-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1080.172', 'ambient_counter': 172, 'ambient_order': 1080, 'ambient_tex': 'C_{30}.C_6^2', 'central': False, 'central_factor': False, 'centralizer_order': 24, 'characteristic': False, 'core_order': 108, 'counter': 8, 'cyclic': False, 'direct': None, 'hall': 6, 'label': '1080.172.5.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '5.a1', 'outer_equivalence': True, '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': 5, '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': '216.74', 'subgroup_hash': 74, 'subgroup_order': 216, 'subgroup_tex': 'C_6.C_6^2', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '1080.172', 'aut_centralizer_order': 4, 'aut_label': '5.a1', 'aut_quo_index': None, 'aut_stab_index': 5, 'aut_weyl_group': '1728.47845', 'aut_weyl_index': 20, 'centralizer': '45.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1'], 'contains': ['10.a1', '10.b1', '10.c1', '15.a1'], 'core': '10.a1', 'coset_action_label': None, 'count': 5, 'diagramx': [5540, -1, 5502, -1], 'generators': [3, 24, 36, 18, 2, 720], 'label': '1080.172.5.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '5.a1', 'old_label': '5.a1', 'projective_image': '90.5', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '5.a1', 'subgroup_fusion': None, 'weyl_group': '9.2'}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '72.36', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [12, 6, 8], 'aut_gens': [[1, 3, 18], [150, 119, 99], [145, 161, 90], [151, 83, 99]], 'aut_group': '3456.in', 'aut_hash': 5403615404265510450, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3456, 'aut_permdeg': 13, 'aut_perms': [610167622, 526266840, 47946361], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [3, 1, 2, 1], [3, 3, 8, 1], [4, 1, 4, 1], [6, 1, 2, 1], [6, 1, 4, 1], [6, 3, 8, 1], [6, 3, 16, 1], [12, 1, 8, 1], [12, 3, 32, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4.\\SL(3,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '72.37', 'autcent_hash': 37, 'autcent_nilpotent': True, 'autcent_order': 72, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_4\\times C_3^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '48.29', 'autcentquo_hash': 29, 'autcentquo_nilpotent': False, 'autcentquo_order': 48, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 3], [3, 1, 2], [3, 3, 8], [4, 1, 4], [6, 1, 6], [6, 3, 24], [12, 1, 8], [12, 3, 32]], 'center_label': '24.9', 'center_order': 24, 'central_product': True, 'central_quotient': '9.2', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 74, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['27.3', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [3, 3, 2, 4], [4, 1, 2, 2], [6, 1, 2, 3], [6, 3, 2, 12], [12, 1, 4, 2], [12, 3, 4, 8]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 12, 'exponents_of_order': [3, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '6.2', 'frattini_quotient': '36.14', 'hash': 74, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [3, 3, 1], 'inner_gens': [[1, 75, 18], [145, 3, 18], [1, 3, 18]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 9, 'inner_split': True, 'inner_tex': 'C_3^2', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 72], [3, 16]], 'label': '216.74', 'linC_count': 432, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 4, 'linQ_dim': 8, 'linQ_dim_count': 4, 'linR_count': 8, 'linR_degree': 7, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6.C6^2', 'ngens': 6, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 12, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 88, 'number_divisions': 36, 'number_normal_subgroups': 56, 'number_subgroup_autclasses': 30, 'number_subgroup_classes': 88, 'number_subgroups': 152, 'old_label': None, 'order': 216, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 3], [3, 26], [4, 4], [6, 78], [12, 104]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [4, 3, 2, 2, 2, 4, 4, 2], 'outer_gen_pows': [0, 0, 0, 151, 0, 0, 0, 0], 'outer_gens': [[86, 76, 126], [85, 147, 18], [1, 3, 126], [150, 11, 99], [146, 87, 126], [85, 52, 27], [146, 195, 27], [1, 3, 135]], 'outer_group': '384.17972', 'outer_hash': 17972, 'outer_nilpotent': False, 'outer_order': 384, 'outer_permdeg': 12, 'outer_perms': [262903087, 261648120, 7, 379566, 40284847, 176163142, 22, 6], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'D_4\\times \\GL(2,3)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 18], [4, 8], [6, 4], [12, 2]], 'representations': {'PC': {'code': 292842257541766825, 'gens': [1, 2, 4], 'pres': [6, -3, -2, -3, -2, -2, -3, 901, 31, 2702, 69, 88]}, 'GLZN': {'d': 2, 'p': 36, 'gens': [47089, 1166401, 1632995, 47305, 638209, 46981]}, 'Perm': {'d': 15, 'gens': [1516838400, 87178291200, 138838, 276480, 482630400, 80884]}}, 'schur_multiplier': [3, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6, 12], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6.C_6^2', 'transitive_degree': 72, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '72.36', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 120, 'aut_gen_orders': [12, 8, 30, 6, 12], 'aut_gens': [[1, 6, 72], [603, 42, 196], [975, 30, 148], [713, 66, 74], [17, 42, 1012], [975, 66, 916]], 'aut_group': None, 'aut_hash': 8203041546889010050, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 34560, 'aut_permdeg': 36, 'aut_perms': [9958971235211374035182506467157966939858, 103712400469541577631153979940955899616011, 371869508589964100043584403391432018452544, 278557059630071156172164384767455477357350, 113159332264301409239668097968449449044042], 'aut_phi_ratio': 120.