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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1200.1038', 'ambient_counter': 1038, 'ambient_order': 1200, 'ambient_tex': 'C_{10}^2:D_6', 'central': True, 'central_factor': False, 'centralizer_order': 1200, 'characteristic': False, 'core_order': 20, 'counter': 44, 'cyclic': False, 'direct': True, 'hall': 0, 'label': '1200.1038.60.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '60.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '60.12', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 12, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 60, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_{30}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '20.5', 'subgroup_hash': 5, 'subgroup_order': 20, 'subgroup_tex': 'C_2\\times C_{10}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1200.1038', 'aut_centralizer_order': 3840, 'aut_label': '60.a1', 'aut_quo_index': 1, 'aut_stab_index': 7, 'aut_weyl_group': '24.5', 'aut_weyl_index': 26880, 'centralizer': '1.a1', 'complements': ['20.a1'], 'conjugacy_class_count': 7, 'contained_in': ['12.a1', '20.b1', '30.a1', '30.c1'], 'contains': ['120.a1', '300.a1'], 'core': '60.a1', 'coset_action_label': None, 'count': 7, 'diagramx': [5122, 6406, 4363, 7228], 'generators': [20, 8, 600], 'label': '1200.1038.60.a1', 'mobius_quo': -2, 'mobius_sub': 30, 'normal_closure': '60.a1', 'normal_contained_in': ['12.a1', '20.b1', '30.a1'], 'normal_contains': ['120.a1', '300.a1'], 'normalizer': '1.a1', 'old_label': '60.a1', 'projective_image': '60.12', 'quotient_action_image': '1.1', 'quotient_action_kernel': '60.12', 'quotient_action_kernel_order': 60, 'quotient_fusion': None, 'short_label': '60.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '20.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 12], 'aut_gens': [[1, 2], [10, 13], [10, 15]], 'aut_group': '24.5', 'aut_hash': 5, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24, 'aut_permdeg': 7, 'aut_perms': [127, 857], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [5, 1, 4, 1], [10, 1, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4\\times S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.5', 'autcent_hash': 5, 'autcent_nilpotent': False, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4\\times S_3', '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], [5, 1, 4], [10, 1, 12]], 'center_label': '20.5', 'center_order': 20, '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', '5.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [5, 1, 4, 1], [10, 1, 4, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 6, 'exponent': 10, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '20.5', 'hash': 5, '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, 20]], 'label': '20.5', 'linC_count': 72, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 6, 'linQ_dim': 5, 'linQ_dim_count': 6, 'linR_count': 12, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C10', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 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': 20, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 20, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [5, 4], [10, 12]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 12], 'outer_gen_pows': [0, 0], 'outer_gens': [[10, 13], [10, 15]], 'outer_group': '24.5', 'outer_hash': 5, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 7, 'outer_perms': [127, 857], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4\\times S_3', '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], [4, 4]], 'representations': {'PC': {'code': 2179, 'gens': [1, 2], 'pres': [3, -2, -2, -5, 16]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [141602720900, 706303489327]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [13320, 4001]}, 'Perm': {'d': 9, 'gens': [40320, 720, 96]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 10], '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_{10}', 'transitive_degree': 20, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '80.52', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 420, 'aut_gen_orders': [12, 60, 14, 20], 'aut_gens': [[1, 2, 4, 40], [20, 603, 632, 81], [620, 902, 12, 61], [600, 162, 25, 581], [601, 723, 4, 460]], 'aut_group': None, 'aut_hash': 6303684600578070319, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 645120, 'aut_permdeg': 124, 'aut_perms': [49552358874655333975904882779028105196038244422852280079573935493269201325608815542677275410307235468907001214199586985253188273906857537951994502843697882871355100783258663126847818058776741756837166852702, 847933205457994788273903431030094671739552978647435870368831119477252655873275811628188048941902983564340906625998524941653833182053349637507804269044866635958848881071782008906163514564115220470911234792133, 295511561730690055415813040865561781456623357772427086125856483554361955946503965416195094775172827338094576570332488948671704688893449244846004907593550077649918990736713600212365419431166489189565865132601, 336093381091594903590176216940994136475809141602095073099088614740588981764083728253520449225967837033044797290450411056717954077565633221676059372422475292449990211528556419545535886433540453177389861148155], 'aut_phi_ratio': 2016.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1], [2, 15, 8, 1], [3, 2, 1, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 8, 1], [6, 2, 7, 1], [10, 1, 28, 1], [10, 2, 14, 1], [10, 2, 56, 1], [10, 15, 32, 1], [15, 2, 4, 2], [15, 2, 16, 1], [30, 2, 28, 2], [30, 2, 112, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times S_3\\times C_2^3.