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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '10368.ca', 'ambient_counter': 53, 'ambient_order': 10368, 'ambient_tex': '(C_2\\times C_6^3):S_4', 'central': False, 'central_factor': False, 'centralizer_order': 2592, 'characteristic': False, 'core_order': 2, 'counter': 1052, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '10368.ca.2592.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '2592.a1', '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': 2592, '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': '10368.ca', 'aut_centralizer_order': None, 'aut_label': '2592.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '4.b1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['864.a1', '864.b1', '864.c1', '864.d1', '864.e1', '864.f1', '1296.b1', '1296.c1'], 'contains': ['5184.a1', '5184.b1'], 'core': '5184.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [6552, -1, 5481, -1], 'generators': [39597, 24], 'label': '10368.ca.2592.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '648.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.b1', 'old_label': '2592.a1', 'projective_image': '5184.kx', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '2592.a1', '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 36, 'aut_gen_orders': [6, 6, 6, 4, 6, 4], 'aut_gens': [[46136603123293, 23813564907848], [112549213349443, 23819875393932], [27741983704157, 155894068493266], [176623824000786, 24074148678980], [152158294741630, 167651039026399], [151709027878476, 47601342853910], [134983173823341, 167824920220526]], 'aut_group': '41472.mf', 'aut_hash': 6240288420315592407, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 41472, 'aut_permdeg': 224, 'aut_perms': [14100244591781225663492140293699579098840431319505129220557804216680667589697918161217046302364288450895475487852970808750940949436059132764465951213793650434476157465865739619266453199703176875148569917090639288499990467998506278958028670664881879680315320157173360027884178360765446851389838543731142925632625375627085621124840448841147564352025284406107453014975993755029037108552699430482237566333861967312830385594296997866767, 25188982327200982976111221341061216185196639778589013087058056885868297473980155858679978857592359246003619428460739433136319621182980855414008724343273829970633982883116077931694033927307045266032371378958453465202360787074957985723427963938365802648802802971594206688716804421429208602247436830516040641609249619447288842616413563927518392774068799800436585701710471851802635732871942134563865608284647088125007058554469880612202, 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10356458577348041488279665379700631269889130141001757507106563126206211825907241667166732006305667322416477626181919327092602139807943853476827092678705937362414447045399835934198356392508036851708521590179628802125883972239093030767535631668626747880294662632056819673834714653439028623117044371519549312357270614481414826029288862426687439192727366640997783196403966598178994957869390353587963143513309966695816731717781269740971, 9981390643251909790978443080316668474913721656982710565898714893292795064820705848251233965483448317642758700790773419563327998098826244086434403897687013633968476443030482609158443929339815171258562543921164202653967057308213377039780077189837211120424455483577274818184789626086492757685406687916001975759984049972265089968173591850848023773080725476827464735083704508452354783895023971919234703632719420771153143683499356533916], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 2, 1], [2, 6, 1, 1], [2, 108, 1, 1], [2, 108, 2, 1], [2, 216, 1, 1], [3, 6, 1, 1], [3, 8, 1, 1], [3, 12, 1, 1], [3, 288, 1, 1], [4, 108, 1, 1], [4, 108, 2, 1], [4, 216, 1, 2], [4, 432, 1, 1], [6, 6, 1, 1], [6, 8, 1, 1], [6, 8, 2, 1], [6, 12, 1, 4], [6, 12, 4, 1], [6, 24, 1, 3], [6, 24, 2, 4], [6, 216, 1, 1], [6, 216, 2, 2], [6, 288, 1, 1], [6, 288, 2, 1], [8, 432, 1, 1], [9, 288, 2, 1], [12, 216, 1, 1], [12, 216, 2, 2], [12, 432, 1, 1], [12, 432, 2, 1], [18, 288, 2, 1], [18, 288, 4, 1], [24, 432, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6^3.(C_2^3\\times S_4)', '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': 36, 'autcentquo_group': '10368.mp', 'autcentquo_hash': 5188747272954890844, 'autcentquo_nilpotent': False, 'autcentquo_order': 10368, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_6^3:(C_2\\times S_4)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 4, 2], [2, 6, 1], [2, 108, 3], [2, 216, 1], [3, 6, 1], [3, 8, 1], [3, 12, 1], [3, 288, 1], [4, 108, 3], [4, 216, 2], [4, 432, 1], [6, 6, 1], [6, 8, 3], [6, 12, 8], [6, 24, 11], [6, 216, 5], [6, 288, 3], [8, 432, 1], [9, 288, 2], [12, 216, 5], [12, 432, 3], [18, 288, 6], [24, 432, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '5184.kx', 'commutator_count': 1, 'commutator_label': '2592.da', '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', '3.1', '3.1'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 53, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 1, 2], [2, 6, 1, 1], [2, 108, 1, 3], [2, 216, 1, 1], [3, 6, 1, 1], [3, 8, 1, 1], [3, 12, 1, 1], [3, 288, 1, 1], [4, 108, 1, 3], [4, 216, 1, 2], [4, 432, 1, 1], [6, 6, 1, 1], [6, 8, 1, 3], [6, 12, 1, 4], [6, 12, 2, 2], [6, 24, 1, 9], [6, 24, 2, 1], [6, 216, 1, 3], [6, 216, 2, 1], [6, 288, 1, 3], [8, 432, 1, 1], [9, 288, 1, 2], [12, 216, 1, 3], [12, 216, 2, 1], [12, 432, 1, 1], [12, 432, 2, 1], [18, 288, 1, 6], [24, 432, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 468, 'exponent': 72, 'exponents_of_order': [7, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 1, 6], [12, 0, 8], [24, 1, 6]], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '1296.3492', 'hash': 1506925402112939753, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [12, 18], 'inner_gens': [[46136603123293, 62968599984503], [132629363568936, 23813564907848]], 'inner_hash': 5571395750252258158, 'inner_nilpotent': False, 'inner_order': 5184, 'inner_split': True, 'inner_tex': 'C_6^3:S_4', 'inner_used': [1, 2], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 4], [2, 2], [3, 4], [4, 4], [6, 12], [8, 14], [12, 18], [24, 11]], 'label': '10368.ca', '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*C6^3):S4', 'ngens': 2, '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, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 46, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 69, 'number_divisions': 62, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 1084, 'number_subgroup_classes': 1558, 'number_subgroups': 73872, 'old_label': None, 'order': 10368, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 555], [3, 314], [4, 1188], [6, 2334], [8, 432], [9, 576], [12, 2376], [18, 1728], [24, 864]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[46136603145592, 23813564907848], [46136603123293, 23813564946945], [27494252761693, 44722507295048]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 7, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [3, 4], [4, 4], [6, 8], [8, 14], [12, 12], [24, 13], [48, 1]], 'representations': {'PC': {'code': '26772446599649967360236403606980247562862905539456447040462679073797905847738193595795443151384652972502968328017199993155172620166782310300479503636820371837385285', 'gens': [1, 2, 3, 4, 6, 8, 10], 'pres': [11, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 3, 26334, 65429, 51228, 56630, 91687, 34872, 57555, 17174, 11025, 124, 33004, 33015, 11026, 612617, 136438, 42003, 42410, 12919, 192, 465702, 38825, 68404, 675847, 210162, 8158, 260, 458576, 373051, 1647, 316820, 158431, 13264, 328, 69717, 139424, 11681]}, 'Perm': {'d': 17, 'gens': [46136603123293, 23813564907848]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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\\times C_6^3):S_4', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
