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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '139968.hx', 'ambient_counter': 206, 'ambient_order': 139968, 'ambient_tex': 'C_3^4.C_{12}^2:D_6', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': False, 'core_order': 36, 'counter': 339, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '139968.hx.36._.GT', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '36.GT', 'outer_equivalence': False, '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': 36, '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': '3888.fl', 'subgroup_hash': None, 'subgroup_order': 3888, 'subgroup_tex': 'C_3^3.(C_6\\times S_4)', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '139968.hx', 'aut_centralizer_order': None, 'aut_label': None, 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': None, 'complements': None, 'conjugacy_class_count': 1, 'contained_in': None, 'contains': None, 'core': None, 'coset_action_label': None, 'count': 36, 'diagramx': None, 'generators': [2432923018148448000, 44885565975961182171380735872653, 183180, 1863222707917826278649933938129, 969543720820463348800880640000, 194383, 2439304469539660800, 54292741358404679685619384320000, 1644860432396280847591468800], 'label': '139968.hx.36._.GT', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': None, 'normal_contained_in': None, 'normal_contains': None, 'normalizer': None, 'old_label': '36.GT', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '36._.GT', 'subgroup_fusion': None, 'weyl_group': None}
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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': 36, 'aut_gen_orders': [12, 18, 6, 6], 'aut_gens': [[1, 6, 108, 648], [217, 1254, 2484, 648], [3409, 6, 2160, 3348], [2377, 114, 108, 648], [1, 3354, 108, 648]], 'aut_group': '3888.fl', 'aut_hash': 3044566718417071494, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3888, 'aut_permdeg': 63, 'aut_perms': [830612640461771533641624694756461587350547575226542480167034786097010318810260779134817, 6443743887424598440296883830308049795495716909882723373512772025823720668335943936908, 767497761907956226711626478826200381700896597963076445576997636140190632484461809939178, 32593332953609159815276087395728358370211284550998388407040484609979940713583098625603], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 1], [2, 54, 1, 2], [3, 2, 1, 2], [3, 3, 1, 2], [3, 4, 1, 1], [3, 6, 1, 5], [3, 12, 1, 3], [4, 54, 1, 2], [6, 3, 1, 2], [6, 6, 1, 9], [6, 12, 1, 15], [6, 18, 1, 1], [6, 27, 1, 4], [6, 54, 1, 7], [6, 108, 1, 6], [9, 72, 1, 3], [9, 144, 1, 3], [12, 54, 1, 4], [12, 108, 1, 6], [18, 216, 1, 3]], 'aut_supersolvable': False, 'aut_tex': 'C_3^3.(C_6\\times S_4)', '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': 36, 'autcentquo_group': '3888.fl', 'autcentquo_hash': 3044566718417071494, 'autcentquo_nilpotent': False, 'autcentquo_order': 3888, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^3.(C_6\\times S_4)', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 9, 1], [2, 27, 1], [2, 54, 2], [3, 2, 2], [3, 3, 2], [3, 4, 1], [3, 6, 5], [3, 12, 3], [4, 54, 2], [6, 3, 2], [6, 6, 9], [6, 12, 15], [6, 18, 1], [6, 27, 4], [6, 54, 7], [6, 108, 6], [9, 72, 3], [9, 144, 3], [12, 54, 4], [12, 108, 6], [18, 216, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '3888.fl', 'commutator_count': 1, 'commutator_label': '324.59', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 142, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 9, 1, 1], [2, 27, 1, 1], [2, 54, 1, 2], [3, 2, 1, 2], [3, 3, 2, 1], [3, 4, 1, 1], [3, 6, 1, 1], [3, 6, 2, 2], [3, 12, 1, 1], [3, 12, 2, 1], [4, 54, 1, 2], [6, 3, 2, 1], [6, 6, 1, 3], [6, 6, 2, 3], [6, 12, 1, 3], [6, 12, 2, 6], [6, 18, 1, 1], [6, 27, 2, 2], [6, 54, 1, 1], [6, 54, 2, 3], [6, 108, 1, 2], [6, 108, 2, 2], [9, 72, 1, 1], [9, 72, 2, 1], [9, 144, 1, 1], [9, 144, 2, 1], [12, 54, 2, 2], [12, 108, 1, 2], [12, 108, 2, 2], [18, 216, 1, 1], [18, 216, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 432, 'exponent': 36, 'exponents_of_order': [5, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 8], [12, 1, 1]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '432.745', 'hash': 3044566718417071494, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [12, 18, 6, 6], 'inner_gens': [[1, 714, 2484, 648], [3397, 6, 2484, 1836], [2161, 2166, 108, 648], [1, 3354, 108, 648]], 'inner_hash': 3044566718417071494, 'inner_nilpotent': False, 'inner_order': 3888, 'inner_split': True, 'inner_tex': 'C_3^3.