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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1344.8525', 'ambient_counter': 8525, 'ambient_order': 1344, 'ambient_tex': 'D_{56}.D_6', 'central': False, 'central_factor': False, 'centralizer_order': 336, 'characteristic': True, 'core_order': 8, 'counter': 203, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1344.8525.168.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '168.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '168.50', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 50, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 168, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'S_3\\times D_{14}', 'simple': False, 'solvable': True, 'special_labels': ['U1'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '8.2', 'subgroup_hash': 2, 'subgroup_order': 8, 'subgroup_tex': 'C_2\\times C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.8525', 'aut_centralizer_order': None, 'aut_label': '168.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '4.d1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['24.a1', '56.a1', '84.a1', '84.c1', '84.d1', '84.e1', '84.f1'], 'contains': ['336.a1', '336.b1', '336.c1'], 'core': '168.a1', 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [672, 24], 'label': '1344.8525.168.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '168.a1', 'normal_contained_in': ['24.a1', '56.a1', '84.a1'], 'normal_contains': ['336.a1', '336.b1', '336.c1'], 'normalizer': '1.a1', 'old_label': '168.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '168.a1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '8.2', '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], 'aut_gens': [[1, 2], [1, 3], [5, 3]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [1, 22], '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]], 'aut_supersolvable': True, 'aut_tex': 'D_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': 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]], 'center_label': '8.2', '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': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 2, '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, 8]], 'label': '8.2', 'linC_count': 12, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 4, 'linQ_dim': 3, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C4', 'ngens': 2, '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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 8, 'number_divisions': 6, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 4]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 3], [5, 3]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [1, 22], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2]], 'representations': {'PC': {'code': 34, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16426]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [251, 504]}, 'Perm': {'d': 6, 'gens': [22, 120, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], '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_4', 'transitive_degree': 8, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 168, 'aut_gen_orders': [12, 14, 24, 12, 12, 6, 24, 24], 'aut_gens': [[1, 2, 4, 96], [217, 58, 4, 912], [769, 18, 44, 96], [541, 86, 52, 1056], [385, 714, 20, 1056], [409, 42, 20, 912], [385, 82, 68, 864], [565, 694, 20, 480], [133, 30, 68, 336]], 'aut_group': None, 'aut_hash': 3795850347135509252, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 64512, 'aut_permdeg': 104, 'aut_perms': [934652821525523799032621486596013198225506309690550686060218346348127973654279102156880651784935513203561789444685859470647429082619602368917718940414486496505266512, 734634206253474648014723412283299012083049893952639191105041638655717524289023171096393361603284285565411994452431945055428870412549139525689469119172471414139499035, 9071075179168653911063566219494854054455489018144374894545490128371737507177240991802104502165882934011715064085704265026923332012130066024531651931128776779380661453, 414493737417758442259859792280762432927267188321879708118359774511176284764653252232702604280452896265243686183554947978530796589604643653718238237362340175127412099, 1182715582985287740450713634385080172740365044607150336122613522028111523122460756160166513449225364125347475115849374539046525118469576826942809958981056009091353580, 445211133586153430442457687634178090202343091580442744966740988059364845157124412762117243207802004614305562419868223073368272299535097097238788867991742341122792896, 9440353117423416861021743256726612768389529694667793983744706363186447019400851677982924009491389075439043073130513691968057759582975647390274207475619486912364794125, 4152817559221255976543958694591104782651349076640009288175049678997941357111253861513080642842444034588000638671978139116372327887804048518909056936351024061374167713], 'aut_phi_ratio': 168.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 28, 2, 1], [2, 42, 2, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 12, 4, 1], [4, 28, 2, 1], [4, 42, 2, 1], [6, 2, 1, 1], [6, 4, 1, 1], [6, 56, 2, 1], [7, 2, 3, 1], [8, 2, 2, 1], [8, 4, 1, 1], [8, 84, 2, 1], [12, 2, 2, 1], [12, 4, 1, 1], [12, 56, 2, 1], [14, 2, 3, 1], [14, 2, 6, 1], [21, 4, 3, 1], [24, 4, 2, 2], [28, 4, 3, 2], [28, 24, 12, 1], [42, 4, 3, 1], [42, 4, 6, 1], [56, 4, 6, 2], [84, 4, 6, 2], [168, 4, 12, 2]], 'aut_supersolvable': True, 'aut_tex': '(C_2\\times C_{84}).C_6.C_2^6', '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': None, 'autcentquo_hash': 866793471638122205, 'autcentquo_nilpotent': False, 'autcentquo_order': 4032, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': '(C_6\\times F_7).C_2^4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 28, 2], [2, 42, 2], [3, 2, 1], [4, 2, 2], [4, 12, 4], [4, 28, 2], [4, 42, 2], [6, 2, 1], [6, 4, 1], [6, 56, 2], [7, 2, 3], [8, 2, 2], [8, 4, 1], [8, 84, 2], [12, 2, 2], [12, 4, 1], [12, 56, 2], [14, 2, 9], [21, 4, 3], [24, 4, 4], [28, 4, 6], [28, 24, 12], [42, 4, 9], [56, 4, 12], [84, 4, 12], [168, 4, 24]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '672.1142', 'commutator_count': 1, 'commutator_label': '84.6', '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', '7.