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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1120.703', 'ambient_counter': 703, 'ambient_order': 1120, 'ambient_tex': 'C_5\\times D_{14}:D_4', 'central': False, 'central_factor': False, 'centralizer_order': 40, 'characteristic': False, 'core_order': 28, 'counter': 64, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1120.703.40.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '40.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '40.10', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 10, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 40, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_5\\times D_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '28.3', 'subgroup_hash': 3, 'subgroup_order': 28, 'subgroup_tex': 'D_{14}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1120.703', 'aut_centralizer_order': 64, 'aut_label': '40.a1', 'aut_quo_index': 2, 'aut_stab_index': 4, 'aut_weyl_group': '84.7', 'aut_weyl_index': 256, 'centralizer': '28.a1', 'complements': ['28.h1', '28.e1'], 'conjugacy_class_count': 4, 'contained_in': ['8.a1', '20.c1', '20.f1', '20.g1'], 'contains': ['80.a1', '80.e1', '280.e1'], 'core': '40.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [7811, 577, 7554, 8591], 'generators': [447, 160, 4], 'label': '1120.703.40.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '40.a1', 'normal_contained_in': ['8.a1', '20.c1'], 'normal_contains': ['80.a1'], 'normalizer': '1.a1', 'old_label': '40.a1', 'projective_image': '560.140', 'quotient_action_image': '2.1', 'quotient_action_kernel': '20.5', 'quotient_action_kernel_order': 20, 'quotient_fusion': None, 'short_label': '40.a1', 'subgroup_fusion': None, 'weyl_group': '28.3'}
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gps_groups • Show schema
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{'Agroup': True, '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': 42, 'aut_gen_orders': [6, 14], 'aut_gens': [[1, 2], [1, 6], [11, 2]], 'aut_group': '84.7', 'aut_hash': 7, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 84, 'aut_permdeg': 9, 'aut_perms': [6055, 203047], 'aut_phi_ratio': 7.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 7, 2, 1], [7, 2, 3, 1], [14, 2, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times F_7', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '42.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 42, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 7, 2], [7, 2, 3], [14, 2, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '14.1', 'commutator_count': 1, 'commutator_label': '7.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '7.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [['14.1', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 7, 1, 2], [7, 2, 3, 1], [14, 2, 3, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 14, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[2, 1, 3]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '28.3', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 7], 'inner_gens': [[1, 26], [5, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 14, 'inner_split': True, 'inner_tex': 'D_7', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 6]], 'label': '28.3', 'linC_count': 3, 'linC_degree': 2, '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': 'D14', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 10, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 10, 'number_subgroups': 28, 'old_label': None, 'order': 28, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 15], [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': [[15, 10]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [6, 2]], 'representations': {'PC': {'code': 28266197, 'gens': [1, 2], 'pres': [3, -2, -2, -7, 157, 16, 218]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [26761470676238614, 122882691976452762]}, 'GLFp': {'d': 2, 'p': 7, 'gens': [351, 2059, 2064]}, 'Perm': {'d': 9, 'gens': [5166, 1, 50646]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{14}', 'transitive_degree': 14, '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': '40.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [6, 12, 2, 12, 28, 12, 6], 'aut_gens': [[1, 2, 4, 8], [1, 566, 4, 76], [565, 767, 564, 136], [1, 1007, 564, 552], [561, 843, 564, 268], [565, 43, 564, 8], [565, 367, 564, 476], [1, 326, 4, 312]], 'aut_group': None, 'aut_hash': 983146866434854081, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 21504, 'aut_permdeg': 64, 'aut_perms': [47616133688167446286561750681806213378791876418476550182997239694912512991642248379609483, 78277847545416524760432835805733102767268699508529897749831874525958842729097047650902530, 89258325688350311836242998151928955172345436898457696190588269282758048224761901288228749, 82155303163836730848685944837601731059943473693224714561058437101106728367973701388352911, 100092180648949008889702480262820033060078445726748553570128058064726554385854691614492613, 88194991377233086330839995638285262303258732665075613168477500864588405085339489248518683, 113054564671386329847943329088837954354052770003021819016364621083832006248145380544775615], 'aut_phi_ratio': 56.0, '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], [2, 14, 4, 1], [4, 4, 1, 1], [4, 28, 2, 1], [5, 1, 4, 1], [7, 2, 3, 1], [10, 1, 4, 1], [10, 1, 8, 1], [10, 2, 8, 1], [10, 4, 4, 1], [10, 14, 16, 1], [14, 2, 3, 1], [14, 2, 6, 1], [14, 4, 6, 2], [20, 4, 4, 1], [20, 28, 8, 1], [28, 4, 6, 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_2^4\\times C_7:C_3).C_2^6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '256.56082', 'autcent_hash': 56082, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '84.