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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '448.659', 'ambient_counter': 659, 'ambient_order': 448, 'ambient_tex': 'C_{56}.(C_2\\times C_4)', 'central': False, 'central_factor': False, 'centralizer_order': 224, 'characteristic': True, 'core_order': 56, 'counter': 26, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '448.659.8.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '8.b1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '8.5', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 5, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^3', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '56.8', 'subgroup_hash': 8, 'subgroup_order': 56, 'subgroup_tex': 'C_2\\times C_{28}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '448.659', 'aut_centralizer_order': 896, 'aut_label': '8.b1', 'aut_quo_index': 21, 'aut_stab_index': 1, 'aut_weyl_group': '24.15', 'aut_weyl_index': 896, 'centralizer': '2.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '4.b1.a1', '4.b1.b1', '4.c1.a1', '4.c1.b1', '4.c1.c1', '4.c1.d1'], 'contains': ['16.a1.a1', '16.c1.a1', '16.e1.a1', '56.b1.a1'], 'core': '8.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4335, 5516, 4811, 4095, 4334, 4776, 4601, 3056], 'generators': [2, 240, 224, 56], 'label': '448.659.8.b1.a1', 'mobius_quo': 0, 'mobius_sub': -8, 'normal_closure': '8.b1.a1', 'normal_contained_in': ['4.a1.a1', '4.b1.b1', '4.b1.a1', '4.c1.a1', '4.c1.b1', '4.c1.d1', '4.c1.c1'], 'normal_contains': ['16.a1.a1', '16.c1.a1', '16.e1.a1', '56.b1.a1'], 'normalizer': '1.a1.a1', 'old_label': '8.b1.a1', 'projective_image': '112.29', 'quotient_action_image': '2.1', 'quotient_action_kernel': '4.2', 'quotient_action_kernel_order': 4, 'quotient_fusion': None, 'short_label': '8.b1.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': '56.8', '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], [1, 26], [29, 2], [29, 47]], 'aut_group': '48.45', 'aut_hash': 45, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 48, 'aut_permdeg': 9, 'aut_perms': [24, 10824, 120964], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 1, 4, 1], [7, 1, 6, 1], [14, 1, 6, 1], [14, 1, 12, 1], [28, 1, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_6\\times D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '48.45', 'autcent_hash': 45, 'autcent_nilpotent': True, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_6\\times D_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, 3], [4, 1, 4], [7, 1, 6], [14, 1, 18], [28, 1, 24]], 'center_label': '56.8', 'center_order': 56, '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', '7.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 8, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['4.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2], [7, 1, 6, 1], [14, 1, 6, 3], [28, 1, 12, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 24, 'exponent': 28, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '28.4', 'hash': 8, '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, 56]], 'label': '56.8', 'linC_count': 576, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 4, 'linQ_dim': 8, 'linQ_dim_count': 4, 'linR_count': 24, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C28', 'ngens': 4, 'nilpotency_class': 1, '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': 56, 'number_divisions': 12, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 16, 'number_subgroups': 16, 'old_label': None, 'order': 56, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [4, 4], [7, 6], [14, 18], [28, 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': [[1, 26], [29, 2], [29, 47]], 'outer_group': '48.45', 'outer_hash': 45, 'outer_nilpotent': True, 'outer_order': 48, 'outer_permdeg': 9, 'outer_perms': [24, 10824, 120964], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6\\times D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 7], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [6, 4], [12, 2]], 'representations': {'PC': {'code': 41811557, 'gens': [1, 2], 'pres': [4, -2, -2, -2, -7, 21, 34]}, 'GLFp': {'d': 2, 'p': 29, 'gens': [243896, 682920]}, 'Perm': {'d': 13, 'gens': [11612160, 479001600, 4320, 3669120]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 28], '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_{28}', 'transitive_degree': 56, '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': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 168, 'aut_gen_orders': [28, 24, 24, 24, 6, 12], 'aut_gens': [[1, 4, 112], [211, 230, 112], [223, 302, 394], [53, 438, 170], [223, 318, 394], [153, 190, 112], [151, 148, 394]], 'aut_group': None, 'aut_hash': 1947352312369200806, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 21504, 'aut_permdeg': 224, 'aut_perms': [5245381074930004935410139745775154681956547247322486555645266250915006421033109210562891372960436191853722408627714861446062490185396720433337371096186208639640536229825811858605990020304143833419720934946180900610666583449667989985266812143520026123698085065027945881340446512144550528415111876519905330946204404493081649447159617639466915061912514979100916977436282315672187620248431396419030307773373194891320866609639220742246, 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44242059854152824433978139593008282497967342606331670307459485796757219283202733713042842054920118553100354162443474764633247764517724497363852953120952459520162166407758042553009174227622282311004615659659577295607373720288355224805556853909512364309423390374218847574582332809794751545254468715831086874219279370680906435339882537575305904108984289904091012703343708447799138561772711906826693872981016296633024020092190249461040, 25463398768056602855652002912910465508528817883166978311766138824279896837172639463549832034661152954575635423972489293789809830865053248291389408776736973092519749823305403718744162610089980497255920130911717110236520817843333804331199944292139025973412712856132344323619782180092168933590878285148386964191733544306791260123922735917716956603953435957709342330676090396884092760779156594437203201750503724016373326861977656732010, 36449858973116064843935183075568559433681966230551892394448992075183357502297362429361316537803778913108391709603394194748984314306605553116748272054864579272044844697648353344567862009517749982693380715705949336010423291972134452155018209324169833170872195782027137077885764186028617643709222166660628273233324757578649713509290314436022445191893170954943004096709040155596134211385797031729239313426618286285340823871815101732351], 'aut_phi_ratio': 112.