-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '648.407', 'ambient_counter': 407, 'ambient_order': 648, 'ambient_tex': 'S_3\\times \\He_3:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 18, 'characteristic': False, 'core_order': 54, 'counter': 12, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '648.407.6.d1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '6.d1', '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': 6, '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': '108.32', 'subgroup_hash': 32, 'subgroup_order': 108, 'subgroup_tex': 'C_3^2:C_{12}', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '648.407', 'aut_centralizer_order': 18, 'aut_label': '6.d1', 'aut_quo_index': None, 'aut_stab_index': 12, 'aut_weyl_group': '48.51', 'aut_weyl_index': 216, 'centralizer': '36.a1', 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['2.b1', '3.b1'], 'contains': ['12.a1', '18.d1', '18.f1', '18.i1', '18.k1'], 'core': '12.a1', 'coset_action_label': None, 'count': 12, 'diagramx': [6340, -1, 4322, -1], 'generators': [6, 72, 8, 216, 12], 'label': '648.407.6.d1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.b1', 'old_label': '6.d1', 'projective_image': '108.39', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.d1', 'subgroup_fusion': None, 'weyl_group': '12.4'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '36.8', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [4, 6, 2, 4, 3, 2, 2, 2], 'aut_gens': [[1, 3, 36], [24, 28, 36], [14, 11, 36], [14, 3, 72], [13, 16, 36], [1, 75, 36], [2, 33, 36], [1, 3, 72], [1, 21, 36]], 'aut_group': '576.8325', 'aut_hash': 8325, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 576, 'aut_permdeg': 13, 'aut_perms': [1516929840, 1528491601, 490654086, 2091681360, 48, 1, 482671446, 482671441], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 8, 1], [3, 2, 1, 1], [3, 2, 8, 1], [4, 3, 2, 1], [6, 1, 8, 1], [6, 2, 1, 1], [6, 2, 8, 1], [12, 3, 16, 1]], 'aut_supersolvable': False, 'aut_tex': 'D_6\\times \\GL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '96.189', 'autcent_hash': 189, 'autcent_nilpotent': False, 'autcent_order': 96, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2\\times \\GL(2,3)', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 1, 8], [3, 2, 9], [4, 3, 2], [6, 1, 8], [6, 2, 9], [12, 3, 16]], 'center_label': '18.5', 'center_order': 18, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 32, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.1', 1], ['3.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 4], [3, 2, 1, 1], [3, 2, 2, 4], [4, 3, 2, 1], [6, 1, 2, 4], [6, 2, 1, 1], [6, 2, 2, 4], [12, 3, 4, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 12, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '54.12', 'hash': 32, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [1, 2, 3], 'inner_gens': [[1, 3, 36], [1, 3, 72], [1, 75, 36]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 36], [2, 18]], 'label': '108.32', 'linC_count': 288, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 48, 'linQ_dim': 6, 'linQ_dim_count': 24, 'linR_count': 144, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^2:C12', '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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 54, 'number_divisions': 25, 'number_normal_subgroups': 30, 'number_subgroup_autclasses': 22, 'number_subgroup_classes': 52, 'number_subgroups': 80, 'old_label': None, 'order': 108, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 1], [3, 26], [4, 6], [6, 26], [12, 48]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [3, 2, 2, 2, 4, 4], 'outer_gen_pows': [0, 0, 0, 0, 0, 0], 'outer_gens': [[14, 29, 36], [25, 15, 36], [1, 21, 36], [2, 33, 36], [12, 11, 36], [13, 16, 36]], 'outer_group': '96.189', 'outer_hash': 189, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 10, 'outer_perms': [45504, 2672094, 1, 2805415, 2189071, 1217880], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times \\GL(2,3)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [4, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 11], [4, 12]], 'representations': {'PC': {'code': 6289916115, 'gens': [1, 2, 5], 'pres': [5, -3, -2, -2, -3, -3, 26, 42, 609]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101732552038707, 41624322709028323]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [13516898, 37770312, 22711195, 5426393, 23068812]}, 