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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1600.908', 'ambient_counter': 908, 'ambient_order': 1600, 'ambient_tex': 'C_2^4.C_{100}', 'central': False, 'central_factor': False, 'centralizer_order': 800, 'characteristic': False, 'core_order': 400, 'counter': 9, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1600.908.4.e1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '4.e1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '4.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '400.45', 'subgroup_hash': 45, 'subgroup_order': 400, 'subgroup_tex': 'C_2^2\\times C_{100}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1600.908', 'aut_centralizer_order': 32, 'aut_label': '4.e1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '320.1628', 'aut_weyl_index': 64, 'centralizer': '2.a1', 'complements': [], 'conjugacy_class_count': 2, 'contained_in': ['2.a1'], 'contains': ['8.b1', '8.d1', '8.e1', '8.k1', '20.e1'], 'core': '4.e1', 'coset_action_label': None, 'count': 2, 'diagramx': [5428, 2738, 3990, 1503], 'generators': [1, 320, 4, 704, 402, 800], 'label': '1600.908.4.e1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '4.e1', 'normal_contained_in': ['2.a1'], 'normal_contains': ['8.b1', '8.d1', '8.e1', '20.e1'], 'normalizer': '1.a1', 'old_label': '4.e1', 'projective_image': '16.10', 'quotient_action_image': '2.1', 'quotient_action_kernel': '2.1', 'quotient_action_kernel_order': 2, 'quotient_fusion': None, 'short_label': '4.e1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '400.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 60, 'aut_gen_orders': [60, 10, 20], 'aut_gens': [[1, 2, 4], [202, 203, 132], [201, 202, 236], [202, 1, 134]], 'aut_group': None, 'aut_hash': 7579317348600198797, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3840, 'aut_permdeg': 28, 'aut_perms': [220896397505575905079044381961, 123737130505005152087100527340, 220896397505573405951664094696], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [4, 1, 8, 1], [5, 1, 4, 1], [10, 1, 4, 1], [10, 1, 24, 1], [20, 1, 32, 1], [25, 1, 20, 1], [50, 1, 20, 1], [50, 1, 120, 1], [100, 1, 160, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^3.(C_{20}\\times S_4)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 60, 'autcent_group': None, 'autcent_hash': 7579317348600198797, 'autcent_nilpotent': False, 'autcent_order': 3840, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^3.(C_{20}\\times S_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], [2, 1, 7], [4, 1, 8], [5, 1, 4], [10, 1, 28], [20, 1, 32], [25, 1, 20], [50, 1, 140], [100, 1, 160]], 'center_label': '400.45', 'center_order': 400, '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', '2.1', '2.1', '2.1', '5.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 45, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['25.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [4, 1, 2, 4], [5, 1, 4, 1], [10, 1, 4, 7], [20, 1, 8, 4], [25, 1, 20, 1], [50, 1, 20, 7], [100, 1, 40, 4]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 5425, 'exponent': 100, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '10.2', 'frattini_quotient': '40.14', 'hash': 45, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 400]], 'label': '400.45', 'linC_count': 3472000, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 23, 'linQ_degree_count': 96, 'linQ_dim': 23, 'linQ_dim_count': 96, 'linR_count': 960, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2*C100', 'ngens': 6, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 12, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 400, 'number_divisions': 36, 'number_normal_subgroups': 81, 'number_subgroup_autclasses': 27, 'number_subgroup_classes': 81, 'number_subgroups': 81, 'old_label': None, 'order': 400, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 7], [4, 8], [5, 4], [10, 28], [20, 32], [25, 20], [50, 140], [100, 160]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 60, 'outer_gen_orders': [4, 60, 20], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[203, 2, 399], [203, 1, 151], [203, 202, 287]], 'outer_group': None, 'outer_hash': 7579317348600198797, 'outer_nilpotent': False, 'outer_order': 3840, 'outer_permdeg': 28, 'outer_perms': [146136261379122839030041761978, 61771695763985183064757273984, 169787659421651158803549072990], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^3.(C_{20}\\times S_4)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 33, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4, 25], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 8], [8, 4], [20, 8], [40, 4]], 'representations': {'PC': {'code': 695892076249103, 'gens': [1, 2, 3], 'pres': [6, -2, -2, -2, -2, -5, -5, 50, 69, 178]}, 'GLZN': {'d': 2, 'p': 75, 'gens': [13500032, 19576396, 20671924, 423001, 18140668, 10267524]}, 'Perm': {'d': 33, 'gens': [936442772735978285826048000000, 263130836933693530167218012160000000, 265252859812191058636308480000000, 15431352467814595952640000, 305291636072840466137088000000, 2481794093513708107384416]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 100], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times C_{100}', 'transitive_degree': 400, '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': '400.