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 8525, '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, 1], [2, 28, 1, 2], [2, 42, 1, 2], [3, 2, 1, 1], [4, 2, 1, 2], [4, 12, 1, 4], [4, 28, 1, 2], [4, 42, 1, 2], [6, 2, 1, 1], [6, 4, 1, 1], [6, 56, 1, 2], [7, 2, 3, 1], [8, 2, 2, 1], [8, 4, 1, 1], [8, 84, 1, 2], [12, 2, 2, 1], [12, 4, 1, 1], [12, 56, 1, 2], [14, 2, 3, 1], [14, 2, 6, 1], [21, 4, 3, 1], [24, 4, 2, 2], [28, 4, 3, 2], [28, 24, 3, 4], [42, 4, 3, 1], [42, 4, 6, 1], [56, 4, 6, 2], [84, 4, 6, 2], [168, 4, 12, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 14938560, 'exponent': 168, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[4, 0, 24]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '336.219', 'hash': 8525, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 2, 12, 14], 'inner_gens': [[1, 2, 76, 1296], [1, 2, 92, 96], [25, 10, 4, 96], [241, 2, 4, 96]], 'inner_hash': 1142, 'inner_nilpotent': False, 'inner_order': 672, 'inner_split': False, 'inner_tex': 'D_{12}:D_{14}', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 4, 'irrQ_degree': 96, 'irrQ_dim': 96, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 36], [4, 74]], 'label': '1344.8525', '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': 'D56.D6', 'ngens': 8, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 37, 'number_characteristic_subgroups': 50, 'number_conjugacy_classes': 126, 'number_divisions': 50, 'number_normal_subgroups': 128, 'number_subgroup_autclasses': 240, 'number_subgroup_classes': 496, 'number_subgroups': 4020, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 143], [3, 2], [4, 192], [6, 118], [7, 6], [8, 176], [12, 120], [14, 18], [21, 12], [24, 16], [28, 312], [42, 36], [56, 48], [84, 48], [168, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 2, 6], 'outer_gen_pows': [0, 0, 0, 672, 756], 'outer_gens': [[1, 2, 68, 96], [1, 2, 76, 96], [1, 2, 76, 144], [25, 674, 4, 96], [685, 686, 4, 1056]], 'outer_group': '96.231', 'outer_hash': 231, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 13, 'outer_perms': [41064, 744, 24, 479001600, 3628803], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4\\times C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 42, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 2], [6, 8], [8, 1], [12, 6], [16, 1], [24, 2], [48, 1], [96, 1]], 'representations': {'PC': {'code': 28107751486745081963795891475843132904563924571587878441, 'gens': [1, 2, 3, 7], 'pres': [8, 2, 2, 2, 2, 2, 3, 2, 7, 393, 1826, 1114, 66, 1795, 1419, 91, 1612, 116, 1549, 72582, 166, 73735]}, 'Perm': {'d': 42, 'gens': [816438677858699394067303498450164068538106826776, 8701402215898541265430894655720055625, 26606506113808516086687099890417507, 35105502784793169281575077356067807, 43068376768692878804566590895678317, 51832097802223708870126852512026509, 303916116658416027343136804044800000000, 35085403596887862568986022075080631753113600000000]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{56}.D_6', 'transitive_degree': 672, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 42, 'aut_gen_orders': [2, 2, 6, 42], 'aut_gens': [[1, 2, 4], [1, 86, 4], [1, 2, 116], [1, 2, 68], [29, 38, 4]], 'aut_group': '1008.906', 'aut_hash': 906, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1008, 'aut_permdeg': 14, 'aut_perms': [7, 120, 2405934840, 14530119856], 'aut_phi_ratio': 21.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [2, 7, 2, 1], [2, 21, 2, 1], [3, 2, 1, 1], [6, 2, 1, 1], [6, 14, 2, 1], [7, 2, 3, 1], [14, 2, 3, 1], [14, 6, 6, 1], [21, 4, 3, 1], [42, 4, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_6\\times F_7', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '252.26', 'autcentquo_hash': 26, 'autcentquo_nilpotent': False, 'autcentquo_order': 252, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [2, 7, 2], [2, 21, 2], [3, 2, 1], [6, 2, 1], [6, 14, 2], [7, 2, 3], [14, 2, 3], [14, 6, 6], [21, 4, 3], [42, 4, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '84.8', 'commutator_count': 1, 'commutator_label': '21.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 50, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['14.1', 1], ['2.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [2, 7, 1, 2], [2, 21, 1, 2], [3, 2, 1, 1], [6, 2, 1, 1], [6, 14, 1, 2], [7, 2, 3, 1], [14, 2, 3, 1], [14, 6, 3, 2], [21, 4, 3, 1], [42, 4, 3, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1344, 'exponent': 42, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[4, 1, 3]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '168.50', 'hash': 50, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [2, 2, 21], 'inner_gens': [[1, 2, 116], [1, 2, 52], [57, 122, 4]], 'inner_hash': 8, 'inner_nilpotent': False, 'inner_order': 84, 'inner_split': True, 'inner_tex': 'S_3\\times D_7', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 16], [4, 6]], 'label': '168.50', 'linC_count': 39, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 12, 'linQ_dim': 8, 'linQ_dim_count': 12, 'linR_count': 39, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'S3*D14', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 13, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 30, 'number_divisions': 18, 'number_normal_subgroups': 28, 'number_subgroup_autclasses': 40, 'number_subgroup_classes': 64, 'number_subgroups': 336, 'old_label': None, 'order': 168, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 63], [3, 2], [6, 30], [7, 6], [14, 42], [21, 12], [42, 12]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[85, 86, 4], [1, 86, 124]], 'outer_group': '12.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [720, 28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [6, 4], [12, 2]], 'representations': {'PC': {'code': 14390349381887488258187, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -3, -7, 1742, 397, 42, 1283, 1048, 78, 1809]}, 'GLZN': {'d': 2, 'p': 14, 'gens': [2773, 2843, 35685, 2757, 4222]}, 'Perm': {'d': 12, 'gens': [3669840, 7, 1, 30, 47255760]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times D_{14}', 'transitive_degree': 42, 'wreath_data': None, 'wreath_product': False}
