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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1120.784', 'ambient_counter': 784, 'ambient_order': 1120, 'ambient_tex': 'C_7\\times C_2^3.D_{10}', 'central': False, 'central_factor': False, 'centralizer_order': 28, 'characteristic': False, 'core_order': 80, 'counter': 47, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1120.784.14.e1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '14.e1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '14.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 14, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{14}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '80.12', 'subgroup_hash': 12, 'subgroup_order': 80, 'subgroup_tex': 'C_{10}.D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1120.784', 'aut_centralizer_order': 24, 'aut_label': '14.e1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '320.1638', 'aut_weyl_index': 48, 'centralizer': '40.a1.a1', 'complements': ['80.d1.a1', '80.c1.a1', '80.c1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['2.e1.a1', '7.a1.a1'], 'contains': ['28.c1.a1', '28.d1.a1', '28.e1.b1', '70.e1.a1'], 'core': '14.e1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3311, 578, 6101, 9218, 7438, 245, 6150, 2898], 'generators': [41, 4, 560, 2, 16], 'label': '1120.784.14.e1.a1', 'mobius_quo': 0, 'mobius_sub': 1, 'normal_closure': '14.e1.a1', 'normal_contained_in': ['2.e1.a1', '7.a1.a1'], 'normal_contains': ['28.c1.a1', '28.d1.a1', '28.e1.b1'], 'normalizer': '1.a1.a1', 'old_label': '14.e1.a1', 'projective_image': '280.38', 'quotient_action_image': '2.1', 'quotient_action_kernel': '7.1', 'quotient_action_kernel_order': 7, 'quotient_fusion': None, 'short_label': '14.e1.a1', 'subgroup_fusion': None, 'weyl_group': '40.13'}
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gps_groups • Show schema
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{'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': 2, 'aut_exponent': 20, 'aut_gen_orders': [2, 2, 2, 4, 10], 'aut_gens': [[1, 4, 16], [3, 4, 16], [11, 4, 16], [3, 6, 16], [1, 4, 48], [1, 76, 16]], 'aut_group': '320.1638', 'aut_hash': 1638, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 320, 'aut_permdeg': 13, 'aut_perms': [5167, 7, 23, 47174407, 1923275536], 'aut_phi_ratio': 10.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 2, 1], [4, 10, 2, 2], [5, 2, 2, 1], [10, 2, 2, 3], [20, 2, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^4\\times F_5', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 2], [4, 10, 4], [5, 2, 2], [10, 2, 6], [20, 2, 8]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '20.4', 'commutator_count': 1, 'commutator_label': '10.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 12, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 2, 1], [4, 10, 1, 2], [4, 10, 2, 1], [5, 2, 2, 1], [10, 2, 2, 3], [20, 2, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 20, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '20.4', 'hash': 12, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 2, 5], 'inner_gens': [[1, 12, 16], [9, 4, 64], [1, 36, 16]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 20, 'inner_split': True, 'inner_tex': 'D_{10}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 18]], 'label': '80.12', 'linC_count': 32, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 10, 'linQ_dim': 8, 'linQ_dim_count': 4, 'linR_count': 26, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C10.D4', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 12, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 26, 'number_divisions': 13, 'number_normal_subgroups': 17, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 26, 'number_subgroups': 58, 'old_label': None, 'order': 80, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [4, 44], [5, 4], [10, 12], [20, 16]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[3, 4, 16], [1, 6, 16], [1, 4, 48]], 'outer_group': '16.10', 'outer_hash': 10, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_4', 'pc_rank': 3, '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, 4], [4, 2], [8, 3]], 'representations': {'PC': {'code': 26313725350915, 'gens': [1, 3, 5], 'pres': [5, -2, -2, -2, -2, -5, 10, 182, 42, 414]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [122109387504, 14536340629, 58658042680, 135202078093, 94271882503]}, 'Perm': {'d': 13, 'gens': [522588247, 83553840, 41040, 1037836800, 37]}}, '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_{10}.D_4', 'transitive_degree': 80, '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': '56.13', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 60, 'aut_gen_orders': [12, 12, 30, 10, 2, 10, 60], 'aut_gens': [[1, 2, 8, 80], [561, 562, 620, 240], [5, 615, 588, 404], [5, 50, 8, 880], [561, 626, 8, 1040], [561, 630, 636, 80], [561, 610, 8, 1040], [561, 631, 572, 724]], 'aut_group': None, 'aut_hash': 5837632494092988874, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 15360, 'aut_permdeg': 50, 'aut_perms': [18965454407775376502411152476002708222185302373968712178966887318, 7485743674254324754202192537226282782320193309141766655642104391, 12333535348248209936247138610578892690558351942575449214002203556, 25769080156806506245558185943207089584876516533567955326044649019, 29210757915746799363570092290227341582234909917908545046024080278, 14814974884060028685425149604778466768756798523693666807381427899, 27458708942417364011923412553789622964242769827031281823477263692], 'aut_phi_ratio': 40.