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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '960.10807', 'ambient_counter': 10807, 'ambient_order': 960, 'ambient_tex': 'C_2^2:Q_8\\times C_{30}', 'central': False, 'central_factor': False, 'centralizer_order': 480, 'characteristic': True, 'core_order': 16, 'counter': 139, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '960.10807.60.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '60.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '60.13', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 13, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 60, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times C_{30}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '16.14', 'subgroup_hash': 14, 'subgroup_order': 16, 'subgroup_tex': 'C_2^4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '960.10807', 'aut_centralizer_order': 1024, 'aut_label': '60.a1', 'aut_quo_index': 3, 'aut_stab_index': 1, 'aut_weyl_group': '32.27', 'aut_weyl_index': 1024, 'centralizer': '2.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['12.a1', '20.a1', '30.a1', '30.b1'], 'contains': ['120.a1', '120.b1', '120.c1', '120.i1', '120.l1'], 'core': '60.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [6391, 6862, 7357, 6769], 'generators': [1, 2, 488, 480], 'label': '960.10807.60.a1', 'mobius_quo': 0, 'mobius_sub': 2, 'normal_closure': '60.a1', 'normal_contained_in': ['12.a1', '20.a1', '30.a1', '30.b1'], 'normal_contains': ['120.a1', '120.b1', '120.c1'], 'normalizer': '1.a1', 'old_label': '60.a1', 'projective_image': '120.47', 'quotient_action_image': '2.1', 'quotient_action_kernel': '30.4', 'quotient_action_kernel_order': 30, 'quotient_fusion': None, 'short_label': '60.a1', '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 420, 'aut_gen_orders': [6, 3], 'aut_gens': [[1, 2, 4, 8], [12, 4, 5, 3], [10, 14, 7, 5]], 'aut_group': '20160.a', 'aut_hash': 3764836782182912467, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20160, 'aut_permdeg': 8, 'aut_perms': [5193, 5760], 'aut_phi_ratio': 2520.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 15, 1]], 'aut_supersolvable': False, 'aut_tex': 'A_8', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 420, 'autcent_group': '20160.a', 'autcent_hash': 3764836782182912467, 'autcent_nilpotent': False, 'autcent_order': 20160, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'A_8', '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, 15]], 'center_label': '16.14', 'center_order': 16, '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'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 14, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 4]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [4], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '16.14', 'hash': 14, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1, 1], 'inner_gens': [[1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 8]], '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, 16]], 'label': '16.14', 'linC_count': 840, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 840, 'linQ_dim': 4, 'linQ_dim_count': 840, 'linR_count': 840, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^4', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 16, 'number_divisions': 16, 'number_normal_subgroups': 67, 'number_subgroup_autclasses': 5, 'number_subgroup_classes': 67, 'number_subgroups': 67, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 15]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 420, 'outer_gen_orders': [6, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[12, 4, 5, 3], [10, 14, 7, 5]], 'outer_group': '20160.a', 'outer_hash': 3764836782182912467, 'outer_nilpotent': False, 'outer_order': 20160, 'outer_permdeg': 8, 'outer_perms': [5193, 5760], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'A_8', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3, 4], 'pres': [4, -2, 2, 2, 2]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [7233746, 7115648, 35812976, 7115160]}, 'GLFp': {'d': 4, 'p': 2, 'gens': [33837, 18465, 27183, 18467]}, 'Perm': {'d': 8, 'gens': [5040, 120, 6, 1]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^4', 'transitive_degree': 16, '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': '240.208', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [4, 4, 4, 2, 4, 4, 4, 4], 'aut_gens': [[1, 2, 4, 16], [489, 483, 252, 24], [481, 491, 484, 944], [481, 11, 245, 273], [481, 2, 484, 504], [489, 11, 244, 696], [489, 482, 253, 312], [481, 491, 733, 784], [489, 490, 724, 473]], 'aut_group': None, 'aut_hash': 1578825338858570690, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 32768, 'aut_permdeg': 30, 'aut_perms': [40625978246357329016067542538085, 125498127070006875660656952168249, 195755592584107088828852366957058, 221567793276023425239363156909129, 218687630188130555494334083735605, 40701735695439733699550905671715, 166004957809407173367373123091271, 201382718939134187120518644828726], 'aut_phi_ratio': 128.