-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1728.32842', 'ambient_counter': 32842, 'ambient_order': 1728, 'ambient_tex': 'C_6^2.(C_6\\times D_4)', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 108, 'counter': 62, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1728.32842.8.p1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '8.p1.b1', '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': 8, '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': '216.168', 'subgroup_hash': 168, 'subgroup_order': 216, 'subgroup_tex': 'C_2\\times C_3^2:C_{12}', 'supersolvable': False, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '1728.32842', 'aut_centralizer_order': 4, 'aut_label': '8.p1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '288.1031', 'aut_weyl_index': 16, 'centralizer': '288.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.e1.b1', '4.k1.b1', '4.m1.b1'], 'contains': ['16.b1.b1', '16.m1.b1', '24.bx1.b1', '72.bs1.b1'], 'core': '16.b1.b1', 'coset_action_label': None, 'count': 2, 'diagramx': [6315, -1, 6531, -1, 5634, -1, 3416, -1], 'generators': [438, 576, 48, 864, 8, 948], 'label': '1728.32842.8.p1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.e1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.b1', 'old_label': '8.p1.b1', 'projective_image': '288.889', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.p1.b1', 'subgroup_fusion': None, 'weyl_group': '144.186'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '24.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [2, 8, 6, 6], 'aut_gens': [[1, 12, 36], [7, 12, 192], [1, 72, 204], [149, 12, 36], [61, 12, 36]], 'aut_group': '576.8658', 'aut_hash': 8658, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 576, 'aut_permdeg': 13, 'aut_perms': [83558287, 55643056, 4126469767, 1137725296], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [3, 1, 2, 1], [3, 4, 2, 1], [3, 4, 4, 1], [4, 9, 4, 1], [6, 1, 2, 1], [6, 4, 2, 1], [6, 4, 4, 1], [6, 9, 2, 2], [12, 9, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_9:C_2^3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '144.182', 'autcentquo_hash': 182, 'autcentquo_nilpotent': False, 'autcentquo_order': 144, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_9:C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 9, 2], [3, 1, 2], [3, 4, 6], [4, 9, 4], [6, 1, 2], [6, 4, 6], [6, 9, 4], [12, 9, 8]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '36.9', 'commutator_count': 1, 'commutator_label': '9.2', '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': 168, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['36.9', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [3, 1, 2, 1], [3, 4, 1, 2], [3, 4, 2, 2], [4, 9, 2, 2], [6, 1, 2, 1], [6, 4, 1, 2], [6, 4, 2, 2], [6, 9, 2, 2], [12, 9, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 12, 'exponents_of_order': [3, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '216.168', 'hash': 168, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [4, 3, 3], 'inner_gens': [[1, 96, 60], [169, 12, 36], [13, 12, 36]], 'inner_hash': 9, 'inner_nilpotent': False, 'inner_order': 36, 'inner_split': False, 'inner_tex': 'C_3^2:C_4', 'inner_used': [1, 2], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 24], [4, 12]], 'label': '216.168', 'linC_count': 4, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 12, 'linQ_dim': 6, 'linQ_dim_count': 12, 'linR_count': 24, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*C3^2:C12', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, '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': 14, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 36, 'number_divisions': 20, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 42, 'number_subgroup_classes': 60, 'number_subgroups': 248, 'old_label': None, 'order': 216, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 19], [3, 26], [4, 36], [6, 62], [12, 72]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 3], 'outer_gens': [[7, 12, 192], [5, 12, 36], [109, 12, 36], [1, 72, 204]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 6], [8, 4]], 'representations': {'PC': {'code': 320972801006877777297597645, 'gens': [1, 4, 5], 'pres': [6, 2, 2, 3, 3, 2, 3, 12, 31, 2307, 297, 1804, 2710, 88, 3029, 2603]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [108470311038820273, 125101743017174152]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [29293137, 14412736, 1017496, 38138223, 24637050, 28816400]}, 'Perm': {'d': 11, 'gens': [3710160, 1, 30, 131760, 8115120, 1128960]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 12], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_3^2:C_{12}', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '24.