-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '960.9542', 'ambient_counter': 9542, 'ambient_order': 960, 'ambient_tex': 'C_{60}.C_2^4', 'central': False, 'central_factor': False, 'centralizer_order': 40, 'characteristic': True, 'core_order': 120, 'counter': 83, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '960.9542.8.e1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '8.e1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '8.3', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 3, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '120.23', 'subgroup_hash': 23, 'subgroup_order': 120, 'subgroup_tex': 'C_5\\times D_{12}', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '960.9542', 'aut_centralizer_order': 80, 'aut_label': '8.e1', 'aut_quo_index': 2, 'aut_stab_index': 1, 'aut_weyl_group': '192.1103', 'aut_weyl_index': 80, 'centralizer': '24.b1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['4.b1.a1', '4.z1.a1', '4.bb1.a1'], 'contains': ['16.b1.a1', '16.b1.b1', '16.d1.a1', '24.t1.a1', '40.e1.a1'], 'core': '8.e1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [1116, 1157, 8428, 1262, 1501, 804, 8440, 9002], 'generators': [27, 192, 48, 504, 32], 'label': '960.9542.8.e1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '8.e1.a1', 'normal_contained_in': ['4.b1.a1'], 'normal_contains': ['16.b1.a1', '16.b1.b1', '16.d1.a1', '40.e1.a1'], 'normalizer': '1.a1.a1', 'old_label': '8.e1.a1', 'projective_image': '480.1123', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.e1.a1', 'subgroup_fusion': None, 'weyl_group': '24.14'}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 4, 12], 'aut_gens': [[1, 2], [1, 82], [1, 38], [1, 26], [111, 2]], 'aut_group': '192.1103', 'aut_hash': 1103, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 192, 'aut_permdeg': 11, 'aut_perms': [362887, 1, 11640, 3991697], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 2, 1], [3, 2, 1, 1], [4, 2, 1, 1], [5, 1, 4, 1], [6, 2, 1, 1], [10, 1, 4, 1], [10, 6, 8, 1], [12, 2, 2, 1], [15, 2, 4, 1], [20, 2, 4, 1], [30, 2, 4, 1], [60, 2, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4^2:D_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '16.10', 'autcent_hash': 10, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '12.4', 'autcentquo_hash': 4, 'autcentquo_nilpotent': False, 'autcentquo_order': 12, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 6, 2], [3, 2, 1], [4, 2, 1], [5, 1, 4], [6, 2, 1], [10, 1, 4], [10, 6, 8], [12, 2, 2], [15, 2, 4], [20, 2, 4], [30, 2, 4], [60, 2, 8]], 'center_label': '10.2', 'center_order': 10, 'central_product': True, 'central_quotient': '12.4', 'commutator_count': 1, 'commutator_label': '6.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 23, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['24.6', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 2], [3, 2, 1, 1], [4, 2, 1, 1], [5, 1, 4, 1], [6, 2, 1, 1], [10, 1, 4, 1], [10, 6, 4, 2], [12, 2, 2, 1], [15, 2, 4, 1], [20, 2, 4, 1], [30, 2, 4, 1], [60, 2, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 18, 'exponent': 60, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[2, 0, 8]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '60.11', 'hash': 23, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6], 'inner_gens': [[1, 22], [101, 2]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'D_6', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 4, 'irrep_stats': [[1, 20], [2, 25]], 'label': '120.23', 'linC_count': 8, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 12, 'linQ_dim': 8, 'linQ_dim_count': 12, 'linR_count': 20, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5*D12', 'ngens': 5, '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': 14, 'number_conjugacy_classes': 45, 'number_divisions': 16, 'number_normal_subgroups': 18, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 32, 'number_subgroups': 68, 'old_label': None, 'order': 120, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 13], [3, 2], [4, 2], [5, 4], [6, 2], [10, 52], [12, 4], [15, 8], [20, 8], [30, 8], [60, 16]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4], 'outer_gen_pows': [6, 0, 0], 'outer_gens': [[31, 2], [1, 62], [1, 26]], 'outer_group': '16.10', 'outer_hash': 10, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [120, 5040, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 3], [4, 5], [8, 3], [16, 1]], 'representations': {'PC': {'code': 3378759755618187271, 'gens': [1, 2], 'pres': [5, -2, -2, -2, -3, -5, 221, 26, 662, 42, 1763, 78]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [5471, 13311]}, 'Perm': {'d': 12, 'gens': [40280400, 80196480, 33, 127370880, 5760]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times D_{12}', 'transitive_degree': 60, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 60, 'aut_gen_orders': [4, 4, 4, 20, 4, 4, 4, 2], 'aut_gens': [[1, 2, 4, 96], [25, 562, 68, 912], [1, 50, 28, 336], [49, 506, 772, 672], [49, 554, 844, 144], [49, 50, 244, 720], [25, 562, 476, 288], [73, 530, 820, 672], [1, 2, 860, 912]], 'aut_group': None, 'aut_hash': 5847347909673561214, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 15360, 'aut_permdeg': 36, 'aut_perms': [365347223059961355309501428406104385736271, 306736440428684102883410054746056272919500, 143916011655617118833605859357523320016248, 148418686559765272388409731928532605725749, 114070573906702359597662280509065716739310, 44585318717381938965858760039188030095684, 371680084161279150933014742425588249172345, 70641481942879119281479794469409038266023], 'aut_phi_ratio': 60.