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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '26244.nl', 'ambient_counter': 350, 'ambient_order': 26244, 'ambient_tex': 'C_9^2.C_3^2.S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': 729, 'characteristic': True, 'core_order': 27, 'counter': 1218, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '26244.nl.972.d1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '972.d1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '972.795', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 795, 'quotient_metabelian': False, 'quotient_nilpotent': False, 'quotient_order': 972, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': '(C_3\\times C_9):S_3^2', 'simple': False, 'solvable': True, 'special_labels': ['C7', 'C7'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '27.2', 'subgroup_hash': 2, 'subgroup_order': 27, 'subgroup_tex': 'C_3\\times C_9', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '26244.nl', 'aut_centralizer_order': None, 'aut_label': '972.d1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '36.c1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['324.e1', '324.h1', '324.fr1', '324.ft1', '324.fv1', '324.fx1', '324.fz1', '324.ga1', '324.gb1', '324.gc1', '324.gd1', '486.l1', '486.cx1', '486.dz1'], 'contains': ['2916.b1', '2916.c1', '2916.k1'], 'core': '972.d1', 'coset_action_label': None, 'count': 1, 'diagramx': [8486, 6337, 7254, 6505], 'generators': [324, 20412], 'label': '26244.nl.972.d1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '972.d1', 'normal_contained_in': ['324.e1', '324.h1'], 'normal_contains': ['2916.b1', '2916.c1'], 'normalizer': '1.a1', 'old_label': '972.d1', 'projective_image': '26244.nl', 'quotient_action_image': '36.12', 'quotient_action_kernel': '27.5', 'quotient_action_kernel_order': 27, 'quotient_fusion': None, 'short_label': '972.d1', 'subgroup_fusion': None, 'weyl_group': '36.12'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '27.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 6, 'aut_gen_orders': [3, 3, 2, 2, 6], 'aut_gens': [[1, 3], [1, 23], [1, 12], [2, 24], [2, 4], [11, 15]], 'aut_group': '108.28', 'aut_hash': 28, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 108, 'aut_permdeg': 11, 'aut_perms': [4556304, 11088384, 1, 169567, 26386807], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 1, 6, 1], [9, 1, 18, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^2:D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '108.28', 'autcent_hash': 28, 'autcent_nilpotent': False, 'autcent_order': 108, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^2: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], [3, 1, 8], [9, 1, 18]], 'center_label': '27.2', 'center_order': 27, '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': ['3.1', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 4], [9, 1, 6, 3]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 4, 'exponent': 9, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '9.2', 'hash': 2, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 3], [1, 3]], '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, 27]], 'label': '27.2', 'linC_count': 216, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 9, 'linQ_dim': 8, 'linQ_dim_count': 9, 'linR_count': 54, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*C9', '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': 27, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 27, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 8], [9, 18]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [3, 3, 2, 2, 6], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 23], [1, 12], [2, 24], [2, 4], [11, 15]], 'outer_group': '108.28', 'outer_hash': 28, 'outer_nilpotent': False, 'outer_order': 108, 'outer_permdeg': 11, 'outer_perms': [4556304, 11088384, 1, 169567, 26386807], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3^2:D_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 12, 'pgroup': 3, 'primary_abelian_invariants': [3, 9], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4], [6, 3]], 'representations': {'PC': {'code': 34, 'gens': [1, 2], 'pres': [3, -3, 3, -3, 22]}, 'GLFp': {'d': 2, 'p': 19, 'gens': [34311, 75456]}, 'Perm': {'d': 12, 'gens': [357120, 79833600, 80884]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 9], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times C_9', 'transitive_degree': 27, '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': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 54, 'aut_gen_orders': [18, 6, 3, 3, 3, 27], 'aut_gens': [[1, 6, 36, 108, 324, 972], [8941, 8994, 180, 108, 9072, 19764], [8761, 8754, 252, 108, 18144, 25920], [73, 114, 36, 108, 324, 10368], [1, 6, 36, 108, 324, 9720], [8749, 18150, 36, 108, 324, 972], [2593, 5190, 360, 17604, 324, 972]], 'aut_group': '26244.nl', 'aut_hash': 1966067676738836325, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 26244, 'aut_permdeg': 108, 'aut_perms': [96707966278880676810054438480132325685179745078899080723262647436044078410908706869908056162828168709991339316740022926153819113343889232954162891842629495512205666489673, 601138744233548867943811170788589668044640093056820679043869440161516862407203280688542720624661603520795831085803835535077631555527374953112563510110117402983773750828554779, 1275696731176136356514093814976942756593090156267705068869262594029199681649083949593174201190233529372432517675429616627046411213946791654568636836783357098224855672101529544, 25211378975253700494990324449461013246517390813247247334285506148874050573830295357755934041700626586156488537108602585871120468624270020835199859345604029458, 270617200972847197826758092349431824413178803027092166418400128016856790575507955186161099467247080997763835421009534878261587347929518947170646432448607957731039261019763464, 246881563294797847015752049810377897223453452260675884081877773400193245731564999210052056378090142607103655760806943846777525818958828766001908124603763310712136124247485458], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 27, 1, 1], [2, 81, 1, 1], [2, 243, 1, 1], [3, 2, 1, 1], [3, 3, 1, 2], [3, 6, 1, 3], [3, 18, 1, 2], [3, 36, 1, 2], [3, 54, 1, 2], [6, 54, 1, 1], [6, 81, 1, 4], [6, 243, 1, 2], [6, 486, 1, 7], [9, 6, 1, 3], [9, 9, 1, 6], [9, 12, 1, 3], [9, 18, 1, 6], [9, 36, 1, 3], [9, 54, 1, 8], [9, 108, 1, 11], [18, 81, 1, 6], [18, 162, 1, 3], [18, 243, 1, 12], [18, 486, 1, 14], [27, 54, 1, 9], [27, 108, 1, 9], [27, 324, 1, 9], [54, 486, 1, 9]], 'aut_supersolvable': True, 'aut_tex': 'C_9^2.C_3^2.S_3^2', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 54, 'autcentquo_group': '26244.nl', 'autcentquo_hash': 1966067676738836325, 'autcentquo_nilpotent': False, 'autcentquo_order': 26244, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_9^2.C_3^2.S_3^2', 'cc_stats': [[1, 1, 1], [2, 27, 1], [2, 81, 1], [2, 243, 1], [3, 2, 1], [3, 3, 2], [3, 6, 3], [3, 18, 2], [3, 36, 2], [3, 54, 2], [6, 54, 1], [6, 81, 4], [6, 243, 2], [6, 486, 7], [9, 6, 3], [9, 9, 6], [9, 12, 3], [9, 18, 6], [9, 36, 3], [9, 54, 8], [9, 108, 11], [18, 81, 6], [18, 162, 3], [18, 243, 12], [18, 486, 14], [27, 54, 9], [27, 108, 9], [27, 324, 9], [54, 486, 9]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '26244.nl', 'commutator_count': 1, 'commutator_label': '2187.4282', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 350, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 27, 1, 1], [2, 81, 1, 1], [2, 243, 1, 1], [3, 2, 1, 1], [3, 3, 2, 1], [3, 6, 1, 1], [3, 6, 2, 1], [3, 18, 1, 2], [3, 36, 1, 2], [3, 54, 1, 2], [6, 54, 1, 1], [6, 81, 2, 2], [6, 243, 2, 1], [6, 486, 1, 5], [6, 486, 2, 1], [9, 6, 1, 1], [9, 6, 2, 1], [9, 9, 6, 1], [9, 12, 1, 1], [9, 12, 2, 1], [9, 18, 6, 1], [9, 36, 1, 1], [9, 36, 2, 1], [9, 54, 2, 4], [9, 108, 1, 3], [9, 108, 2, 4], [18, 81, 6, 1], [18, 162, 1, 1], [18, 162, 2, 1], [18, 243, 6, 2], [18, 486, 2, 4], [18, 486, 6, 1], [27, 54, 1, 1], [27, 54, 2, 1], [27, 54, 6, 1], [27, 108, 1, 1], [27, 108, 2, 1], [27, 108, 6, 1], [27, 324, 1, 3], [27, 324, 2, 3], [54, 486, 1, 1], [54, 486, 2, 1], [54, 486, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 229267584, 'exponent': 54, 'exponents_of_order': [8, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[54, 1, 2], [108, 1, 1]], 'familial': False, 'frattini_label': '81.11', 'frattini_quotient': '324.166', 'hash': 1966067676738836325, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 54, 'inner_gen_orders': [18, 6, 3, 3, 3, 27], 'inner_gens': [[1, 30, 72, 108, 324, 2268], [13, 6, 252, 108, 18144, 17172], [73, 114, 36, 108, 324, 1620], [1, 6, 36, 108, 324, 9720], [1, 9402, 36, 108, 324, 972], [25921, 11022, 360, 17604, 324, 972]], 'inner_hash': 1966067676738836325, 'inner_nilpotent': False, 'inner_order': 26244, 'inner_split': True, 'inner_tex': 'C_9^2.C_3^2.S_3^2', 'inner_used': [1, 2, 3, 5, 6], 'irrC_degree': 54, 'irrQ_degree': 54, 'irrQ_dim': 54, 'irrR_degree': 54, 'irrep_stats': [[1, 12], [2, 30], [3, 24], [4, 12], [6, 32], [12, 22], [18, 4], [36, 2], [54, 2], [108, 1]], 'label': '26244.nl', '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': False, 'metacyclic': False, 'monomial': True, 'name': 'C9^2.C3^2.S3^2', 'ngens': 10, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 141, 'number_characteristic_subgroups': 64, 'number_conjugacy_classes': 141, 'number_divisions': 68, 'number_normal_subgroups': 64, 'number_subgroup_autclasses': 1481, 'number_subgroup_classes': 1481, 'number_subgroups': 42166, 'old_label': None, 'order': 26244, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 351], [3, 242], [6, 4266], [9, 1944], [18, 10692], [27, 4374], [54, 4374]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': True, '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': 6, 'perfect': False, 'permutation_degree': 81, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 14], [4, 14], [6, 4], [8, 4], [12, 8], [18, 4], [24, 5], [36, 6], [54, 2], [72, 2], [108, 1]], 'representations': {'PC': {'code': '305825073163014883826273526050510691833853207277851053586148194618824927523632768715423347326802771964252500079811967', 'gens': [1, 3, 5, 6, 7, 8], 'pres': [10, -2, -3, -2, -3, -3, -3, -3, 3, -3, -3, 20, 1091, 902, 82, 963, 3604, 2124, 634, 211706, 103986, 181447, 155537, 228987, 10837, 3647, 7257, 317, 524888, 524898, 349948, 358, 1749609, 291629]}, 'Perm': {'d': 81, 'gens': [5000805750885579674414839437642421872981416461791358384333512089451649779912307570909461671829402110356660071723501680649, 5512279422367949750937680046822789821584194793978246317730686293813378056073121147573297734742903145995524448021649843943, 4581870275908479512961833203883814240424562572532372070850820828033355969286125849676827488311230268823217452884539402134, 5278031688636638763284884731735354160305178733432149069299329879847211324287029352734554907668644314399924609464876120801, 4939303161648624213318830474425946102426500135575059055587840187450531270721565625215460074512411234613440729903716012277, 3913679520160220097963926740263158529998383604400359325340600486384757243137007690807883985695834829455077262584553732480, 1015008853484147425387036597558201477895788136343551492283680227744880108084211272617628357367901352519220699789303283684, 4081136774549442530690664792623363003975374129587800383415061863170383791437053148881373157992938186145150783904521584568, 860347231724571879966574045127164366432971027451106882133575121396456793947024900959173532384847541632178422906401158546]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_9^2.C_3^2.S_3^2', 'transitive_degree': 81, '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': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [3, 4, 4, 2, 3, 3, 6, 6, 2, 2], 'aut_gens': [[1, 2, 6, 36, 108], [3, 2, 330, 36, 108], [1, 338, 682, 36, 108], [1, 16, 658, 36, 108], [1, 348, 344, 36, 864], [1, 660, 32, 36, 108], [339, 326, 330, 36, 108], [29, 24, 658, 36, 756], [675, 16, 42, 36, 864], [1, 4, 354, 36, 108], [5, 4, 728, 72, 864]], 'aut_group': '7776.jy', 'aut_hash': 6858566785119505195, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 7776, 'aut_permdeg': 18, 'aut_perms': [1320926013395304, 232213416434904, 43502486238144, 4769988440453, 1016050297800, 3643680530672160, 4700328888396749, 591094110453627, 127025516233680, 1679316272318053], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 6, 4, 1], [3, 12, 4, 1], [6, 3, 2, 1], [6, 9, 2, 1], [6, 18, 1, 1], [6, 18, 2, 1], [6, 18, 4, 1], [6, 27, 2, 1], [9, 1, 6, 1], [9, 2, 6, 1], [9, 6, 8, 1], [9, 12, 8, 1], [18, 3, 6, 1], [18, 9, 6, 1], [18, 18, 6, 1], [18, 18, 8, 1], [18, 27, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times C_3^3:\\GL(2,3)', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 3, 'autcent_group': '3.