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 1, 4, 1], [2, 2, 4, 1], [3, 1, 2, 1], [4, 2, 4, 2], [4, 4, 4, 2], [5, 1, 4, 1], [6, 1, 2, 3], [6, 1, 8, 1], [6, 2, 8, 1], [10, 1, 4, 3], [10, 1, 16, 1], [10, 2, 16, 1], [12, 2, 8, 2], [12, 4, 8, 2], [15, 1, 8, 1], [20, 2, 16, 2], [20, 4, 16, 2], [30, 1, 8, 3], [30, 1, 32, 1], [30, 2, 32, 1], [60, 2, 32, 2], [60, 4, 32, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_2^8.C_2^6.C_2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': None, 'autcent_hash': 2851693483551186743, 'autcent_nilpotent': True, 'autcent_order': 16384, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_4\\times C_2^8.C_2^3', '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, 7], [2, 2, 4], [3, 1, 2], [4, 2, 8], [4, 4, 8], [5, 1, 4], [6, 1, 14], [6, 2, 8], [10, 1, 28], [10, 2, 16], [12, 2, 16], [12, 4, 16], [15, 1, 8], [20, 2, 32], [20, 4, 32], [30, 1, 56], [30, 2, 32], [60, 2, 64], [60, 4, 64]], 'center_label': '120.47', 'center_order': 120, 'central_product': True, 'central_quotient': '8.5', 'commutator_count': 1, 'commutator_label': '4.2', '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': 10807, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['32.29', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 2, 1, 4], [3, 1, 2, 1], [4, 2, 1, 4], [4, 2, 2, 2], [4, 4, 1, 8], [5, 1, 4, 1], [6, 1, 2, 7], [6, 2, 2, 4], [10, 1, 4, 7], [10, 2, 4, 4], [12, 2, 2, 4], [12, 2, 4, 2], [12, 4, 2, 8], [15, 1, 8, 1], [20, 2, 4, 4], [20, 2, 8, 2], [20, 4, 4, 8], [30, 1, 8, 7], [30, 2, 8, 4], [60, 2, 8, 4], [60, 2, 16, 2], [60, 4, 8, 8]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 7862400, 'exponent': 60, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '240.208', 'hash': 10807, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [1, 2, 2, 2], 'inner_gens': [[1, 2, 4, 16], [1, 2, 492, 16], [1, 490, 4, 496], [1, 2, 484, 16]], 'inner_hash': 5, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'C_2^3', 'inner_used': [2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 240], [2, 180]], 'label': '960.10807', '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': 'C2^2:Q8*C30', '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, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 40, 'number_characteristic_subgroups': 52, 'number_conjugacy_classes': 420, 'number_divisions': 104, 'number_normal_subgroups': 388, 'number_subgroup_autclasses': 208, 'number_subgroup_classes': 644, 'number_subgroups': 900, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 15], [3, 2], [4, 48], [5, 4], [6, 30], [10, 60], [12, 96], [15, 8], [20, 192], [30, 120], [60, 384]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [4, 4, 4, 2, 4, 2, 4, 4], 'outer_gen_pows': [0, 0, 0, 0, 0, 240, 0, 0], 'outer_gens': [[489, 483, 252, 24], [481, 491, 484, 944], [481, 11, 245, 273], [481, 2, 484, 504], [489, 11, 244, 696], [489, 482, 253, 312], [481, 491, 733, 784], [489, 490, 724, 473]], 'outer_group': None, 'outer_hash': 1926513769237116296, 'outer_nilpotent': True, 'outer_order': 4096, 'outer_permdeg': 512, 'outer_perms': 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'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^8.C_2^4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 3, 5], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 24], [4, 26], [8, 26], [16, 10], [32, 2]], 'representations': {'PC': {'code': 40550065741427112744954556055559, 'gens': [1, 2, 3, 5], 'pres': [8, -2, -2, -2, -2, -2, -2, -3, -5, 5914, 66, 4980, 116, 141, 222]}, 'GLZN': {'d': 2, 'p': 88, 'gens': [30666285, 31010893, 684443, 6133257, 45664499, 685345, 49875609, 44295745]}, 'Perm': {'d': 22, 'gens': [51212944467962686080, 107047691251113271680, 4987804804190208000, 107048048350191742080, 10080, 96, 4302991486080, 107047688359772160000]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 30], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2:Q_8\\times C_{30}', 'transitive_degree': 480, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '60.13', '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, 12], 'aut_gens': [[1, 2], [30, 33], [1, 22], [30, 15]], 'aut_group': '48.35', 'aut_hash': 35, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 48, 'aut_permdeg': 9, 'aut_perms': [1, 24, 46107], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [3, 1, 2, 1], [5, 1, 4, 1], [6, 1, 6, 1], [10, 1, 12, 1], [15, 1, 8, 1], [30, 1, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '48.35', 'autcent_hash': 35, 'autcent_nilpotent': False, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4\\times D_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, 3], [3, 1, 2], [5, 1, 4], [6, 1, 6], [10, 1, 12], [15, 1, 8], [30, 1, 24]], 'center_label': '60.13', 'center_order': 60, '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', '3.1', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 13, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['3.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [5, 1, 4, 1], [6, 1, 2, 3], [10, 1, 4, 3], [15, 1, 8, 1], [30, 1, 8, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 24, 'exponent': 30, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '60.13', 'hash': 13, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 60]], 'label': '60.13', 'linC_count': 576, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 6, 'linQ_dim': 6, 'linQ_dim_count': 6, 'linR_count': 24, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C30', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 60, 'number_divisions': 16, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 20, 'number_subgroups': 20, 'old_label': None, 'order': 60, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 3], [3, 2], [5, 4], [6, 6], [10, 12], [15, 8], [30, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 12], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[30, 33], [1, 22], [30, 15]], 'outer_group': '48.35', 'outer_hash': 35, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 9, 'outer_perms': [1, 24, 46107], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4\\times D_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 4], [8, 4]], 'representations': {'PC': {'code': 8521223, 'gens': [1, 2], 'pres': [4, -2, -2, -3, -5, 21, 46]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [24992906222183252, 58415888370979138]}, 'GLFp': {'d': 2, 'p': 31, 'gens': [178762, 893760]}, 'Perm': {'d': 12, 'gens': [39916800, 362880, 10080, 96]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 30], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_{30}', 'transitive_degree': 60, 'wreath_data': None, 'wreath_product': False}