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 1], [2, 6, 2, 1], [2, 12, 2, 1], [2, 20, 1, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 4, 1, 1], [4, 6, 2, 1], [4, 20, 1, 1], [4, 60, 2, 1], [5, 2, 2, 1], [6, 2, 1, 1], [6, 4, 1, 1], [6, 8, 1, 1], [6, 40, 1, 1], [8, 10, 2, 1], [8, 20, 1, 1], [8, 60, 2, 1], [10, 2, 2, 1], [10, 4, 2, 1], [10, 4, 4, 1], [10, 12, 4, 1], [10, 12, 8, 1], [12, 2, 2, 1], [12, 4, 1, 1], [12, 8, 1, 1], [12, 40, 1, 1], [15, 4, 2, 1], [20, 4, 2, 2], [20, 4, 4, 1], [20, 12, 4, 1], [24, 20, 2, 2], [30, 4, 2, 1], [30, 8, 2, 1], [30, 8, 4, 1], [60, 4, 4, 1], [60, 8, 2, 1], [60, 8, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{15}:(C_2^3.C_2^6.C_2)', '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': 60, 'autcentquo_group': '960.11377', 'autcentquo_hash': 11377, 'autcentquo_nilpotent': False, 'autcentquo_order': 960, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times D_6\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 4, 1], [2, 6, 2], [2, 12, 2], [2, 20, 1], [3, 2, 1], [4, 2, 2], [4, 4, 1], [4, 6, 2], [4, 20, 1], [4, 60, 2], [5, 2, 2], [6, 2, 1], [6, 4, 1], [6, 8, 1], [6, 40, 1], [8, 10, 2], [8, 20, 1], [8, 60, 2], [10, 2, 2], [10, 4, 6], [10, 12, 12], [12, 2, 2], [12, 4, 1], [12, 8, 1], [12, 40, 1], [15, 4, 2], [20, 4, 8], [20, 12, 4], [24, 20, 4], [30, 4, 2], [30, 8, 6], [60, 4, 4], [60, 8, 6]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '480.1123', 'commutator_count': 1, 'commutator_label': '60.4', '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', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 9542, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 1], [2, 6, 1, 2], [2, 12, 1, 2], [2, 20, 1, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 4, 1, 1], [4, 6, 1, 2], [4, 20, 1, 1], [4, 60, 1, 2], [5, 2, 2, 1], [6, 2, 1, 1], [6, 4, 1, 1], [6, 8, 1, 1], [6, 40, 1, 1], [8, 10, 2, 1], [8, 20, 1, 1], [8, 60, 1, 2], [10, 2, 2, 1], [10, 4, 2, 1], [10, 4, 4, 1], [10, 12, 2, 2], [10, 12, 4, 2], [12, 2, 2, 1], [12, 4, 1, 1], [12, 8, 1, 1], [12, 40, 1, 1], [15, 4, 2, 1], [20, 4, 2, 2], [20, 4, 4, 1], [20, 12, 2, 2], [24, 20, 2, 2], [30, 4, 2, 1], [30, 8, 2, 1], [30, 8, 4, 1], [60, 4, 4, 1], [60, 8, 2, 1], [60, 8, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 16248960, 'exponent': 120, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[8, -1, 4]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '240.202', 'hash': 9542, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 2, 12, 10], 'inner_gens': [[1, 2, 76, 144], [1, 2, 44, 96], [25, 58, 4, 864], [49, 2, 196, 96]], 'inner_hash': 1123, 'inner_nilpotent': False, 'inner_order': 480, 'inner_split': True, 'inner_tex': 'D_{30}:C_2^3', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 32, 'irrQ_dim': 64, 'irrR_degree': 16, 'irrep_stats': [[1, 16], [2, 44], [4, 24], [8, 6]], 'label': '960.9542', 'linC_count': 128, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 64, 'linQ_dim': 14, 'linQ_dim_count': 64, 'linR_count': 32, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C60.C2^4', 'ngens': 8, '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': 44, 'number_characteristic_subgroups': 76, 'number_conjugacy_classes': 90, 'number_divisions': 52, 'number_normal_subgroups': 136, 'number_subgroup_autclasses': 364, 'number_subgroup_classes': 516, 'number_subgroups': 2792, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 63], [3, 2], [4, 160], [5, 4], [6, 54], [8, 160], [10, 172], [12, 56], [15, 8], [20, 80], [24, 80], [30, 56], [60, 64]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [0, 0, 480, 0], 'outer_gens': [[1, 2, 68, 96], [1, 2, 52, 96], [25, 482, 4, 96], [1, 2, 4, 288]], 'outer_group': '32.45', 'outer_hash': 45, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 10, 'outer_perms': [5040, 362880, 120, 17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 10], [8, 9], [16, 4], [32, 1]], 'representations': {'PC': {'code': 3979068386868250292594582861399756575104547551448989715, 'gens': [1, 2, 3, 7], 'pres': [8, 2, 2, 2, 2, 2, 3, 2, 5, 384, 1826, 538, 66, 1795, 1419, 91, 1612, 116, 1549, 8070, 12118, 166, 12311]}, 'Perm': {'d': 24, 'gens': [26120790521626347994344, 58725434311664498340481, 85897012702764369254400, 112422500799302284778880, 137588906839382955129600, 165442458588883993440000, 3, 6504]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{60}.C_2^4', 'transitive_degree': 240, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 6], [3, 2]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [5, 9], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [4, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', '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': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [4, 2, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [4, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 6], [5, 2]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 1]], 'label': '8.3', 'linC_count': 1, 'linC_degree': 2, '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': 'D4', 'ngens': 2, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 5, 'number_divisions': 5, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 10, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 5], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [2], 'outer_gens': [[3, 2]], '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': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1]], 'representations': {'PC': {'code': 294, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 37, 16]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [29, 56, 24], 'family': 'COPlus'}, {'d': 1, 'q': 4, 'gens': [7, 16, 1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [12, 55, 56]}, 'Perm': {'d': 4, 'gens': [6, 16, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4', 'transitive_degree': 4, 'wreath_data': ['C_2', 'C_2', '2T1'], 'wreath_product': True}
