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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '6912.hm', 'ambient_counter': 195, 'ambient_order': 6912, 'ambient_tex': '(C_2\\times D_6^2):S_4', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': False, 'core_order': 2, 'counter': 1254, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '6912.hm.216.bb1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '216.bb1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 216, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '32.46', 'subgroup_hash': 46, 'subgroup_order': 32, 'subgroup_tex': 'C_2^2\\times D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '6912.hm', 'aut_centralizer_order': None, 'aut_label': '216.bb1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '432.b1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['72.bk1', '72.bz1', '108.c1', '108.k1', '108.z1'], 'contains': ['432.l1', '432.m1', '432.q1', '432.bh1', '432.bj1', '432.bv1', '432.ce1', '432.ej1'], 'core': '3456.a1', 'coset_action_label': None, 'count': 108, 'diagramx': [1779, -1, 2334, -1], 'generators': [3776, 32140648232, 28003, 33178492109, 40319], 'label': '6912.hm.216.bb1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '54.a1', 'old_label': '216.bb1', 'projective_image': '3456.de', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '216.bb1', 'subgroup_fusion': None, 'weyl_group': '8.5'}
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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': 4, 'aut_exponent': 24, 'aut_gen_orders': [12, 4], 'aut_gens': [[2565, 517, 1537, 2577], [517, 2593, 1553, 529], [2597, 2561, 1569, 529]], 'aut_group': '3072.dx', 'aut_hash': 2019763691454065160, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3072, 'aut_permdeg': 16, 'aut_perms': [11404422904581, 2678277315556], 'aut_phi_ratio': 192.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [2, 2, 8, 1], [4, 2, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^6:(C_2\\times S_4)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '1536.408640869', 'autcent_hash': 4599629986281706675, 'autcent_nilpotent': False, 'autcent_order': 1536, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^6:S_4', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 2, 8], [4, 2, 4]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 46, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 2, 1, 8], [4, 2, 1, 4]], 'element_repr_type': 'GLZq', 'elementary': 2, 'eulerian_function': 105, 'exponent': 4, 'exponents_of_order': [5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '16.14', 'hash': 46, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [1, 1, 2, 2], 'inner_gens': [[2565, 517, 1537, 2577], [2565, 517, 1537, 2577], [2565, 517, 1537, 2609], [2565, 517, 1569, 2577]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 4]], 'label': '32.46', 'linC_count': 192, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 192, 'linQ_dim': 4, 'linQ_dim_count': 192, 'linR_count': 192, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2*D4', 'ngens': 4, 'nilpotency_class': 2, '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': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 20, 'number_divisions': 20, 'number_normal_subgroups': 78, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 118, 'number_subgroups': 158, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 23], [4, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [513, 2577, 513, 513, 513, 513, 513, 513, 513], 'outer_gens': [[2561, 517, 1537, 2577], [2565, 517, 3601, 2577], [2561, 2565, 1537, 2577], [2597, 549, 3585, 2577], [2597, 517, 1541, 2577], [2597, 549, 1537, 529], [2597, 517, 1537, 2581], [2565, 517, 3585, 2577], [2565, 517, 1541, 2577]], 'outer_group': '768.1090070', 'outer_hash': 1090070, 'outer_nilpotent': False, 'outer_order': 768, 'outer_permdeg': 12, 'outer_perms': [41909165, 41141520, 727204, 182610863, 138989536, 135359400, 101693640, 182610840, 138989520], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^5:S_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 4]], 'representations': {'PC': {'code': 542728, 'gens': [1, 2, 3, 4], 'pres': [5, -2, 2, 2, 2, -2, 133, 