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 2, 2, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 2, 2, 1], [7, 2, 3, 1], [8, 4, 4, 1], [8, 28, 8, 1], [14, 2, 3, 1], [14, 2, 6, 1], [14, 4, 6, 1], [28, 2, 6, 2], [28, 4, 6, 1], [56, 4, 24, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_{14}\\times D_4).C_6.C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 84, 'autcentquo_group': '1344.6612', 'autcentquo_hash': 6612, 'autcentquo_nilpotent': False, 'autcentquo_order': 1344, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\wr C_2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 3], [4, 1, 2], [4, 2, 3], [7, 2, 3], [8, 4, 4], [8, 28, 8], [14, 2, 9], [14, 4, 6], [28, 2, 12], [28, 4, 6], [56, 4, 24]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, 'central_quotient': '112.29', 'commutator_count': 1, 'commutator_label': '28.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 659, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [4, 1, 2, 1], [4, 2, 1, 3], [7, 2, 3, 1], [8, 4, 1, 4], [8, 28, 2, 4], [14, 2, 3, 1], [14, 2, 6, 1], [14, 4, 3, 2], [28, 2, 6, 2], [28, 4, 3, 2], [56, 4, 6, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1344, 'exponent': 56, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [[4, 0, 12]], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '56.12', 'hash': 659, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 28, 'inner_gen_orders': [2, 28, 2], 'inner_gens': [[1, 108, 112], [233, 4, 336], [1, 228, 112]], 'inner_hash': 29, 'inner_nilpotent': False, 'inner_order': 112, 'inner_split': False, 'inner_tex': 'C_2\\times D_{28}', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 48, 'irrQ_dim': 48, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 52], [4, 14]], 'label': '448.659', 'linC_count': 12, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 8, 'linQ_dim': 14, 'linQ_dim_count': 4, 'linR_count': 6, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C56.(C2*C4)', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 17, 'number_characteristic_subgroups': 21, 'number_conjugacy_classes': 82, 'number_divisions': 30, 'number_normal_subgroups': 67, 'number_subgroup_autclasses': 54, 'number_subgroup_classes': 102, 'number_subgroups': 292, 'old_label': None, 'order': 448, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 7], [4, 8], [7, 6], [8, 240], [14, 42], [28, 48], [56, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 2, 12], 'outer_gen_pows': [112, 0, 0, 0, 56], 'outer_gens': [[1, 116, 112], [1, 284, 336], [3, 4, 112], [339, 286, 112], [141, 188, 170]], 'outer_group': '192.1410', 'outer_hash': 1410, 'outer_nilpotent': True, 'outer_order': 192, 'outer_permdeg': 13, 'outer_perms': [83502840, 40320, 83502720, 2424, 1437006268], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5:C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 23, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [6, 8], [8, 1], [12, 4], [48, 1]], 'representations': {'PC': {'code': 28596804166799201627270647134501755700, 'gens': [1, 3, 6], 'pres': [7, -2, -2, -2, -2, -7, -2, -2, 14, 1576, 2270, 58, 9187, 80, 11204, 522, 3547, 124]}, 'Perm': {'d': 23, 'gens': [1243880575127355513727, 19591789436470056960, 2511781832821677252480, 3700931334296409601920, 4826897590801599573120, 5997609414251742833280, 973]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{56}.(C_2\\times C_4)', 'transitive_degree': 112, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 84, 'aut_gen_orders': [3, 3], 'aut_gens': [[1, 2, 4], [4, 5, 3], [2, 4, 1]], 'aut_group': '168.42', 'aut_hash': 42, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 168, 'aut_permdeg': 7, 'aut_perms': [4361, 244], 'aut_phi_ratio': 42.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1]], 'aut_supersolvable': False, 'aut_tex': '\\PSL(2,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '168.42', 'autcent_hash': 42, 'autcent_nilpotent': False, 'autcent_order': 168, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\PSL(2,7)', '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]], 'center_label': '8.5', 'center_order': 8, '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'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 3]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '8.5', 'hash': 5, '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, 8]], 'label': '8.5', 'linC_count': 28, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 28, 'linQ_dim': 3, 'linQ_dim_count': 28, 'linR_count': 28, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 8, 'number_divisions': 8, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 16, 'number_subgroups': 16, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 84, 'outer_gen_orders': [3, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[4, 5, 3], [2, 4, 1]], 'outer_group': '168.42', 'outer_hash': 42, 'outer_nilpotent': False, 'outer_order': 168, 'outer_permdeg': 7, 'outer_perms': [4361, 244], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\PSL(2,7)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3], 'pres': [3, -2, 2, 2]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [16482, 16322, 3362]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [8156, 13286, 13933]}, 'Perm': {'d': 6, 'gens': [120, 6, 1]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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_2^3', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}
