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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1152.157872', 'ambient_counter': 157872, 'ambient_order': 1152, 'ambient_tex': '(C_2^3\\times C_6):S_4', 'central': False, 'central_factor': False, 'centralizer_order': 192, 'characteristic': False, 'core_order': 1, 'counter': 311, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1152.157872.288.x1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '288.x1', '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': 288, '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': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1152.157872', 'aut_centralizer_order': 3456, 'aut_label': '288.x1', 'aut_quo_index': None, 'aut_stab_index': 12, 'aut_weyl_group': '6.1', 'aut_weyl_index': 41472, 'centralizer': '6.a1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['96.bb1', '144.m1', '144.s1', '144.bi2'], 'contains': ['576.e1'], 'core': '1152.a1', 'coset_action_label': None, 'count': 12, 'diagramx': [8543, -1, 2734, -1], 'generators': [25430670727, 37885438096], 'label': '1152.157872.288.x1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '18.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '6.a1', 'old_label': '288.x1', 'projective_image': '1152.157872', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '288.x1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', '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]], 'center_label': '4.2', 'center_order': 4, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', '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, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, '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, 12, 12, 6], 'aut_gens': [[89744619000, 7, 16, 519321600, 840, 199348168320, 294111336960, 381365832960, 211802935680], [457018370040, 16, 23, 212680380360, 1440, 199348168320, 387592894080, 381365832960, 287883630720], [536174070480, 23, 16, 207008526240, 840, 375138126720, 205575874560, 211802935680, 287883630720], [457018370663, 23, 16, 38403953280, 1440, 294111336960, 387592894080, 375138126720, 205575874560], [631375679663, 23, 16, 181581927840, 840, 387592894080, 199348168320, 205575874560, 300338398080], [536174070480, 23, 7, 355419590400, 1440, 375138126720, 205575874560, 211802935680, 287883630720]], 'aut_group': None, 'aut_hash': 7003357016634712411, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 248832, 'aut_permdeg': 144, 'aut_perms': [104340879963189040563072928509637816104762085032619564135612963480991929547212892358421746257966809010846933461616769891022583121400842388452931909922783870154725295047222030375664750909761965805710479809760921515956933382881824127162636652494828837, 5153062247688080429644397283044171157530745128271461813191588442473698249545347528646086678953645431483173025425708885633979033380096524701937766897433313414314314228084321226944946689615267881037511927302378412261374772292541386162462866592643983737, 1671347569703372168259455370161988830425959551113258446149688477350251214374071368298392836023219297089058229938203178592513869208670415667374903416895694142660280813243828942573471865119831680378472438192069576520766825674478730496987889324320525919, 3202713934296131188142044317875385922219543178641666512012569914760471027350724536779736506835769292752654476787614434473931202728648650324884303572264130762149856139389469567694612833202020016381615244125260435275650067283937732896000923555493868243, 4729240114961920637363161061938546681652568596830374587790519415000522222725001667812998173489531788747867002327123896383134503600783963924443846692425562428984675563496174051524207544536708351531137867175747499077595134074317429787701889829872465675], 'aut_phi_ratio': 648.