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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '360.113', 'ambient_counter': 113, 'ambient_order': 360, 'ambient_tex': 'C_6.D_{30}', 'central': True, 'central_factor': False, 'centralizer_order': 360, 'characteristic': False, 'core_order': 2, 'counter': 93, 'cyclic': True, 'direct': True, 'hall': 0, 'label': '360.113.180.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '180.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '180.17', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 17, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 180, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_6.D_{15}', 'simple': True, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '360.113', 'aut_centralizer_order': 34560, 'aut_label': '180.a1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '1.1', 'aut_weyl_index': 69120, 'centralizer': '1.a1.a1', 'complements': ['2.b1.a1', '2.b1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['36.a1.a1', '60.a1.a1', '60.a1.b1', '60.a1.c1', '60.a1.d1', '90.a1.a1'], 'contains': ['360.a1.a1'], 'core': '180.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5245, 3560, 3498, 7376, 6003, 8081, 2610, 6031], 'generators': [12], 'label': '360.113.180.a1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '180.a1.a1', 'normal_contained_in': ['36.a1.a1', '60.a1.d1', '60.a1.c1', '60.a1.a1', '60.a1.b1', '90.a1.a1'], 'normal_contains': ['360.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '180.a1.a1', 'projective_image': '180.17', 'quotient_action_image': '1.1', 'quotient_action_kernel': '180.17', 'quotient_action_kernel_order': 180, 'quotient_fusion': None, 'short_label': '180.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', '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]], 'center_label': '2.1', 'center_order': 2, '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'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', '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': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, '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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 120, 'aut_gen_orders': [8, 12, 8, 8, 8], 'aut_gens': [[1, 4, 24], [23, 262, 344], [109, 4, 168], [199, 246, 320], [135, 140, 64], [235, 140, 224]], 'aut_group': None, 'aut_hash': 1973075154544504877, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 69120, 'aut_permdeg': 180, 'aut_perms': [122432953420446127126577931336419880374228178076345184022460353319286849044159401365579502855742354024817394682461425583407397523689213245616989729811389479265659411687688209072669301728912640182054070404693452113502126339280040409922869745612848170275525898698543923549084730719722345075212728177585948063031174103534189339874606, 47503152666027134083656619809839196401894444240629512868457237933954643991316470984396532610532966597026467417492073015833976976565496142467550278910980052368182435698914237978235068108559028185573474430920152565957395145517921491661586500780241384006631292622781624736516467272446517696424588899450520661439068354435143264880224, 49705556478742988215578820323585267783091512926072611470767392779864100104139600814021655081734974773952058129096445522149711885444059841043803855176360475920953485832479251431134397706362687352471479071482231682774850165495033150865640099036996026631902743300164659210437346121938932473924656932106561701998951284781908935508866, 51695148980001289120915940712145241586270651775477868690501376353233077089051276993892457339502409657944493140104618317409231965822213162689769863992314083965759846969380816143827583165104827518908030094823873400261319079125125141469771381072935617582224357504379013820891018465775496290410463531169564809427615508285514082265897, 53118327011780929108639050928069194949040657714654147913745154593984458928207985333539080846432611514797130751869150948976717820171746846779937565097920140725738698892237729440023751043898897588780008542944126447907066907246286461260813811517942629369909078356262495868421248513445370952363509267350341303437294131166079753270522], 'aut_phi_ratio': 720.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [3, 2, 4, 1], [4, 45, 4, 1], [5, 2, 2, 1], [6, 2, 4, 1], [6, 2, 8, 1], [10, 2, 2, 1], [10, 2, 4, 1], [15, 2, 16, 1], [30, 2, 16, 1], [30, 2, 32, 1]], 'aut_supersolvable': False, 'aut_tex': '\\PSU(3,2).C_{30}.C_2.C_2^4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.3', 'autcent_hash': 3, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': '8640.be', 'autcentquo_hash': 9203803121985669655, 'autcentquo_nilpotent': False, 'autcentquo_order': 8640, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\times C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 3], [3, 2, 4], [4, 45, 4], [5, 2, 2], [6, 2, 12], [10, 2, 6], [15, 2, 16], [30, 2, 48]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '90.9', 'commutator_count': 1, 'commutator_label': '45.