(C_6\\times S_4)', 'inner_used': [1, 2], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 12], [2, 12], [3, 12], [4, 3], [6, 26], [12, 19]], 'label': '3888.fl', 'linC_count': 93, 'linC_degree': 12, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 13, 'linQ_dim': 12, 'linQ_dim_count': 13, 'linR_count': 13, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^3.(C6*S4)', 'ngens': 9, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 84, 'number_characteristic_subgroups': 38, 'number_conjugacy_classes': 84, 'number_divisions': 56, 'number_normal_subgroups': 38, 'number_subgroup_autclasses': 710, 'number_subgroup_classes': 710, 'number_subgroups': 11098, 'old_label': None, 'order': 3888, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 147], [3, 80], [4, 108], [6, 1392], [9, 648], [12, 864], [18, 648]], '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': 4, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [3, 4], [4, 5], [6, 14], [8, 1], [12, 13], [24, 7]], 'representations': {'PC': {'code': '17332017364350600297170854971909930236157727789920379073816802666887113347888075071157', 'gens': [1, 3, 6, 8], 'pres': [9, 2, 3, 2, 3, 3, 2, 3, 2, 3, 18, 17506, 19280, 46262, 74, 4755, 2820, 138, 3244, 134141, 22379, 10724, 158, 27222, 4560, 22057, 7162, 214, 40850, 11699]}, 'Perm': {'d': 22, 'gens': [53787474035045498409, 5257481346106226372]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3.(C_6\\times S_4)', 'transitive_degree': 36, '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': '4.2', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 72, 'aut_gen_orders': [24, 24, 12, 18, 24], 'aut_gens': [[9452749799372436341698560647132, 327489444396113751585501298129], [64039056266236725977497361318079, 18600236553878537514604614575416], [19241097655203179377482789068139, 1872885571326460410879425426689], [17705334610171084348646275559270, 35390502785897071486036021032869], [37261742929183709440494283719953, 18600236553878537514604614590669], [64082611768763955507370353527536, 10999836698777491030055148698636]], 'aut_group': None, 'aut_hash': 2958339236527833249, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1679616, 'aut_permdeg': 432, 'aut_perms': 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'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 108, 1, 1], [2, 324, 1, 1], [2, 972, 1, 1], [2, 2916, 1, 1], [3, 2, 1, 1], [3, 6, 1, 2], [3, 6, 3, 1], [3, 8, 1, 1], [3, 12, 1, 3], [3, 12, 3, 1], [3, 16, 1, 1], [3, 24, 1, 3], [3, 24, 3, 2], [3, 48, 1, 2], [3, 48, 3, 2], [3, 2592, 1, 1], [4, 54, 1, 2], [4, 324, 1, 1], [4, 972, 1, 1], [6, 6, 1, 3], [6, 6, 3, 1], [6, 12, 1, 5], [6, 12, 3, 2], [6, 24, 1, 9], [6, 24, 3, 5], [6, 48, 1, 7], [6, 48, 2, 1], [6, 48, 3, 5], [6, 48, 6, 1], [6, 216, 1, 1], [6, 324, 2, 1], [6, 648, 1, 5], [6, 648, 3, 1], [6, 1296, 1, 1], [6, 1296, 3, 1], [6, 1944, 1, 2], [6, 3888, 1, 1], [6, 5832, 1, 1], [6, 7776, 1, 1], [8, 648, 1, 1], [8, 1944, 1, 1], [9, 2592, 2, 1], [9, 5184, 1, 1], [9, 5184, 2, 1], [12, 54, 2, 2], [12, 54, 6, 1], [12, 108, 1, 4], [12, 108, 2, 4], [12, 108, 3, 1], [12, 108, 6, 3], [12, 216, 1, 2], [12, 216, 2, 2], [12, 216, 3, 1], [12, 216, 6, 2], [12, 648, 1, 2], [12, 648, 3, 1], [12, 1296, 1, 1], [12, 1296, 3, 1], [12, 1944, 1, 2], [12, 3888, 1, 1], [18, 7776, 2, 1], [24, 648, 2, 2], [24, 648, 4, 1], [24, 648, 6, 1], [24, 648, 12, 1], [24, 1944, 2, 2], [24, 1944, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^5.C_4^2.C_3^3.C_2^4', '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': 72, 'autcentquo_group': None, 'autcentquo_hash': 2958339236527833249, 'autcentquo_nilpotent': False, 'autcentquo_order': 1679616, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^5.C_4^2.C_3^3.C_2^4', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 108, 1], [2, 324, 1], [2, 972, 1], [2, 2916, 1], [3, 2, 1], [3, 6, 5], [3, 8, 1], [3, 12, 6], [3, 16, 1], [3, 24, 9], [3, 48, 8], [3, 2592, 1], [4, 54, 2], [4, 324, 1], [4, 972, 1], [6, 6, 6], [6, 12, 11], [6, 24, 24], [6, 48, 30], [6, 216, 1], [6, 324, 2], [6, 648, 8], [6, 1296, 4], [6, 