58]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [35931074, 7233746, 7233586, 7115160]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [34876415, 29555524, 28816400, 14811013, 28233362]}, 'GLZq': {'d': 2, 'q': 8, 'gens': [2561, 529, 2565, 1537, 545]}, 'Perm': {'d': 8, 'gens': [1567, 5160, 11536, 23, 11520]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times D_4', 'transitive_degree': 16, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [6, 6, 4, 12, 12, 2, 6], 'aut_gens': [[4142, 25434270655, 18498, 28003, 414, 32140668740, 26824106177, 40319, 14493436958, 722, 3254], [36177, 13412083247, 21821, 12316, 414, 19679384369, 26352732222, 40319, 13499166561, 24338, 20933], [3776, 1037873121, 28003, 5329, 36177, 32140673368, 26824122083, 40319, 26352730579, 37965, 6620], [4142, 13412081121, 23616, 5167, 36543, 6267324740, 13499142096, 40319, 26352720702, 722, 8580], [3776, 26911202059, 11536, 16703, 414, 6267329368, 26824117459, 40319, 14493436958, 37965, 1448], [36543, 2043380216, 28003, 21821, 414, 19679346914, 14493455423, 40319, 26824099280, 29514, 26259], [414, 87105540, 34990, 28003, 4142, 6267330873, 13499168483, 40319, 26352731662, 288, 1448], [4142, 12940689659, 18498, 12316, 414, 20237841860, 26824112656, 40319, 26352739041, 39597, 3194]], 'aut_group': None, 'aut_hash': 1505175068374294371, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 110592, 'aut_permdeg': 72, 'aut_perms': [22135980794948693807590373440647446601496129381036527636255194999137670324722315491140161030929322210892, 45867085406409487574600901748073314094047386957726231832612604267813722333625942366483602286318622033584, 39233560766626797193421166721221456823124067113267260087806176546751845077695603975809472336452582904433, 47130066738493011773303637444727809452262082546083289120262929166981641306160454238037283988134948675110, 34891239178665805588881990081260828935852675148395801964680871615202993516710379357279102639469885092517, 40312255513403226938511804027771398146748821918867752882226420201917772283080448759827603311561683643215, 40530044337639181707516346759809486585031829989661322993389077454443019573188414521387553687260248485071], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 1], [2, 9, 2, 1], [2, 12, 1, 1], [2, 12, 2, 1], [2, 12, 4, 1], [2, 54, 1, 1], [2, 72, 2, 1], [2, 108, 1, 1], [2, 108, 2, 1], [3, 2, 2, 1], [3, 4, 1, 1], [3, 32, 1, 1], [3, 64, 2, 1], [3, 128, 1, 1], [4, 12, 1, 1], [4, 24, 1, 1], [4, 36, 4, 1], [4, 48, 1, 1], [4, 72, 2, 1], [4, 108, 1, 1], [4, 216, 1, 1], [4, 432, 1, 1], [6, 2, 2, 1], [6, 4, 1, 1], [6, 12, 2, 1], [6, 24, 1, 1], [6, 24, 2, 2], [6, 24, 4, 2], [6, 32, 1, 1], [6, 48, 2, 1], [6, 64, 2, 1], [6, 96, 4, 1], [6, 128, 1, 1], [6, 144, 2, 1], [6, 192, 4, 1], [6, 288, 2, 1], [8, 144, 2, 1], [12, 24, 2, 1], [12, 48, 1, 1], [12, 48, 2, 1], [12, 72, 4, 1], [12, 96, 1, 1], [12, 96, 2, 2], [12, 144, 2, 1], [24, 288, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6^2.(C_2^4\\times A_4).C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '13824.fp', 'autcentquo_hash': 4107807662957182902, 'autcentquo_nilpotent': False, 'autcentquo_order': 13824, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'D_6^2.(C_2^2\\times S_4)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 6, 1], [2, 9, 2], [2, 12, 7], [2, 54, 1], [2, 72, 2], [2, 108, 3], [3, 2, 2], [3, 4, 1], [3, 32, 1], [3, 64, 2], [3, 128, 1], [4, 12, 1], [4, 24, 1], [4, 36, 4], [4, 48, 1], [4, 72, 2], [4, 108, 1], [4, 216, 1], [4, 432, 1], [6, 2, 2], [6, 4, 1], [6, 12, 2], [6, 24, 13], [6, 32, 1], [6, 48, 2], [6, 64, 2], [6, 96, 4], [6, 128, 1], [6, 144, 2], [6, 192, 4], [6, 288, 2], [8, 144, 2], [12, 24, 2], [12, 48, 3], [12, 72, 4], [12, 96, 5], [12, 144, 2], [24, 288, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '3456.de', 'commutator_count': 1, 'commutator_label': '864.4609', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 195, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 1], [2, 9, 1, 2], [2, 12, 1, 7], [2, 54, 1, 1], [2, 72, 1, 2], [2, 108, 1, 3], [3, 2, 1, 2], [3, 4, 1, 1], [3, 32, 1, 1], [3, 64, 1, 2], [3, 128, 1, 1], [4, 12, 1, 1], [4, 24, 1, 1], [4, 36, 1, 4], [4, 48, 1, 1], [4, 72, 1, 2], [4, 108, 1, 1], [4, 216, 1, 1], [4, 432, 1, 1], [6, 2, 1, 2], [6, 4, 1, 1], [6, 