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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '5184.de', 'ambient_counter': 83, 'ambient_order': 5184, 'ambient_tex': '(C_3\\times C_6^3):D_4', 'central': False, 'central_factor': False, 'centralizer_order': 72, 'characteristic': False, 'core_order': 4, 'counter': 193, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '5184.de.72.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '72.c1', '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': 72, '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': '72.50', 'subgroup_hash': 50, 'subgroup_order': 72, 'subgroup_tex': 'C_2\\times C_6^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '5184.de', 'aut_centralizer_order': None, 'aut_label': '72.c1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '72.c1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['24.b1', '36.a1', '36.l1', '36.r1'], 'contains': ['144.h1', '144.o1', '144.bi1', '216.b1'], 'core': '1296.a1', 'coset_action_label': None, 'count': 36, 'diagramx': None, 'generators': [3, 1728, 1008, 24, 1840], 'label': '5184.de.72.c1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '2.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '18.e1', 'old_label': '72.c1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '72.c1', 'subgroup_fusion': None, 'weyl_group': '4.2'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '72.50', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 168, 'aut_gen_orders': [2, 4, 4, 2, 2, 3, 4], 'aut_gens': [[1, 2, 12], [1, 10, 60], [43, 57, 30], [1, 58, 20], [37, 54, 44], [1, 30, 40], [1, 30, 68], [1, 50, 68]], 'aut_group': '8064.co', 'aut_hash': 8143655284198067037, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 8064, 'aut_permdeg': 15, 'aut_perms': [40031, 469062703732, 18101, 287404625241, 4166, 2232, 22530], 'aut_phi_ratio': 336.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1], [3, 1, 8, 1], [6, 1, 56, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(2,3)\\times \\GL(3,2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 168, 'autcent_group': '8064.co', 'autcent_hash': 8143655284198067037, 'autcent_nilpotent': False, 'autcent_order': 8064, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(2,3)\\times \\GL(3,2)', '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, 7], [3, 1, 8], [6, 1, 56]], 'center_label': '72.50', 'center_order': 72, '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', '2.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 50, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['3.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 1, 2, 4], [6, 1, 2, 28]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 13, 'exponent': 6, 'exponents_of_order': [3, 2], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '72.50', 'hash': 50, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 12], [1, 2, 12], [1, 2, 12]], '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, 72]], 'label': '72.50', 'linC_count': 17472, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 1008, 'linQ_dim': 5, 'linQ_dim_count': 1008, 'linR_count': 1008, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*C6^2', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 72, 'number_divisions': 40, 'number_normal_subgroups': 96, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 96, 'number_subgroups': 96, 'old_label': None, 'order': 72, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 7], [3, 8], [6, 56]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 168, 'outer_gen_orders': [2, 4, 4, 2, 2, 3, 4], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[1, 10, 60], [43, 57, 30], [1, 58, 20], [37, 54, 44], [1, 30, 40], [1, 30, 68], [1, 50, 68]], 'outer_group': '8064.co', 'outer_hash': 8143655284198067037, 'outer_nilpotent': False, 'outer_order': 8064, 'outer_permdeg': 15, 'outer_perms': [40031, 469062703732, 18101, 287404625241, 4166, 2232, 22530], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,3)\\times \\GL(3,2)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 32]], 'representations': {'PC': {'code': 40894789, 'gens': [1, 2, 4], 'pres': [5, -2, -2, -3, -2, -3, 26, 58]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [141602720900, 705558874143, 705685889511]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [17150691, 29071310, 28836015, 7612888, 23068812]}, 'Perm': {'d': 12, 'gens': [39916800, 362880, 5040, 240, 4]}}, 'schur_multiplier': [2, 2, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6, 6], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_6^2', 'transitive_degree': 72, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [24, 8, 8, 24, 24, 8], 'aut_gens': [[1, 2, 8, 48, 288, 1728], [3043, 978, 3480, 1416, 1720, 1744], [3619, 4230, 3656, 648, 1104, 3104], [4463, 2946, 1736, 1216, 3992, 32], [299, 442, 3592, 88, 1592, 4704], [2475, 4430, 1752, 1240, 4976, 16], [777, 3682, 4152, 4768, 520, 224]], 'aut_group': None, 'aut_hash': 8145239932073915313, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1990656, 'aut_permdeg': 144, 'aut_perms': [3561022814342274011072066412972684321110904508082827268750009736703445765459369700306994232578872654054327005875208152619921420801178829421953148000436474415656769518020415464046680306973453611300180252685726227317973068324809902699700153637361060217, 