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 84, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 4, 1], [2, 14, 4], [4, 4, 1], [4, 28, 2], [5, 1, 4], [7, 2, 3], [10, 1, 12], [10, 2, 8], [10, 4, 4], [10, 14, 16], [14, 2, 9], [14, 4, 12], [20, 4, 4], [20, 28, 8], [28, 4, 6], [35, 2, 12], [70, 2, 36], [70, 4, 48], [140, 4, 24]], 'center_label': '20.5', 'center_order': 20, 'central_product': True, 'central_quotient': '56.12', 'commutator_count': 1, 'commutator_label': '28.4', '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': 703, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['224.132', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 4, 1, 1], [2, 14, 1, 4], [4, 4, 1, 1], [4, 28, 1, 2], [5, 1, 4, 1], [7, 2, 3, 1], [10, 1, 4, 3], [10, 2, 4, 2], [10, 4, 4, 1], [10, 14, 4, 4], [14, 2, 3, 3], [14, 4, 3, 2], [14, 4, 6, 1], [20, 4, 4, 1], [20, 28, 4, 2], [28, 4, 6, 1], [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': 20832, 'exponent': 140, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 5, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '280.37', 'hash': 703, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 2, 1, 14], 'inner_gens': [[1, 566, 4, 568], [565, 2, 4, 892], [1, 2, 4, 8], [561, 246, 4, 8]], 'inner_hash': 12, 'inner_nilpotent': False, 'inner_order': 56, 'inner_split': False, 'inner_tex': 'C_2\\times D_{14}', 'inner_used': [1, 2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 40], [2, 150], [4, 30]], 'label': '1120.703', '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': 'C5*D14:D4', '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': 220, 'number_divisions': 44, 'number_normal_subgroups': 74, 'number_subgroup_autclasses': 128, 'number_subgroup_classes': 260, 'number_subgroups': 1276, 'old_label': None, 'order': 1120, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 67], [4, 60], [5, 4], [7, 6], [10, 268], [14, 66], [20, 240], [28, 24], [35, 24], [70, 264], [140, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 4, 12], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 2, 4, 572], [565, 2, 4, 572], [5, 2, 4, 8], [1, 2, 4, 904], [1, 843, 564, 968]], 'outer_group': '384.19882', 'outer_hash': 19882, 'outer_nilpotent': True, 'outer_order': 384, 'outer_permdeg': 15, 'outer_perms': [12933084264, 186814293864, 3628800, 3720240, 93884354643], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_4\\times C_{12}', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 8], [6, 4], [8, 6], [12, 4], [24, 4], [48, 4]], 'representations': {'PC': {'code': 465222441220956046247987884425588642839, 'gens': [1, 2, 3, 4], 'pres': [7, -2, -2, -2, 2, -2, -5, -7, 7925, 15907, 12498, 80, 11491, 102, 4044, 250, 23533]}, 'Perm': {'d': 20, 'gens': [6423384156548400, 1037847600, 83503440, 33, 90720, 1037927520, 134491780578355200]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times D_{14}:D_4', 'transitive_degree': 280, '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': '20.5', '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, 4], 'aut_gens': [[1, 2], [1, 22], [1, 34], [11, 2]], 'aut_group': '32.25', 'aut_hash': 25, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 32, 'aut_permdeg': 8, 'aut_perms': [10800, 11529, 5880], '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], [5, 1, 4, 1], [10, 1, 4, 1], [10, 2, 8, 1], [20, 2, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times D_4', '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': 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], [5, 1, 4], [10, 1, 4], [10, 2, 8], [20, 2, 4]], 'center_label': '10.2', 'center_order': 10, 'central_product': True, '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', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 10, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['5.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [4, 2, 1, 1], [5, 1, 4, 1], [10, 1, 4, 1], [10, 2, 4, 2], [20, 2, 4, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 18, 'exponent': 20, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 5], 'faithful_reps': [[2, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '20.5', 'hash': 10, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 22], [21, 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': 8, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 20], [2, 5]], 'label': '40.10', 'linC_count': 4, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 4, 'linQ_dim': 6, 'linQ_dim_count': 4, 'linR_count': 10, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5*D4', 'ngens': 4, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 25, 'number_divisions': 10, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 16, 'number_subgroups': 20, 'old_label': None, 'order': 40, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 5], [4, 2], [5, 4], [10, 20], [20, 8]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [2, 0], 'outer_gens': [[11, 2], [1, 34]], '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': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 4], [8, 1]], 'representations': {'PC': {'code': 1473908227, 'gens': [1, 2], 'pres': [4, -2, -2, -2, -5, 177, 21, 34]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [25038659807093174, 58438750843733785]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [13320, 132, 4001]}, 'Perm': {'d': 9, 'gens': [40320, 90720, 96, 41040]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times D_4', 'transitive_degree': 20, 'wreath_data': None, 'wreath_product': False}
