, 'old_label': None, 'order': 1568, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 3], [4, 116], [7, 48], [8, 8], [14, 144], [28, 864], [56, 384]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 6, 6], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 58, 112], [785, 2, 112], [1, 870, 112], [1, 786, 1232], [1, 790, 1456]], 'outer_group': '288.1045', 'outer_hash': 1045, 'outer_nilpotent': True, 'outer_order': 288, 'outer_permdeg': 16, 'outer_perms': [6227020800, 1307674368000, 5040, 39916944, 362883], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_6^2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 7], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 2], [6, 6], [12, 19], [24, 10], [48, 6]], 'representations': {'PC': {'code': 96176250732955190976096639203466690782009, 'gens': [1, 2, 6], 'pres': [7, -2, -2, -2, -2, -7, -2, -7, 392, 11733, 36, 2270, 58, 2915, 80, 3364, 124]}, 'GLZN': {'d': 2, 'p': 87, 'gens': [37976253, 46319827, 10536064, 491550, 658519, 18438112, 38851736]}, 'Perm': {'d': 26, 'gens': [15568541872220315641471320, 80288049, 32991990780808352400307200, 128193976, 128193960, 175795200, 49151793490183855357286400]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 28], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{14}^2.D_4', 'transitive_degree': 224, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '4.1', '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], [3]], '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], [4, 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], [4, 1, 2]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, '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': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [2], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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, 4]], 'label': '4.1', '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': 'C4', 'ngens': 1, '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': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 1], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3]], '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': 4, 'pgroup': 2, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -2, -2, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [56, 15], 'family': 'CSOPlus'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [21]}, 'Perm': {'d': 4, 'gens': [22, 7]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