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 4, 1, 1], [4, 4, 1, 1], [4, 10, 4, 1], [4, 20, 2, 1], [5, 2, 2, 1], [7, 1, 6, 1], [10, 2, 2, 1], [10, 2, 4, 1], [10, 4, 4, 2], [14, 1, 6, 1], [14, 1, 12, 1], [14, 2, 12, 1], [14, 4, 6, 1], [20, 4, 4, 1], [28, 4, 6, 1], [28, 10, 24, 1], [28, 20, 12, 1], [35, 2, 12, 1], [70, 2, 12, 1], [70, 2, 24, 1], [70, 4, 24, 2], [140, 4, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{15}:(C_2^6.C_2^4)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '384.20169', 'autcent_hash': 20169, 'autcent_nilpotent': True, 'autcent_order': 384, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6\\times C_6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '40.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 40, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 4, 1], [4, 4, 1], [4, 10, 4], [4, 20, 2], [5, 2, 2], [7, 1, 6], [10, 2, 6], [10, 4, 8], [14, 1, 18], [14, 2, 12], [14, 4, 6], [20, 4, 4], [28, 4, 6], [28, 10, 24], [28, 20, 12], [35, 2, 12], [70, 2, 36], [70, 4, 48], [140, 4, 24]], 'center_label': '28.4', 'center_order': 28, 'central_product': True, 'central_quotient': '40.13', 'commutator_count': 1, 'commutator_label': '20.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 784, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['160.156', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 4, 1, 1], [4, 4, 1, 1], [4, 10, 2, 2], [4, 20, 1, 2], [5, 2, 2, 1], [7, 1, 6, 1], [10, 2, 2, 3], [10, 4, 2, 2], [10, 4, 4, 1], [14, 1, 6, 3], [14, 2, 6, 2], [14, 4, 6, 1], [20, 4, 4, 1], [28, 4, 6, 1], [28, 10, 12, 2], [28, 20, 6, 2], [35, 2, 12, 1], [70, 2, 12, 3], [70, 4, 12, 2], [70, 4, 24, 1], [140, 4, 24, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 28728, 'exponent': 140, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '280.38', 'hash': 784, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 2, 10, 1], 'inner_gens': [[1, 2, 12, 80], [1, 2, 632, 80], [5, 578, 8, 80], [1, 2, 8, 80]], 'inner_hash': 13, 'inner_nilpotent': False, 'inner_order': 40, 'inner_split': False, 'inner_tex': 'C_2\\times D_{10}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 56], [2, 154], [4, 28]], 'label': '1120.784', 'linC_count': 1536, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 160, 'linQ_dim': 16, 'linQ_dim_count': 128, 'linR_count': 576, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C7*C2^3.D10', '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, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 28, 'number_characteristic_subgroups': 34, 'number_conjugacy_classes': 238, 'number_divisions': 40, 'number_normal_subgroups': 66, 'number_subgroup_autclasses': 104, 'number_subgroup_classes': 156, 'number_subgroups': 416, 'old_label': None, 'order': 1120, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 11], [4, 84], [5, 4], [7, 6], [10, 44], [14, 66], [20, 16], [28, 504], [35, 24], [70, 264], [140, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 4, 12], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 3, 8, 84], [565, 2, 8, 80], [565, 6, 8, 1040], [1, 2, 24, 80], [5, 3, 8, 724]], 'outer_group': '384.19882', 'outer_hash': 19882, 'outer_nilpotent': True, 'outer_order': 384, 'outer_permdeg': 15, 'outer_perms': [186810624000, 3628800, 24, 3674904, 187293254427], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_4\\times C_{12}', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 7], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [4, 6], [6, 8], [8, 4], [12, 2], [24, 6], [48, 4]], 'representations': {'PC': {'code': 2784954514917482529873619117937, 'gens': [1, 2, 4, 6], 'pres': [7, -2, -2, -2, 2, -5, -2, -7, 36, 339, 8858, 80, 1131, 124]}, 'Perm': {'d': 24, 'gens': [1126433629785835941127, 87096360, 135732245, 6780385003392000, 16, 179424720, 28102451096448368640000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 14], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_7\\times C_2^3.D_{10}', 'transitive_degree': 560, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '14.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 6, 'aut_gen_orders': [6], 'aut_gens': [[1], [3]], 'aut_group': '6.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 6, 'aut_permdeg': 5, 'aut_perms': [27], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [7, 1, 6, 1], [14, 1, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_6', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 6, 'autcent_group': '6.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_6', '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], [2, 1, 1], [7, 1, 6], [14, 1, 6]], 'center_label': '14.2', 'center_order': 14, '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': ['2.1', '7.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [7, 1, 6, 1], [14, 1, 6, 1]], 'element_repr_type': 'PC', 'elementary': 14, 'eulerian_function': 1, 'exponent': 14, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 7], 'faithful_reps': [[1, 0, 6]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '14.2', 'hash': 2, 'hyperelementary': 14, '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': 6, 'irrQ_dim': 6, 'irrR_degree': 2, 'irrep_stats': [[1, 14]], 'label': '14.2', 'linC_count': 6, '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': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C14', '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': 14, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 14, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [7, 6], [14, 6]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[3]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 7], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [6, 2]], 'representations': {'PC': {'code': 185, 'gens': [1], 'pres': [2, -2, -7, 4]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [75621406240170985]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [351, 2064]}, 'Perm': {'d': 9, 'gens': [40320, 4320]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [14], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{14}', 'transitive_degree': 14, 'wreath_data': None, 'wreath_product': False}