'Perm': {'d': 13, 'gens': [40325040, 3, 144, 806400, 518918400]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 12], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2:C_{12}', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'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': 5, 'aut_exponent': 24, 'aut_gen_orders': [2, 3, 3, 6, 3, 8, 2, 3], 'aut_gens': [[1, 2, 24, 72, 216], [17, 10, 24, 72, 216], [9, 2, 24, 72, 216], [1, 554, 456, 504, 216], [13, 554, 504, 192, 216], [1, 2, 24, 552, 216], [5, 622, 624, 480, 432], [13, 602, 624, 288, 432], [1, 74, 456, 72, 216]], 'aut_group': '10368.tp', 'aut_hash': 7867444394908007034, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10368, 'aut_permdeg': 24, 'aut_perms': [26890347575877811590, 19178823381447004134150, 208610218235775843245242, 425631941408615282586438, 157313896508646403879952, 191832233466026981916839, 425019371400320761575835, 285707279535514299156796], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 6, 4, 1], [3, 12, 4, 1], [4, 9, 2, 1], [4, 27, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 3, 4, 1], [6, 6, 4, 1], [6, 12, 4, 1], [6, 18, 8, 1], [12, 9, 4, 1], [12, 18, 2, 1], [12, 18, 4, 1], [12, 27, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(2,3).C_2^6', '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': '2592.fv', 'autcentquo_hash': 9019849488891309561, 'autcentquo_nilpotent': False, 'autcentquo_order': 2592, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\times C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [3, 1, 2], [3, 2, 3], [3, 6, 4], [3, 12, 4], [4, 9, 2], [4, 27, 2], [6, 1, 2], [6, 2, 3], [6, 3, 4], [6, 6, 4], [6, 12, 4], [6, 18, 8], [12, 9, 4], [12, 18, 6], [12, 27, 4]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '108.39', 'commutator_count': 1, 'commutator_label': '81.12', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 407, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['108.11', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 6, 1, 4], [3, 12, 1, 4], [4, 9, 2, 1], [4, 27, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 3, 2, 2], [6, 6, 1, 4], [6, 12, 1, 4], [6, 18, 1, 8], [12, 9, 4, 1], [12, 18, 2, 1], [12, 18, 4, 1], [12, 27, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 9072, 'exponent': 12, 'exponents_of_order': [4, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 0, 4]], 'familial': False, 'frattini_label': '6.2', 'frattini_quotient': '108.39', 'hash': 407, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6, 3, 3, 1], 'inner_gens': [[1, 10, 24, 72, 216], [17, 2, 48, 144, 216], [1, 50, 24, 504, 216], [1, 146, 240, 72, 216], [1, 2, 24, 72, 216]], 'inner_hash': 39, 'inner_nilpotent': False, 'inner_order': 108, 'inner_split': False, 'inner_tex': 'C_3:S_3^2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 6, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 12, 'irrep_stats': [[1, 8], [2, 20], [3, 16], [4, 8], [6, 8]], 'label': '648.407', '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': False, 'metacyclic': False, 'monomial': True, 'name': 'S3*He3:C4', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 21, 'number_characteristic_subgroups': 19, 'number_conjugacy_classes': 60, 'number_divisions': 42, 'number_normal_subgroups': 53, 'number_subgroup_autclasses': 85, 'number_subgroup_classes': 227, 'number_subgroups': 1076, 'old_label': None, 'order': 648, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 7], [3, 80], [4, 72], [6, 236], [12, 252]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 6, 2, 2], 'outer_gen_pows': [0, 0, 0, 6, 6], 'outer_gens': [[13, 2, 48, 144, 216], [1, 2, 48, 72, 432], [1, 14, 48, 408, 216], [1, 2, 600, 408, 216], [1, 2, 72, 48, 216]], 'outer_group': '96.226', 'outer_hash': 226, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 8, 'outer_perms': [7, 10800, 5776, 16680, 11520], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^2\\times S_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 20], [4, 9], [6, 4], [12, 4], [24, 1]], 'representations': {'PC': {'code': 1652280220280824059951344313103, 'gens': [1, 2, 5, 6, 7], 'pres': [7, -2, -2, -2, -3, -3, -3, -3, 141, 36, 422, 58, 451, 851, 3036, 915]}, 'Perm': {'d': 16, 'gens': [127, 180707708177, 7, 269366590080, 1494615991680, 840, 2897640748800]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times \\He_3:C_4', 'transitive_degree': 72, 'wreath_data': None, 'wreath_product': False}