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [12, 4, 2, 6, 2, 4, 2], 'aut_gens': [[1, 2, 24, 144], [1273, 422, 120, 1128], [37, 250, 408, 1128], [673, 490, 120, 1584], [1537, 118, 120, 1128], [1465, 1234, 120, 1584], [937, 1606, 120, 264], [109, 118, 408, 144]], 'aut_group': None, 'aut_hash': 2073078227855066378, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 4608, 'aut_permdeg': 50, 'aut_perms': [16625156418087568116571113096962271568410242872641939985508653955, 25354831028660692467477430922603328935642959535313837626889852264, 11313893963498666990674570472854408658870567325611904697308234956, 30209684710868022248851542306332935299165423069829158306798987831, 20151192419157081322251807553217262791327199386455905085314059667, 28334654872941916024197467381681396102055957034509160104062329896, 25064246211176107856084088165665050100854385379187559222282492364], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 12, 2, 1], [2, 18, 2, 1], [2, 36, 1, 1], [3, 1, 2, 1], [3, 4, 1, 2], [3, 4, 2, 2], [4, 4, 1, 1], [4, 12, 2, 2], [4, 24, 1, 1], [4, 72, 2, 1], [6, 1, 2, 1], [6, 2, 2, 1], [6, 4, 1, 4], [6, 4, 2, 4], [6, 8, 1, 1], [6, 8, 2, 1], [6, 12, 4, 1], [6, 18, 4, 1], [6, 24, 2, 1], [6, 24, 4, 1], [6, 36, 2, 1], [12, 4, 2, 2], [12, 4, 4, 1], [12, 8, 1, 1], [12, 8, 2, 2], [12, 8, 4, 1], [12, 12, 4, 2], [12, 24, 2, 4], [12, 24, 4, 3], [12, 72, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_2\\times D_6:D_6).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': '288.889', 'autcentquo_hash': 889, 'autcentquo_nilpotent': False, 'autcentquo_order': 288, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'D_6\\wr C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 12, 2], [2, 18, 2], [2, 36, 1], [3, 1, 2], [3, 4, 6], [4, 4, 1], [4, 12, 4], [4, 24, 1], [4, 72, 2], [6, 1, 2], [6, 2, 2], [6, 4, 12], [6, 8, 3], [6, 12, 4], [6, 18, 4], [6, 24, 6], [6, 36, 2], [12, 4, 8], [12, 8, 9], [12, 12, 8], [12, 24, 20], [12, 72, 4]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '288.889', 'commutator_count': 1, 'commutator_label': '72.49', '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', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 32842, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['576.5297', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 12, 1, 2], [2, 18, 1, 2], [2, 36, 1, 1], [3, 1, 2, 1], [3, 4, 1, 2], [3, 4, 2, 2], [4, 4, 1, 1], [4, 12, 1, 2], [4, 12, 2, 1], [4, 24, 1, 1], [4, 72, 1, 2], [6, 1, 2, 1], [6, 2, 2, 1], [6, 4, 1, 4], [6, 4, 2, 4], [6, 8, 1, 1], [6, 8, 2, 1], [6, 12, 2, 2], [6, 18, 2, 2], [6, 24, 1, 2], [6, 24, 2, 2], [6, 36, 2, 1], [12, 4, 2, 2], [12, 4, 4, 1], [12, 8, 1, 1], [12, 8, 2, 2], [12, 8, 4, 1], [12, 12, 2, 2], [12, 12, 4, 1], [12, 24, 1, 2], [12, 24, 2, 7], [12, 24, 4, 1], [12, 72, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 349440, 'exponent': 12, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 0, 8], [8, 0, 6]], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '432.754', 'hash': 32842, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 4, 6, 6], 'inner_gens': [[1, 878, 24, 1608], [877, 2, 408, 1128], [1, 1490, 24, 144], [409, 890, 24, 144]], 'inner_hash': 889, 'inner_nilpotent': False, 'inner_order': 288, 'inner_split': True, 'inner_tex': 'D_6\\wr C_2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 4, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 8, 'irrep_stats': [[1, 24], [2, 18], [4, 54], [8, 12]], 'label': '1728.32842', 'linC_count': 8, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 8, 'linQ_dim': 14, 'linQ_dim_count': 96, 'linR_count': 4, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C6^2.(C6*D4)', 'ngens': 9, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 50, 'number_characteristic_subgroups': 34, 'number_conjugacy_classes': 108, 'number_divisions': 63, 'number_normal_subgroups': 62, 'number_subgroup_autclasses': 450, 'number_subgroup_classes': 630, 'number_subgroups': 4756, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 99], [3, 26], [4, 220], [6, 414], [12, 968]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 936], 'outer_gens': [[865, 2, 24, 144], [1, 2, 24, 1008], [1, 10, 24, 144], [937, 506, 120, 720]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 25, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 14], [4, 18], [8, 15], [16, 7], [32, 1]], 'representations': {'PC': {'code': 294051830508081354170695613644852833158281399511968576684060643656038206385705, 'gens': [1, 2, 5, 7], 'pres': [9, 2, 2, 2, 3, 2, 3, 2, 2, 3, 7776, 15805, 46, 74, 9193, 1372, 130, 18158, 1319, 101310, 35547, 24972, 186, 107143, 13840, 25945, 214, 101096, 25289, 23354]}, 'Perm': {'d': 25, 'gens': [628421542361789143161736, 1374977352166435658995221, 1948059198073120809600000, 45360, 2569517761302938246169623, 2647329283975505718758400, 3294774614019626683545600, 311, 56]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_6^2.(C_6\\times D_4)', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
