Formats: - HTML - YAML - JSON - 2026-07-22T01:14:57.639507
  • gps_subgroup_search • Show schema
    {'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '3888.dk', 'ambient_counter': 89, 'ambient_order': 3888, 'ambient_tex': 'C_3^3:S_3^2:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 54, 'counter': 35, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '3888.dk.18.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '18.c1', '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': 18, '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': '216.131', 'subgroup_hash': 131, 'subgroup_order': 216, 'subgroup_tex': 'C_6.S_3^2', 'supersolvable': True, 'sylow': 0}
  • gps_subgroup_data • Show schema
    {'ambient': '3888.dk', 'aut_centralizer_order': 4, 'aut_label': '18.c1', 'aut_quo_index': None, 'aut_stab_index': 18, 'aut_weyl_group': '864.4686', 'aut_weyl_index': 72, 'centralizer': '1944.a1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['2.b1', '9.a1'], 'contains': ['36.b1', '36.r1', '54.b1', '54.e1'], 'core': '72.a1', 'coset_action_label': None, 'count': 18, 'diagramx': [9823, -1, 999, -1], 'generators': [101069337600, 288, 12936672288, 373661250631, 100111334400, 80640], 'label': '3888.dk.18.c1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.a1', 'old_label': '18.c1', 'projective_image': '3888.dk', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '18.c1', 'subgroup_fusion': None, 'weyl_group': '216.158'}
  • gps_groups • Show schema
    {'Agroup': True, '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': 12, 'aut_gen_orders': [2, 2, 2, 2, 2, 2, 3, 3, 3], 'aut_gens': [[1, 4, 24, 72], [3, 4, 48, 72], [13, 4, 72, 24], [3, 20, 24, 72], [1, 6, 24, 72], [1, 4, 48, 144], [3, 4, 24, 72], [9, 4, 24, 72], [25, 28, 24, 72], [49, 196, 24, 72]], 'aut_group': '1728.46671', 'aut_hash': 46671, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1728, 'aut_permdeg': 13, 'aut_perms': [7620480, 482671464, 1, 264, 7983360, 1680, 3, 1005540480, 2075673600], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 4, 1, 1], [3, 4, 2, 2], [4, 9, 4, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 4, 1, 1], [6, 4, 2, 2], [6, 18, 2, 1], [12, 18, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'D_6^2:D_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': 12, 'autcentquo_group': '432.741', 'autcentquo_hash': 741, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3^3:C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 9, 2], [3, 2, 3], [3, 4, 5], [4, 9, 4], [6, 2, 3], [6, 4, 5], [6, 18, 2], [12, 18, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '108.40', 'commutator_count': 1, 'commutator_label': '27.5', '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': 131, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [3, 2, 1, 3], [3, 4, 1, 3], [3, 4, 2, 1], [4, 9, 2, 2], [6, 2, 1, 3], [6, 4, 1, 3], [6, 4, 2, 1], [6, 18, 1, 2], [12, 18, 2, 2]], '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': [[4, 0, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '108.40', 'hash': 131, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6, 3, 3], 'inner_gens': [[1, 20, 48, 72], [9, 4, 48, 144], [49, 52, 24, 72], [1, 148, 24, 72]], 'inner_hash': 40, 'inner_nilpotent': False, 'inner_order': 108, 'inner_split': True, 'inner_tex': 'C_3:S_3^2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 12], [4, 10]], 'label': '216.131', 'linC_count': 2, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 18, 'linQ_dim': 6, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6.S3^2', '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': 16, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 30, 'number_divisions': 24, 'number_normal_subgroups': 27, 'number_subgroup_autclasses': 56, 'number_subgroup_classes': 90, 'number_subgroups': 340, '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': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[13, 4, 72, 24], [1, 22, 48, 72], [1, 20, 24, 72], [3, 20, 24, 144]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [6, 289, 1, 126], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [4, 8], [8, 2]], 'representations': {'PC': {'code': 6910678447847337128263, 'gens': [1, 3, 5, 6], 'pres': [6, -2, -2, -2, -3, -3, -3, 12, 362, 50, 387, 1444, 376, 1313]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101750005089673, 24970019350934389, 91678749264390247]}, 'GLZN': {'d': 2, 'p': 30, 'gens': [27301, 40966, 203117, 513019, 32773, 478283]}, 'Perm': {'d': 13, 'gens': [490654207, 3669137, 7, 1085051520, 1557158400, 840]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6.S_3^2', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
