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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '960.9542', 'ambient_counter': 9542, 'ambient_order': 960, 'ambient_tex': 'C_{60}.C_2^4', 'central': False, 'central_factor': False, 'centralizer_order': 240, 'characteristic': True, 'core_order': 15, 'counter': 410, 'cyclic': True, 'direct': False, 'hall': 15, 'label': '960.9542.64.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '64.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '64.258', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 258, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 64, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_8:C_2^2', 'simple': False, 'solvable': True, 'special_labels': ['L3', 'C6'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '15.1', 'subgroup_hash': 1, 'subgroup_order': 15, 'subgroup_tex': 'C_{15}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '960.9542', 'aut_centralizer_order': 1920, 'aut_label': '64.a1', 'aut_quo_index': 3, 'aut_stab_index': 1, 'aut_weyl_group': '8.2', 'aut_weyl_index': 1920, 'centralizer': '4.a1.a1', 'complements': ['15.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['32.a1.a1', '32.b1.a1', '32.c1.a1', '32.d1.a1', '32.d1.b1', '32.e1.a1', '32.e1.b1', '32.f1.a1'], 'contains': ['192.a1.a1', '320.a1.a1'], 'core': '64.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [682, 4923, 9550, 1967, 1072, 3669, 9274, 7313], 'generators': [32, 192], 'label': '960.9542.64.a1.a1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '64.a1.a1', 'normal_contained_in': ['32.a1.a1'], 'normal_contains': ['192.a1.a1', '320.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '64.a1.a1', 'projective_image': '960.9542', 'quotient_action_image': '4.2', 'quotient_action_kernel': '16.13', 'quotient_action_kernel_order': 16, 'quotient_fusion': None, 'short_label': '64.a1.a1', 'subgroup_fusion': None, 'weyl_group': '4.2'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '15.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1], [11], [7]], 'aut_group': '8.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 6, 'aut_perms': [120, 9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [5, 1, 4, 1], [15, 1, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_4', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [3, 1, 2], [5, 1, 4], [15, 1, 8]], 'center_label': '15.1', 'center_order': 15, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '5.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [5, 1, 4, 1], [15, 1, 8, 1]], 'element_repr_type': 'PC', 'elementary': 15, 'eulerian_function': 1, 'exponent': 15, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2], 'factors_of_order': [3, 5], 'faithful_reps': [[1, 0, 8]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '15.1', 'hash': 1, 'hyperelementary': 15, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 2, 'irrep_stats': [[1, 15]], 'label': '15.1', 'linC_count': 8, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 4, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C15', 'ngens': 2, 'nilpotency_class': 1, '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': 15, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 15, 'order_factorization_type': 11, 'order_stats': [[1, 1], [3, 2], [5, 4], [15, 8]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[11], [7]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 8, 'pgroup': 0, 'primary_abelian_invariants': [3, 5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [4, 1], [8, 1]], 'representations': {'PC': {'code': 87, 'gens': [1], 'pres': [2, -3, -5, 6]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [58415890696506688]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [6351]}, 'Perm': {'d': 8, 'gens': [10080, 96]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [15], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{15}', 'transitive_degree': 15, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'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}
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gps_groups • Show schema
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{'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': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 2, 2, 3, 2, 2], 'aut_gens': [[1, 2, 4, 8], [53, 22, 36, 8], [1, 34, 4, 40], [33, 18, 36, 56], [33, 34, 36, 56], [1, 34, 4, 8], [21, 51, 33, 8], [1, 34, 36, 8], [33, 2, 36, 8]], 'aut_group': '384.20051', 'aut_hash': 20051, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 384, 'aut_permdeg': 10, 'aut_perms': [40320, 1, 126, 55, 288, 45360, 367920, 806400], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 3, 1], [2, 4, 1, 1], [2, 4, 3, 1], [4, 2, 1, 1], [4, 2, 3, 1], [4, 4, 1, 1], [4, 4, 3, 1], [8, 2, 2, 1], [8, 4, 3, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(2,\\mathbb{Z}/4):C_2^2', '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': 6, 'autcentquo_group': '24.14', 'autcentquo_hash': 14, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_6', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 3], [2, 4, 4], [4, 2, 4], [4, 4, 4], [8, 2, 2], [8, 4, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '32.46', 'commutator_count': 1, 'commutator_label': '4.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 258, '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, 3], [2, 4, 1, 4], [4, 2, 1, 4], [4, 4, 1, 4], [8, 2, 2, 1], [8, 4, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 13440, 'exponent': 8, 'exponents_of_order': [6], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [[4, 0, 2]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '16.14', 'hash': 258, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 2, 2, 4], 'inner_gens': [[1, 2, 36, 8], [1, 2, 4, 24], [33, 2, 4, 8], [1, 50, 4, 8]], 'inner_hash': 46, 'inner_nilpotent': True, 'inner_order': 32, 'inner_split': True, 'inner_tex': 'C_2^2\\times D_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 4], [4, 2]], 'label': '64.258', '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': 'D8:C2^2', 'ngens': 4, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 11, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 22, 'number_divisions': 21, 'number_normal_subgroups': 79, 'number_subgroup_autclasses': 54, 'number_subgroup_classes': 129, 'number_subgroups': 205, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 23], [4, 24], [8, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [4, 16], 'outer_gens': [[21, 6, 4, 8], [53, 3, 1, 40]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [6, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 4], [8, 1]], 'representations': {'PC': {'code': 18146306037848, 'gens': [1, 2, 3, 4], 'pres': [6, -2, 2, 2, 2, -2, -2, 650, 297, 69, 730, 88]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [34247624, 14557484, 28410960, 37630912, 28816400, 28233362]}, 'Perm': {'d': 16, 'gens': [12526762711680, 5612462150399, 8402294108878, 5606234750016, 4097506706057, 1313941673647]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_8:C_2^2', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