12, 1, 2], [6, 24, 1, 13], [6, 32, 1, 1], [6, 48, 1, 2], [6, 64, 1, 2], [6, 96, 1, 4], [6, 128, 1, 1], [6, 144, 1, 2], [6, 192, 1, 4], [6, 288, 1, 2], [8, 144, 1, 2], [12, 24, 1, 2], [12, 48, 1, 3], [12, 72, 1, 4], [12, 96, 1, 5], [12, 144, 1, 2], [24, 288, 1, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 645120, 'exponent': 24, 'exponents_of_order': [8, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[16, 1, 2], [32, 1, 1]], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '864.4673', 'hash': 8528456833134353189, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 4, 2, 2, 2, 4, 12, 1, 6, 2, 3], 'inner_gens': [[4142, 25434285179, 18498, 12316, 414, 32140668740, 26824106863, 40319, 14493457761, 722, 3194], [39905, 25434270655, 5329, 12316, 36177, 32140684553, 13499166106, 40319, 26352731662, 34864, 20355], [4142, 25434296987, 18498, 28003, 39905, 32140663778, 26824117823, 40319, 14493457761, 722, 25741], [36177, 25434288064, 18498, 28003, 39905, 32140683569, 26824101696, 40319, 14493436958, 21099, 6620], [4142, 25434285179, 21821, 12316, 414, 32140663778, 26824113342, 40319, 14493436958, 3420, 1032], [4142, 25434295599, 23616, 35152, 36543, 32140668740, 26824121619, 40319, 26352736577, 722, 33905], [36543, 1037860822, 28003, 21821, 39905, 32140682505, 26824106177, 40319, 14493457839, 10805, 32341], [4142, 25434270655, 18498, 28003, 414, 32140668740, 26824106177, 40319, 14493436958, 722, 3254], [36177, 13412047332, 21821, 28003, 414, 6267339569, 26824106863, 40319, 14493436958, 15981, 21517], [4142, 25434268736, 18498, 34990, 3776, 32140668740, 26824104729, 40319, 14493440082, 722, 2676], [414, 25434283222, 5329, 18498, 3776, 32140675287, 26824097858, 40319, 14493455297, 126, 3254]], 'inner_hash': 5423518584134085580, 'inner_nilpotent': False, 'inner_order': 3456, 'inner_split': True, 'inner_tex': 'D_6^2:S_4', 'inner_used': [1, 2, 3, 6, 7, 9], 'irrC_degree': 16, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 16, 'irrep_stats': [[1, 8], [2, 12], [3, 8], [4, 14], [6, 16], [8, 13], [12, 14], [16, 6], [24, 1], [32, 1]], 'label': '6912.hm', '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': None, 'name': '(C2*D6^2):S4', 'ngens': 11, '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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 50, 'number_characteristic_subgroups': 23, 'number_conjugacy_classes': 93, 'number_divisions': 93, 'number_normal_subgroups': 59, 'number_subgroup_autclasses': 2008, 'number_subgroup_classes': 4637, 'number_subgroups': 158754, 'old_label': None, 'order': 6912, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 631], [3, 296], [4, 1128], [6, 2744], [8, 288], [12, 1248], [24, 576]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[4142, 25434285179, 23616, 11536, 414, 32140663778, 26824111843, 40319, 14493457839, 722, 3254], [4142, 25434273540, 23616, 11536, 414, 32140665501, 26824107676, 40319, 14493457839, 39597, 3254], [4142, 25434288064, 18498, 28003, 414, 19679361339, 26352732222, 40319, 13499145758, 39597, 3254], [4142, 25434270655, 18498, 28003, 414, 19679369540, 26352732222, 40319, 13499145758, 39597, 3254]], 'outer_group': '32.46', 'outer_hash': 46, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [16, 23, 1567, 5880], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_4', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [3, 8], [4, 14], [6, 16], [8, 13], [12, 14], [16, 6], [24, 1], [32, 1]], 'representations': {'PC': {'code': '508264333531405241922638342833095331021745078565740141811297005014129133103507156739171566146905360758685313891703956028935882961418216365955626587785386417281345', 'gens': [1, 2, 4, 6, 8, 9], 'pres': [11, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 41184, 28997, 56, 63230, 249747, 48854, 24313, 124, 2644, 1335, 451445, 160396, 34479, 28154, 6583, 192, 299382, 11105, 8344, 23139, 19026, 3197, 27496, 4638, 313640, 237025, 104277, 4419, 2846, 294, 633609, 316820, 8864, 328, 557578, 278805, 7809]}, 'Perm': {'d': 14, 'gens': [4142, 25434270655, 18498, 28003, 414, 32140668740, 26824106177, 40319, 14493436958, 722, 3254]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '(C_2\\times D_6^2):S_4', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