\\PSL(2,7)\\times F_5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '5376.bu', 'autcent_hash': 5218989228857211849, 'autcent_nilpotent': False, 'autcent_order': 5376, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_4\\times C_2^3:\\GL(3,2)', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '120.36', 'autcentquo_hash': 36, 'autcentquo_nilpotent': False, 'autcentquo_order': 120, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 15, 8], [3, 2, 1], [5, 1, 4], [5, 2, 10], [6, 2, 7], [10, 1, 28], [10, 2, 70], [10, 15, 32], [15, 2, 24], [30, 2, 168]], 'center_label': '40.14', 'center_order': 40, 'central_product': True, 'central_quotient': '30.3', 'commutator_count': 1, 'commutator_label': '15.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '5.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1038, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['30.3', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 15, 1, 8], [3, 2, 1, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 4, 2], [6, 2, 1, 7], [10, 1, 4, 7], [10, 2, 2, 7], [10, 2, 4, 14], [10, 15, 4, 8], [15, 2, 4, 2], [15, 2, 8, 2], [30, 2, 4, 14], [30, 2, 8, 14]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 943020, 'exponent': 30, 'exponents_of_order': [4, 2, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1200.1038', 'hash': 1038, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [1, 2, 1, 15], 'inner_gens': [[1, 2, 4, 40], [1, 2, 4, 1160], [1, 2, 4, 40], [1, 82, 4, 40]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 30, 'inner_split': True, 'inner_tex': 'D_{15}', 'inner_used': [2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 80], [2, 280]], 'label': '1200.1038', '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': 'C10^2:D6', '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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 18, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 360, 'number_divisions': 96, 'number_normal_subgroups': 230, 'number_subgroup_autclasses': 72, 'number_subgroup_classes': 600, 'number_subgroups': 2704, 'old_label': None, 'order': 1200, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 127], [3, 2], [5, 24], [6, 14], [10, 648], [15, 48], [30, 336]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 84, 'outer_gen_orders': [6, 12, 4], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 623, 605, 660], [600, 3, 37, 281], [621, 622, 608, 1041]], 'outer_group': None, 'outer_hash': 5444498268710950903, 'outer_nilpotent': False, 'outer_order': 21504, 'outer_permdeg': 16, 'outer_perms': [5231739300480, 4192744924942, 194954014097], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '(C_2^3\\times C_4^2).\\PSL(2,7)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 5], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 8], [4, 24], [8, 32], [16, 16]], 'representations': {'PC': {'code': 16456712895320323478889758919, 'gens': [1, 2, 3, 5], 'pres': [7, -2, -2, -2, -5, -2, -3, -5, 58, 20311, 102, 23532, 166, 23533]}, 'GLZN': {'d': 2, 'p': 66, 'gens': [7187425, 287521, 6612431, 433456, 287519, 12663926, 12362371]}, 'Perm': {'d': 19, 'gens': [6227021545, 40279680, 6267300480, 87091200, 6780291598080000, 3, 6504]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 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_{10}^2:D_6', 'transitive_degree': 240, 'wreath_data': None, 'wreath_product': False}
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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': 60, 'aut_gen_orders': [2, 4, 30], 'aut_gens': [[1, 2], [1, 22], [1, 46], [15, 2]], 'aut_group': '240.195', 'aut_hash': 195, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 240, 'aut_permdeg': 10, 'aut_perms': [6, 120967, 857569], 'aut_phi_ratio': 15.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 15, 2, 1], [3, 2, 1, 1], [5, 2, 2, 1], [6, 2, 1, 1], [10, 2, 2, 1], [15, 2, 4, 1], [30, 2, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6\\times F_5', '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': 60, 'autcentquo_group': '120.36', 'autcentquo_hash': 36, 'autcentquo_nilpotent': False, 'autcentquo_order': 120, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 15, 2], [3, 2, 1], [5, 2, 2], [6, 2, 1], [10, 2, 2], [15, 2, 4], [30, 2, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '30.3', 'commutator_count': 1, 'commutator_label': '15.1', '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': 12, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [['2.1', 1], ['30.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 15, 1, 2], [3, 2, 1, 1], [5, 2, 2, 1], [6, 2, 1, 1], [10, 2, 2, 1], [15, 2, 4, 1], [30, 2, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 30, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[2, 1, 4]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '60.12', 'hash': 12, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [2, 15], 'inner_gens': [[1, 58], [5, 2]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 30, 'inner_split': True, 'inner_tex': 'D_{15}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 14]], 'label': '60.12', 'linC_count': 4, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 3, 'linQ_dim': 6, 'linQ_dim_count': 3, 'linR_count': 4, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D30', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, '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': 9, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 18, 'number_divisions': 10, 'number_normal_subgroups': 11, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 20, 'number_subgroups': 80, 'old_label': None, 'order': 60, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 31], [3, 2], [5, 4], [6, 2], [10, 4], [15, 8], [30, 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': [[31, 2], [1, 14]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [4, 2], [8, 2]], 'representations': {'PC': {'code': 5357497892359, 'gens': [1, 2], 'pres': [4, -2, -2, -3, -5, 465, 21, 674, 46, 771]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125124615165345110, 91678742274876794]}, 'GLFp': {'d': 2, 'p': 29, 'gens': [550123, 682893]}, 'Perm': {'d': 10, 'gens': [40327, 720, 403200, 37]}}, '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_{30}', 'transitive_degree': 30, 'wreath_data': None, 'wreath_product': False}