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 3, 3, 1], [2, 3, 9, 1], [2, 6, 1, 1], [2, 6, 3, 1], [2, 36, 4, 1], [3, 2, 1, 1], [3, 32, 3, 1], [4, 36, 12, 1], [6, 2, 3, 1], [6, 6, 2, 1], [6, 6, 3, 1], [6, 6, 6, 1], [6, 6, 9, 1], [6, 32, 9, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_2^5\\times C_6).C_3^4.C_2^4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.12', 'autcent_hash': 12, 'autcent_nilpotent': False, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '10368.pi', 'autcentquo_hash': 3862291860621009286, 'autcentquo_nilpotent': False, 'autcentquo_order': 10368, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '(C_2^3\\times C_6).S_3^3', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 3, 12], [2, 6, 4], [2, 36, 4], [3, 2, 1], [3, 32, 3], [4, 36, 12], [6, 2, 3], [6, 6, 20], [6, 32, 9]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '288.1036', 'commutator_count': 2, 'commutator_label': '144.194', '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', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 157872, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['288.1036', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 1, 12], [2, 6, 1, 4], [2, 36, 1, 4], [3, 2, 1, 1], [3, 32, 1, 3], [4, 36, 1, 12], [6, 2, 1, 3], [6, 6, 1, 12], [6, 6, 2, 4], [6, 32, 1, 9]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 980, 'exponent': 12, 'exponents_of_order': [7, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1152.157872', 'hash': 157872, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 1, 1, 3, 3, 2, 2, 2, 2], 'inner_gens': [[89744619000, 7, 16, 958728960, 1440, 387592894080, 294111336960, 287883630720, 211802935680], [89744619000, 7, 16, 519321600, 840, 199348168320, 294111336960, 381365832960, 211802935680], [89744619000, 7, 16, 519321600, 840, 199348168320, 294111336960, 381365832960, 211802935680], [89225297400, 7, 16, 519321600, 840, 294111336960, 387592894080, 211802935680, 287883630720], [89744620440, 7, 16, 519321600, 840, 199348168320, 294111336960, 381365832960, 211802935680], [543717982200, 7, 16, 387037284480, 840, 199348168320, 294111336960, 381365832960, 211802935680], [89744619000, 7, 16, 200782270080, 840, 199348168320, 294111336960, 381365832960, 211802935680], [623312121720, 7, 16, 286487268480, 840, 199348168320, 294111336960, 381365832960, 211802935680], [89744619000, 7, 16, 380809900800, 840, 199348168320, 294111336960, 381365832960, 211802935680]], 'inner_hash': 1036, 'inner_nilpotent': False, 'inner_order': 288, 'inner_split': True, 'inner_tex': '(C_2\\times C_6):S_4', 'inner_used': [1, 4, 5, 6, 8], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 16], [3, 24], [6, 24]], 'label': '1152.157872', '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': '(C2^3*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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 16, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 72, 'number_divisions': 68, 'number_normal_subgroups': 81, 'number_subgroup_autclasses': 320, 'number_subgroup_classes': 3024, 'number_subgroups': 24592, 'old_label': None, 'order': 1152, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 207], [3, 98], [4, 432], [6, 414]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 3, 3, 3, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 89744619000, 0, 0, 0], 'outer_gens': [[89744619000, 7, 16, 519321600, 840, 381365832960, 211802935680, 199348168320, 294111336960], [89744619000, 7, 16, 958728960, 840, 387592894080, 294111336960, 287883630720, 211802935680], [89744619000, 16, 7, 519321600, 840, 199348168320, 294111336960, 381365832960, 211802935680], [89744619000, 16, 23, 519321600, 840, 199348168320, 294111336960, 381365832960, 211802935680], [89744619000, 7, 16, 958730400, 1440, 387592894080, 294111336960, 287883630720, 211802935680], [89744619000, 7, 16, 519321600, 840, 300338398080, 375138126720, 199348168320, 294111336960], [89744619007, 7, 16, 519321600, 840, 199348168320, 294111336960, 381365832960, 211802935680], [89744619016, 7, 16, 519321600, 840, 199348168320, 294111336960, 381365832960, 211802935680]], 'outer_group': '864.4673', 'outer_hash': 4673, 'outer_nilpotent': False, 'outer_order': 864, 'outer_permdeg': 10, 'outer_perms': [379560, 374400, 21, 11, 1578360, 1941120, 7, 16], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'S_4\\times S_3^2', 'pc_rank': None, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 16], [3, 24], [6, 16], [12, 4]], 'representations': {'PC': {'code': '347476411094005198023964040122057935916263613810120318226041541712', 'gens': [1, 2, 4, 6, 7, 8, 9], 'pres': [9, -2, -2, -3, -2, -3, -2, 2, 2, 2, 181, 46, 218, 2163, 102, 2164, 54437, 1976, 1175, 3813, 1554, 46663, 5650, 1987, 58328, 1007, 2474]}, 'Perm': {'d': 15, 'gens': [89744619000, 7, 16, 519321600, 840, 199348168320, 294111336960, 381365832960, 211802935680]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '(C_2^3\\times C_6):S_4', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
