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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, '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': 4, 'characteristic': False, 'core_order': 1, 'counter': 629, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '10368.ca.288.bj1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '288.bj1', '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': '36.13', 'subgroup_hash': 13, 'subgroup_order': 36, 'subgroup_tex': 'C_6:S_3', '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': '288.bj1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '2592.a1', 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['72.dn1', '96.n1', '144.f1'], 'contains': ['576.k1', '576.x1', '576.bg1', '864.bd1', '864.bg1'], 'core': '10368.a1', 'coset_action_label': None, 'count': 192, 'diagramx': [9893, -1, 8845, -1], 'generators': [172613619328982, 93118996444315, 128619158169600, 24], 'label': '10368.ca.288.bj1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '48.c1', 'old_label': '288.bj1', 'projective_image': '10368.ca', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '288.bj1', 'subgroup_fusion': None, 'weyl_group': '54.5'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [2, 4, 2, 4, 3, 2, 6, 6], 'aut_gens': [[1, 2, 6], [19, 2, 6], [35, 26, 32], [25, 4, 30], [33, 14, 34], [5, 2, 6], [19, 28, 6], [33, 2, 6], [23, 16, 30]], 'aut_group': '864.4661', 'aut_hash': 4661, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 864, 'aut_permdeg': 11, 'aut_perms': [1, 10889689, 24054000, 20507935, 14605950, 2245249, 19496545, 16180705], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 2, 1], [3, 2, 4, 1], [6, 2, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_3^2:\\GL(2,3)', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '432.734', 'autcentquo_hash': 734, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 9, 2], [3, 2, 4], [6, 2, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '18.4', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 13, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['18.4', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 9, 1, 2], [3, 2, 1, 4], [6, 2, 1, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 21, 'exponent': 6, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '36.13', 'hash': 13, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3, 3], 'inner_gens': [[1, 4, 30], [5, 2, 6], [13, 2, 6]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 18, 'inner_split': False, 'inner_tex': 'C_3:S_3', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4], [2, 8]], 'label': '36.13', 'linC_count': 18, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 18, 'linQ_dim': 4, 'linQ_dim_count': 18, 'linR_count': 18, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6:S3', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 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': 12, 'number_divisions': 12, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 30, 'number_subgroups': 78, 'old_label': None, 'order': 36, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 19], [3, 8], [6, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2], 'outer_gen_pows': [0, 0, 0, 1, 1], 'outer_gens': [[1, 24, 20], [19, 2, 6], [1, 24, 34], [1, 28, 8], [1, 16, 10]], 'outer_group': '48.48', 'outer_hash': 48, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 6, 'outer_perms': [143, 127, 576, 126, 121], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 8, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8]], 'representations': {'PC': {'code': 405218245, 'gens': [1, 2, 3], 'pres': [4, -2, -3, -2, -3, 33, 362, 34, 387]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [12016555, 35931151, 11780031]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [7426, 8156, 8101, 13286]}, 'Perm': {'d': 8, 'gens': [721, 24, 5763, 5760]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6:S_3', 'transitive_degree': 18, '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, 8258288604274745098185548271679974720802939812738572587688518273180654835612451700963633128907910619865009907373178708108003236183282545843958785582607390664610662946827965885901182619415875278114651601135758250277290382859443216367180798253071893102063053931319274752408165519136451805312055597744299962936488633226601935120302640068486169067426842134294963587172985015680778243105739191496810711003965260573182413003676606667868, 25033660614614537826248507923281634551270814248956291602869710576697840158892845147309841643323082775267371841394563362911097577761369967953535892100465920995517216104378396616891387535560510102430463171332104709391834742674596998401351047035538299450351142231572898998717549575806514886674632938682113798903255298097415129552403133564491866348151952157349468088649527020778727483908726609015882649855368755088249248182515694672824, 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}