  • gps_groups • Show 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': 4, 'aut_exponent': 24, 'aut_gen_orders': [6, 3, 2, 2, 12], 'aut_gens': [[100115008562, 193121112394], [262492957494, 461322126948], [100115008614, 193121112822], [262975587842, 448428965521], [269198979894, 460839496548], [268240977313, 460843125354]], 'aut_group': '62208.l', 'aut_hash': 1820680360968750972, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 62208, 'aut_permdeg': 648, 'aut_perms': 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'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 2, 1], [2, 18, 2, 1], [2, 81, 1, 1], [3, 2, 1, 1], [3, 4, 2, 2], [3, 4, 4, 1], [3, 8, 2, 2], [3, 8, 4, 1], [3, 16, 1, 1], [3, 16, 2, 2], [3, 16, 4, 1], [4, 162, 4, 1], [6, 18, 2, 1], [6, 18, 4, 1], [6, 36, 2, 2], [6, 36, 4, 2], [6, 36, 8, 1], [6, 72, 2, 2], [6, 72, 4, 1], [6, 162, 1, 1], [12, 324, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3^5.D_4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '62208.l', 'autcentquo_hash': 1820680360968750972, 'autcentquo_nilpotent': False, 'autcentquo_order': 62208, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3^5.D_4', 'cc_stats': [[1, 1, 1], [2, 9, 2], [2, 18, 2], [2, 81, 1], [3, 2, 1], [3, 4, 8], [3, 8, 8], [3, 16, 9], [4, 162, 4], [6, 18, 6], [6, 36, 20], [6, 72, 8], [6, 162, 1], [12, 324, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '3888.dk', 'commutator_count': 1, 'commutator_label': '486.259', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 89, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 2], [2, 18, 1, 2], [2, 81, 1, 1], [3, 2, 1, 1], [3, 4, 1, 4], [3, 4, 2, 2], [3, 8, 1, 4], [3, 8, 2, 2], [3, 16, 1, 3], [3, 16, 2, 3], [4, 162, 2, 2], [6, 18, 1, 2], [6, 18, 2, 2], [6, 36, 1, 8], [6, 36, 2, 6], [6, 72, 1, 4], [6, 72, 2, 2], [6, 162, 1, 1], [12, 324, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 48, 'exponent': 12, 'exponents_of_order': [5, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 0, 8], [16, 0, 6]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '3888.dk', 'hash': 5387043051655160496, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [6, 12], 'inner_gens': [[100115008562, 361729756237], [13415754873, 193121112394]], 'inner_hash': 5387043051655160496, 'inner_nilpotent': False, 'inner_order': 3888, 'inner_split': True, 'inner_tex': 'C_3^3:S_3^2:C_4', 'inner_used': [1, 2], 'irrC_degree': 8, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 16, 'irrep_stats': [[1, 8], [2, 10], [4, 32], [8, 16], [16, 9]], 'label': '3888.dk', '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': None, 'name': 'C3^3:S3^2:C4', 'ngens': 2, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 28, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 75, 'number_divisions': 54, 'number_normal_subgroups': 29, 'number_subgroup_autclasses': 378, 'number_subgroup_classes': 894, 'number_subgroups': 19656, 'old_label': None, 'order': 3888, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 135], [3, 242], [4, 648], [6, 1566], [12, 1296]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 12936672713, 0, 0], 'outer_gens': [[100115043861, 193121112615], [268240977329, 361250674070], [262017549414, 193121112737], [100115043842, 193121112781]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [126, 55, 289, 288], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 14], [8, 18], [16, 7], [32, 3]], 'representations': {'PC': {'code': '21836155762007731365262521521083754232986598708595860964407876720108632757517292242214866487', 'gens': [1, 3, 5, 6, 8, 9], 'pres': [9, 2, 2, 2, 3, 3, 2, 3, 3, 3, 18, 4330, 44930, 35705, 74, 57603, 23484, 38884, 562, 143213, 21398, 22379, 18014, 158, 81654, 9087, 4560, 3463, 916, 23336, 104993, 2969]}, 'Perm': {'d': 15, 'gens': [100115008562, 193121112394]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3:S_3^2:C_4', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}