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 40, 'aut_gen_orders': [20, 20, 4, 20, 2, 20, 20], 'aut_gens': [[1, 2, 4, 8], [807, 802, 804, 219], [807, 2, 804, 1128], [807, 6, 804, 14], [801, 806, 4, 733], [805, 802, 4, 1214], [3, 802, 804, 1368], [7, 802, 804, 479]], 'aut_group': None, 'aut_hash': 2079054595269466009, 'aut_nilpotency_class': 4, 'aut_nilpotent': True, 'aut_order': 20480, 'aut_permdeg': 52, 'aut_perms': [52531935424299376591101806173863312278528586448303419645962760703368, 27130801701206489994978595240388716913012910671142997227801106611180, 18649339152436907205312120323169926552573593235809878671001059274705, 14735853146644354612690213652714382316018856541355247582795651630738, 24860512106031004689930087189392780600539379354303398438240804489161, 75789428145168134887197601762172032562130327103333067645080940118729, 15191142984979094202300361673022309343743646723977903565120869601292], 'aut_phi_ratio': 32.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 2, 4, 1], [4, 1, 4, 1], [4, 2, 2, 1], [4, 2, 4, 1], [5, 1, 4, 1], [8, 4, 8, 1], [10, 1, 4, 1], [10, 1, 8, 1], [10, 2, 8, 1], [10, 2, 16, 1], [20, 1, 16, 1], [20, 2, 8, 1], [20, 2, 16, 1], [25, 1, 20, 1], [40, 4, 32, 1], [50, 1, 20, 1], [50, 1, 40, 1], [50, 2, 40, 1], [50, 2, 80, 1], [100, 1, 80, 1], [100, 2, 40, 1], [100, 2, 80, 1], [200, 4, 160, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_5:(C_2^4.C_2^5.C_2^3)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 20, 'autcent_group': None, 'autcent_hash': 8909733522074432775, 'autcent_nilpotent': True, 'autcent_order': 2560, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^7\\times C_{20}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '8.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': True, 'autcentquo_order': 8, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 6], [4, 1, 4], [4, 2, 6], [5, 1, 4], [8, 4, 8], [10, 1, 12], [10, 2, 24], [20, 1, 16], [20, 2, 24], [25, 1, 20], [40, 4, 32], [50, 1, 60], [50, 2, 120], [100, 1, 80], [100, 2, 120], [200, 4, 160]], 'center_label': '200.9', 'center_order': 200, 'central_product': True, 'central_quotient': '8.5', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 908, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['25.1', 1], ['64.88', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 6], [4, 1, 2, 2], [4, 2, 1, 2], [4, 2, 2, 2], [5, 1, 4, 1], [8, 4, 2, 4], [10, 1, 4, 3], [10, 2, 4, 6], [20, 1, 8, 2], [20, 2, 4, 2], [20, 2, 8, 2], [25, 1, 20, 1], [40, 4, 8, 4], [50, 1, 20, 3], [50, 2, 20, 6], [100, 1, 40, 2], [100, 2, 20, 2], [100, 2, 40, 2], [200, 4, 40, 4]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 65100, 'exponent': 200, 'exponents_of_order': [6, 2], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '40.9', 'frattini_quotient': '40.14', 'hash': 908, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 1, 2], 'inner_gens': [[1, 2, 4, 12], [1, 2, 4, 808], [1, 2, 4, 8], [5, 802, 4, 8]], 'inner_hash': 5, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'C_2^3', 'inner_used': [1, 2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': None, 'irrep_stats': [[1, 400], [2, 300]], 'label': '1600.908', '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': 'C2^4.C100', 'ngens': 8, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 27, 'number_characteristic_subgroups': 33, 'number_conjugacy_classes': 700, 'number_divisions': 60, 'number_normal_subgroups': 135, 'number_subgroup_autclasses': 117, 'number_subgroup_classes': 285, 'number_subgroups': 435, 'old_label': None, 'order': 1600, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 15], [4, 16], [5, 4], [8, 32], [10, 60], [20, 64], [25, 20], [40, 128], [50, 300], [100, 320], [200, 640]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 20, 'outer_gen_orders': [20, 4, 20, 20, 20, 10, 20], 'outer_gen_pows': [2, 0, 0, 0, 0, 0, 0], 'outer_gens': [[803, 6, 804, 777], [801, 806, 4, 1262], [3, 2, 804, 27], [1, 6, 4, 1304], [3, 802, 804, 314], [801, 802, 4, 1035], [1, 802, 4, 826]], 'outer_group': None, 'outer_hash': 1393058454651024668, 'outer_nilpotent': True, 'outer_order': 2560, 'outer_permdeg': 21, 'outer_perms': [5109159689584594650, 7548398599053358080, 2439304383399252570, 121645100408952993, 2676192210032140896, 7426774508616378810, 121666110859762650], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_5:(C_2^5.C_2^4)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 37, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4, 25], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [4, 12], [8, 8], [16, 4], [20, 8], [40, 8], [80, 4]], 'representations': {'PC': {'code': 876589287634963151844807265484815, 'gens': [1, 2, 3, 4], 'pres': [8, -2, -2, -2, 2, -2, -2, -5, -5, 387, 12939, 91, 116, 141, 334]}, 'Perm': {'d': 37, 'gens': [393267989706071901264716494050164736000000, 754346122852742784426967050475929600000000, 31303376726747018984379784904048640000000, 775012163878126168168455304773894144000000, 1158511675905463936323606862160461824000000, 775012163877494613740331040348569600000000, 15431352467814595952640000, 2481794093513708107384416]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 100], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^4.C_{100}', 'transitive_degree': 400, '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': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [3]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', '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], [4, 1, 2]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [2], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4]], 'label': '4.1', 'linC_count': 2, 'linC_degree': 1, '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': 'C4', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 1], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3]], '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': 1, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -2, -2, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [56, 15], 'family': 'CSOPlus'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [21]}, 'Perm': {'d': 4, 'gens': [22, 7]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