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 2, 1], [3, 1, 2, 1], [3, 3, 8, 1], [4, 1, 2, 1], [4, 5, 2, 1], [5, 2, 2, 1], [6, 1, 2, 1], [6, 3, 8, 1], [6, 5, 4, 1], [6, 15, 16, 1], [10, 2, 2, 1], [12, 1, 4, 1], [12, 3, 16, 1], [12, 5, 4, 1], [12, 15, 16, 1], [15, 2, 4, 1], [15, 6, 16, 1], [20, 2, 4, 1], [30, 2, 4, 1], [30, 6, 16, 1], [60, 2, 8, 1], [60, 6, 32, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2\\times \\AGL(2,3)\\times F_5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '36.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 36, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_6^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': '960.10901', 'autcentquo_hash': 10901, 'autcentquo_nilpotent': False, 'autcentquo_order': 960, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\times \\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 5, 2], [3, 1, 2], [3, 3, 8], [4, 1, 2], [4, 5, 2], [5, 2, 2], [6, 1, 2], [6, 3, 8], [6, 5, 4], [6, 15, 16], [10, 2, 2], [12, 1, 4], [12, 3, 16], [12, 5, 4], [12, 15, 16], [15, 2, 4], [15, 6, 16], [20, 2, 4], [30, 2, 4], [30, 6, 16], [60, 2, 8], [60, 6, 32]], 'center_label': '12.2', 'center_order': 12, 'central_product': True, 'central_quotient': '90.5', 'commutator_count': 1, 'commutator_label': '15.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 172, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['10.1', 1], ['27.3', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 1, 2], [3, 1, 2, 1], [3, 3, 2, 4], [4, 1, 2, 1], [4, 5, 2, 1], [5, 2, 2, 1], [6, 1, 2, 1], [6, 3, 2, 4], [6, 5, 2, 2], [6, 15, 2, 8], [10, 2, 2, 1], [12, 1, 4, 1], [12, 3, 4, 4], [12, 5, 4, 1], [12, 15, 4, 4], [15, 2, 4, 1], [15, 6, 4, 4], [20, 2, 4, 1], [30, 2, 4, 1], [30, 6, 4, 4], [60, 2, 8, 1], [60, 6, 8, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 60, 'exponents_of_order': [3, 3, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[6, 0, 8]], 'familial': False, 'frattini_label': '6.2', 'frattini_quotient': '180.32', 'hash': 172, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [6, 1, 15], 'inner_gens': [[1, 6, 336], [1, 6, 72], [889, 6, 72]], 'inner_hash': 5, 'inner_nilpotent': False, 'inner_order': 90, 'inner_split': False, 'inner_tex': 'C_3^2\\times D_5', 'inner_used': [1, 3], 'irrC_degree': 6, 'irrQ_degree': 48, 'irrQ_dim': 48, 'irrR_degree': 12, 'irrep_stats': [[1, 72], [2, 72], [3, 16], [6, 16]], 'label': '1080.172', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C30.C6^2', '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, 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': 24, 'number_characteristic_subgroups': 27, 'number_conjugacy_classes': 176, 'number_divisions': 54, 'number_normal_subgroups': 77, 'number_subgroup_autclasses': 70, 'number_subgroup_classes': 176, 'number_subgroups': 684, 'old_label': None, 'order': 1080, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 11], [3, 26], [4, 12], [5, 4], [6, 286], [10, 4], [12, 312], [15, 104], [20, 8], [30, 104], [60, 208]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [2, 2, 2, 6, 4, 4], 'outer_gen_pows': [0, 0, 722, 3, 0, 0], 'outer_gens': [[37, 6, 72], [1, 42, 72], [29, 30, 124], [759, 6, 194], [747, 6, 434], [389, 6, 74]], 'outer_group': '384.20055', 'outer_hash': 20055, 'outer_nilpotent': False, 'outer_order': 384, 'outer_permdeg': 14, 'outer_perms': [127, 288, 2566327680, 958729374, 31259294640, 14095025280], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^3\\times \\GL(2,3)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 18], [4, 10], [6, 4], [8, 9], [12, 2], [16, 4], [24, 2], [48, 1]], 'representations': {'PC': {'code': 52353389103677054596951436903447, 'gens': [1, 3, 6], 'pres': [7, -2, -3, -2, -2, -3, 3, -5, 14, 58, 80, 14117, 2028, 166, 42342]}, 'GLZN': {'d': 2, 'p': 45, 'gens': [1458001, 91801, 91144, 91531, 152806, 1731394, 3371662]}, 'Perm': {'d': 18, 'gens': [21010450809600, 1041465600, 11268, 57376, 482630400, 93313, 397620186163200]}}, 'schur_multiplier': [3, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6, 12], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{30}.C_6^2', 'transitive_degree': 180, 'wreath_data': None, 'wreath_product': False}