1944, 2], [6, 3888, 1], [6, 5832, 1], [6, 7776, 1], [8, 648, 1], [8, 1944, 1], [9, 2592, 2], [9, 5184, 3], [12, 54, 10], [12, 108, 33], [12, 216, 21], [12, 648, 5], [12, 1296, 4], [12, 1944, 2], [12, 3888, 1], [18, 7776, 2], [24, 648, 26], [24, 1944, 8]], 'center_label': '1.1', 'center_order': 1, 'central_product': None, 'central_quotient': '139968.hx', 'commutator_count': 1, 'commutator_label': '34992.dq', 'complements_known': False, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 13, 'conjugacy_classes_known': True, 'counter': 206, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': None, 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 108, 1, 1], [2, 324, 1, 1], [2, 972, 1, 1], [2, 2916, 1, 1], [3, 2, 1, 1], [3, 6, 1, 5], [3, 8, 1, 1], [3, 12, 1, 6], [3, 16, 1, 1], [3, 24, 1, 9], [3, 48, 1, 8], [3, 2592, 1, 1], [4, 54, 1, 2], [4, 324, 1, 1], [4, 972, 1, 1], [6, 6, 1, 6], [6, 12, 1, 11], [6, 24, 1, 24], [6, 48, 1, 22], [6, 48, 2, 4], [6, 216, 1, 1], [6, 324, 2, 1], [6, 648, 1, 8], [6, 1296, 1, 4], [6, 1944, 1, 2], [6, 3888, 1, 1], [6, 5832, 1, 1], [6, 7776, 1, 1], [8, 648, 1, 1], [8, 1944, 1, 1], [9, 2592, 1, 2], [9, 5184, 1, 3], [12, 54, 2, 5], [12, 108, 1, 7], [12, 108, 2, 13], [12, 216, 1, 5], [12, 216, 2, 8], [12, 648, 1, 5], [12, 1296, 1, 4], [12, 1944, 1, 2], [12, 3888, 1, 1], [18, 7776, 1, 2], [24, 648, 2, 13], [24, 1944, 2, 4]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 72, 'exponents_of_order': [7, 6], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 12], [24, 1, 30], [48, 0, 6], [48, 1, 15]], 'familial': False, 'frattini_label': '36.14', 'frattini_quotient': '3888.bz', 'hash': 9139085911823018876, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 72, 'inner_gen_orders': [6, 12], 'inner_gens': [[9452749799372436341698560647132, 18031983968555252089907183525229], [19862509018707952552043490234053, 327489444396113751585501298129]], 'inner_hash': 9139085911823018876, 'inner_nilpotent': False, 'inner_order': 139968, 'inner_split': True, 'inner_tex': 'C_3^4.C_{12}^2:D_6', 'inner_used': [1, 2], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 4], [2, 4], [3, 4], [4, 1], [6, 50], [8, 6], [12, 75], [16, 3], [24, 67], [48, 38]], 'label': '139968.hx', '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': 'C3^4.C12^2:D6', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 1, 0, 3, 1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 128, 'number_characteristic_subgroups': 30, 'number_conjugacy_classes': 252, 'number_divisions': 204, 'number_normal_subgroups': 30, 'number_subgroup_autclasses': None, 'number_subgroup_classes': None, 'number_subgroups': None, 'old_label': None, 'order': 139968, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 4323], [3, 3320], [4, 1404], [6, 34800], [8, 2592], [9, 20736], [12, 24840], [18, 15552], [24, 32400]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[9452749799372442764994573901278, 327489444396113751585501465121], [37261742929181229611010786195678, 71692782418306578728456925465121]], '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': None, 'perfect': False, 'permutation_degree': 30, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 4], [2, 4], [3, 4], [4, 1], [6, 26], [8, 6], [12, 51], [16, 3], [24, 57], [48, 44], [96, 4]], 'representations': {'PC': {'code': '1929096842344013017141306491260971016143001970347416109402023296438110568195620996127486763866564462833311992799040882475095711557406208020449867711180057344772432129313058349625862172586466978224106998023931974472682859575986051967172254007523586598479', 'gens': [1, 2, 4, 5, 8, 11, 12, 13], 'pres': [13, 2, 2, 3, 3, 2, 2, 3, 2, 2, 3, 3, 3, 3, 1539408, 3574221, 66, 1317266, 2134, 4485939, 1133200, 442913, 486762, 6624544, 5867, 598693, 186, 2656373, 14058, 220940, 226, 1441446, 13123, 119073, 8906983, 4577060, 289270, 260891, 306, 2451392, 5319777, 1291025, 64644, 346, 5391369, 7413142, 786288, 124861, 8154442, 741335, 391297, 3533, 4852235, 12849420, 2482297, 535443, 486784, 36607]}, 'Perm': {'d': 30, 'gens': [9452749799372436341698560647132, 327489444396113751585501298129]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 96, 'supersolvable': False, 'sylow_subgroups_known': False, 'tex_name': 'C_3^4.C_{12}^2:D_6', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