2346021514521628589103665835038178050238027996555057694567597961547465571630884805933111492309015630446316640460186376836784612033973257942535116614691601635409789397537123349780900408246575030292088060934776855394533501754980648383333125782554158256, 4551357461632521835524086616590157525178081371762412738632524159418117796010569319483631507020727992859731435092035257929263956061222740413410731601865708297589916030684542078330291742291206180707538456694146864330968895609756364747329705308101962688, 41728594890403521804027076876415741115694456738688042524263420115772074344442112451599168609682893461865758905778529478933210944699303657831923141316870401688339214005152684912403587397419010271585021699660817196002637339061199593806668247375946772, 4745420845903492338996617424266582815690542213836935875317262015585230864656687384003950250492596512776254823510321981808927016507954808313197086276867748654398165876688444535217020673540908462458289276495157923073937891762771067932214320386714982865, 4020392708411459161825171076065700876400857698694761270474688593843040458485900695554390031240510549348213684900490545545619647766679091416836305399048243797343770783450068664375922513504881214121380017837081317631671335493614409274263813663180006904], 'aut_phi_ratio': 1152.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 2, 2, 1], [2, 36, 4, 1], [2, 81, 2, 1], [2, 162, 1, 1], [2, 162, 2, 1], [3, 4, 8, 1], [3, 8, 6, 1], [4, 36, 4, 1], [4, 324, 4, 1], [6, 4, 8, 1], [6, 4, 16, 1], [6, 8, 6, 1], [6, 8, 8, 2], [6, 8, 12, 1], [6, 8, 24, 1], [6, 72, 16, 1], [12, 72, 16, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4.Q_8.C_6.C_2^4.C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': None, 'autcentquo_hash': 1694224934503634087, 'autcentquo_nilpotent': False, 'autcentquo_order': 124416, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^4.Q_8.C_6.D_4.C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 3], [2, 36, 4], [2, 81, 2], [2, 162, 3], [3, 4, 8], [3, 8, 6], [4, 36, 4], [4, 324, 4], [6, 4, 24], [6, 8, 58], [6, 72, 16], [12, 72, 16]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': None, 'commutator_count': 1, 'commutator_label': '324.175', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 83, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 36, 1, 4], [2, 81, 1, 2], [2, 162, 1, 3], [3, 4, 1, 8], [3, 8, 1, 6], [4, 36, 1, 4], [4, 324, 1, 4], [6, 4, 1, 24], [6, 8, 1, 58], [6, 72, 1, 16], [12, 72, 1, 16]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 110990880, 'exponent': 12, 'exponents_of_order': [6, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 1, 24]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': None, 'hash': 5774763537450866102, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 4, 3, 6, 6, 3], 'inner_gens': [[1, 30, 8, 3032, 2784, 3472], [29, 2, 1768, 2464, 1096, 1760], [1, 3474, 8, 48, 288, 1728], [4265, 4834, 8, 48, 288, 1728], [4705, 1258, 8, 48, 288, 1728], [3489, 18, 8, 48, 288, 1728]], 'inner_hash': 3490905529419726075, 'inner_nilpotent': False, 'inner_order': 2592, 'inner_split': False, 'inner_tex': 'C_3^4.C_2^3.C_2^2', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 4], [4, 66], [8, 64]], 'label': '5184.de', '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*C6^3):D4', 'ngens': 10, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 21, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 150, 'number_divisions': 150, 'number_normal_subgroups': 111, 'number_subgroup_autclasses': 618, 'number_subgroup_classes': 7121, 'number_subgroups': 199734, 'old_label': None, 'order': 5184, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 799], [3, 80], [4, 1440], [6, 1712], [12, 1152]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 12, 2, 2, 2], 'outer_gen_pows': [888, 0, 28, 28, 28, 28, 28], 'outer_gens': [[1, 1010, 8, 48, 312, 1728], [1, 2, 8, 72, 288, 1728], [893, 146, 1288, 4672, 4896, 2304], [151, 890, 2920, 64, 368, 3072], [25, 26, 4152, 1200, 1632, 224], [1, 26, 4168, 4864, 1072, 3680], [29, 2, 3464, 4784, 992, 3472]], 'outer_group': '768.1088763', 'outer_hash': 1088763, 'outer_nilpotent': False, 'outer_order': 768, 'outer_permdeg': 12, 'outer_perms': [40279680, 87091200, 6, 216086419, 23, 7, 87102720], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^6:D_6', 'pc_rank': None, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 4], [4, 66], [8, 64]], 'representations': {'PC': {'code': '2618972289317599194892768114734787401973146863399294568669926620983804427156414022912431126574005597548227627', 'gens': [1, 2, 4, 6, 8, 10], 'pres': [10, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 240, 601, 51, 35373, 423, 113, 87214, 424, 181925, 73935, 3625, 175, 283366, 162416, 3386, 222727, 43857, 28827, 237, 319688, 5778, 25948, 347209, 88019, 86429]}, 'Perm': {'d': 20, 'gens': [135162536705244720, 262539180096762376, 733605320448030, 262540495466807177, 1087985089865520]}}, 'schur_multiplier': [2, 2, 2, 2, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 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_3\\times C_6^3):D_4', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