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 3, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '2592.fv', 'autcentquo_hash': 9019849488891309561, 'autcentquo_nilpotent': False, 'autcentquo_order': 2592, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\times C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 9, 1], [2, 27, 1], [3, 1, 2], [3, 2, 3], [3, 6, 4], [3, 12, 4], [6, 3, 2], [6, 9, 2], [6, 18, 7], [6, 27, 2], [9, 1, 6], [9, 2, 6], [9, 6, 8], [9, 12, 8], [18, 3, 6], [18, 9, 6], [18, 18, 14], [18, 27, 6]], 'center_label': '9.1', 'center_order': 9, 'central_product': True, 'central_quotient': '108.39', 'commutator_count': 1, 'commutator_label': '81.12', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 795, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['162.44', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 6, 1, 4], [3, 12, 1, 4], [6, 3, 2, 1], [6, 9, 2, 1], [6, 18, 1, 5], [6, 18, 2, 1], [6, 27, 2, 1], [9, 1, 6, 1], [9, 2, 6, 1], [9, 6, 2, 4], [9, 12, 2, 4], [18, 3, 6, 1], [18, 9, 6, 1], [18, 18, 2, 4], [18, 18, 6, 1], [18, 27, 6, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 39312, 'exponent': 18, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[6, 0, 12]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '324.166', 'hash': 795, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3, 6, 3, 1], 'inner_gens': [[1, 4, 354, 36, 108], [5, 2, 654, 36, 108], [661, 326, 6, 72, 108], [1, 2, 78, 36, 108], [1, 2, 6, 36, 108]], 'inner_hash': 39, 'inner_nilpotent': False, 'inner_order': 108, 'inner_split': False, 'inner_tex': 'C_3:S_3^2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 6, 'irrQ_degree': 36, 'irrQ_dim': 36, 'irrR_degree': 12, 'irrep_stats': [[1, 12], [2, 30], [3, 24], [4, 12], [6, 12]], 'label': '972.795', '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': False, 'metacyclic': False, 'monomial': True, 'name': '(C3*C9):S3^2', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 24, 'number_characteristic_subgroups': 23, 'number_conjugacy_classes': 90, 'number_divisions': 42, 'number_normal_subgroups': 47, 'number_subgroup_autclasses': 113, 'number_subgroup_classes': 292, 'number_subgroups': 1452, 'old_label': None, 'order': 972, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 39], [3, 80], [6, 204], [9, 162], [18, 486]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [6, 6, 2, 3, 3], 'outer_gen_pows': [1, 0, 1, 1, 0], 'outer_gens': [[1, 16, 658, 36, 756], [1, 16, 654, 72, 540], [1, 348, 670, 36, 108], [1, 4, 32, 36, 432], [1, 2, 6, 36, 432]], 'outer_group': '72.42', 'outer_hash': 42, 'outer_nilpotent': False, 'outer_order': 72, 'outer_permdeg': 7, 'outer_perms': [747, 28, 2424, 2188, 4], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_3\\times S_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 30, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 14], [4, 14], [8, 4], [18, 4], [36, 2]], 'representations': {'PC': {'code': 126758170481158732105436718756399, 'gens': [1, 2, 3, 5, 6], 'pres': [7, -2, -3, -2, -3, -3, -3, -3, 57, 7436, 6876, 58, 18819, 4714, 438, 166]}, 'Perm': {'d': 30, 'gens': [1, 79480952248588299408682388400, 10106548111225803837422031124344, 19287931234418868667131293493120, 28390607782314919684827619119360, 3, 37591330973438220090252954328800]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_3\\times C_9):S_3^2', 'transitive_degree': 54, 'wreath_data': None, 'wreath_product': False}