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 113, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['180.17', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 4], [4, 45, 2, 2], [5, 2, 2, 1], [6, 2, 1, 12], [10, 2, 2, 3], [15, 2, 4, 4], [30, 2, 4, 12]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 252, 'exponent': 60, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '180.36', 'hash': 113, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [2, 3, 15], 'inner_gens': [[1, 20, 336], [9, 4, 24], [49, 4, 24]], 'inner_hash': 9, 'inner_nilpotent': False, 'inner_order': 90, 'inner_split': False, 'inner_tex': 'C_3:D_{15}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 88]], 'label': '360.113', 'linC_count': 864, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 252, 'linQ_dim': 10, 'linQ_dim_count': 156, 'linR_count': 1440, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6.D30', 'ngens': 6, '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': 13, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 96, 'number_divisions': 42, 'number_normal_subgroups': 63, 'number_subgroup_autclasses': 36, 'number_subgroup_classes': 96, 'number_subgroups': 456, 'old_label': None, 'order': 360, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 3], [3, 8], [4, 180], [5, 4], [6, 24], [10, 12], [15, 32], [30, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [8, 4, 8, 8, 8], 'outer_gen_pows': [0, 120, 0, 0, 0], 'outer_gens': [[23, 262, 344], [109, 4, 168], [199, 246, 320], [135, 140, 64], [235, 140, 224]], 'outer_group': '768.1089283', 'outer_hash': 1089283, 'outer_nilpotent': False, 'outer_order': 768, 'outer_permdeg': 64, 'outer_perms': [2144578431742403566751700000372919374174829596536214789000002434370441963856039399583997, 4159221997540845999976985870722610121041988028354495057987581679735726950198759348480487, 6173741984909934206875180213727397559960911367613973927072272000163001510582322941148379, 8188402904907704526295088709767065298469709620172395562923670793158113009158400383735099, 10202997235859740323348102867446174302550297047709077584287449888111283713493916034444233], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'Q_8.(S_3\\times D_4).C_2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 18], [4, 4], [8, 16]], 'representations': {'PC': {'code': 28282445151010720206114839, 'gens': [1, 3, 5], 'pres': [6, -2, -2, -2, -3, -3, -5, 12, 362, 50, 387, 10084, 118, 10373]}, 'GLZN': {'d': 2, 'p': 30, 'gens': [27301, 40966, 297011, 67618, 27181, 513019]}, 'Perm': {'d': 17, 'gens': [1313905063681, 90744, 90720, 43545603, 4, 23544365644800]}}, 'schur_multiplier': [6], '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_6.D_{30}', 'transitive_degree': 360, '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': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 120, 'aut_gen_orders': [2, 6, 2, 4, 6, 6, 5], 'aut_gens': [[1, 4, 12], [3, 4, 168], [61, 4, 48], [3, 4, 48], [1, 4, 156], [9, 4, 48], [1, 120, 176], [145, 4, 12]], 'aut_group': '17280.fp', 'aut_hash': 7336759155281172877, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 17280, 'aut_permdeg': 16, 'aut_perms': [25517722014, 8371514592414, 415, 169, 3941704892574, 126504000414, 2928], 'aut_phi_ratio': 360.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 4, 1], [4, 45, 2, 1], [5, 2, 2, 1], [6, 2, 4, 1], [10, 2, 2, 1], [15, 2, 16, 1], [30, 2, 16, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{30}:C_{12}:\\GL(2,3)', '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': 120, 'autcentquo_group': '8640.be', 'autcentquo_hash': 9203803121985669655, 'autcentquo_nilpotent': False, 'autcentquo_order': 8640, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\times C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 2, 4], [4, 45, 2], [5, 2, 2], [6, 2, 4], [10, 2, 2], [15, 2, 16], [30, 2, 16]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '90.9', 'commutator_count': 1, 'commutator_label': '45.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 17, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 4], [4, 45, 2, 1], [5, 2, 2, 1], [6, 2, 1, 4], [10, 2, 2, 1], [15, 2, 4, 4], [30, 2, 4, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 168, 'exponent': 60, 'exponents_of_order': [2, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '90.9', 'hash': 17, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [2, 3, 15], 'inner_gens': [[1, 8, 168], [9, 4, 12], [25, 4, 12]], 'inner_hash': 9, 'inner_nilpotent': False, 'inner_order': 90, 'inner_split': False, 'inner_tex': 'C_3:D_{15}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4], [2, 44]], 'label': '180.17', 'linC_count': 432, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 42, 'linQ_dim': 10, 'linQ_dim_count': 18, 'linR_count': 432, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6.D15', 'ngens': 5, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 9, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 48, 'number_divisions': 21, 'number_normal_subgroups': 25, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 36, 'number_subgroups': 156, 'old_label': None, 'order': 180, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 1], [3, 8], [4, 90], [5, 4], [6, 8], [10, 4], [15, 32], [30, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 6, 4, 2, 4], 'outer_gen_pows': [0, 0, 0, 1, 0], 'outer_gens': [[3, 8, 48], [3, 128, 132], [1, 124, 136], [3, 68, 164], [1, 64, 140]], 'outer_group': '192.1480', 'outer_hash': 1480, 'outer_nilpotent': False, 'outer_order': 192, 'outer_permdeg': 18, 'outer_perms': [359903136438864, 4278468223105231, 3026838283310190, 4914020596729320, 1578989323193088], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,3):C_2^2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 9], [4, 2], [8, 8]], 'representations': {'PC': {'code': 53724910819976215, 'gens': [1, 3, 4], 'pres': [5, -2, -2, -3, -3, -5, 10, 122, 3363, 78, 3604]}, 'GLZN': {'d': 2, 'p': 30, 'gens': [27301, 40966, 203123, 27181, 513019]}, 'Perm': {'d': 15, 'gens': [6267345865, 806400, 3, 148, 99672249600]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6.D_{15}', 'transitive_degree': 180, 'wreath_data': None, 'wreath_product': False}
