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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '972.773', 'ambient_counter': 773, 'ambient_order': 972, 'ambient_tex': '(C_3^2\\times \\He_3):C_4', 'central': False, 'central_factor': False, 'centralizer_order': 27, 'characteristic': False, 'core_order': 3, 'counter': 24, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '972.773.36.f1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '36.f1', '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': 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': '27.3', 'subgroup_hash': 3, 'subgroup_order': 27, 'subgroup_tex': '\\He_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '972.773', 'aut_centralizer_order': 9, 'aut_label': '36.f1', 'aut_quo_index': None, 'aut_stab_index': 48, 'aut_weyl_group': '144.182', 'aut_weyl_index': 432, 'centralizer': '36.a1', 'complements': None, 'conjugacy_class_count': 12, 'contained_in': ['12.b1'], 'contains': ['108.d1'], 'core': '324.a1', 'coset_action_label': None, 'count': 48, 'diagramx': [2112, -1, 9832, -1], 'generators': [148, 444], 'label': '972.773.36.f1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.a1', 'old_label': '36.f1', 'projective_image': '324.163', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '36.f1', 'subgroup_fusion': None, 'weyl_group': '9.2'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '9.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [2, 3, 3], 'aut_gens': [[1, 3, 9], [2, 23, 18], [3, 17, 9], [1, 12, 9]], 'aut_group': '432.734', 'aut_hash': 734, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 432, 'aut_permdeg': 9, 'aut_perms': [5354, 17245, 45708], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 3, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^2:\\GL(2,3)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 3, 'autcent_group': '9.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 9, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '48.29', 'autcentquo_hash': 29, 'autcentquo_nilpotent': False, 'autcentquo_order': 48, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\GL(2,3)', 'cc_stats': [[1, 1, 1], [3, 1, 2], [3, 3, 8]], 'center_label': '3.1', 'center_order': 3, 'central_product': False, 'central_quotient': '9.2', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 3, 2, 4]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [[3, 0, 2]], 'familial': True, 'frattini_label': '3.1', 'frattini_quotient': '9.2', 'hash': 3, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [3, 3, 1], 'inner_gens': [[1, 21, 9], [10, 3, 9], [1, 3, 9]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 9, 'inner_split': True, 'inner_tex': 'C_3^2', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 6, 'irrep_stats': [[1, 9], [3, 2]], 'label': '27.3', 'linC_count': 2, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'He3', 'ngens': 2, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 11, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 5, 'number_subgroup_classes': 11, 'number_subgroups': 19, 'old_label': None, 'order': 27, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 26]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 25, 18], [13, 21, 9]], 'outer_group': '48.29', 'outer_hash': 29, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 8, 'outer_perms': [31834, 28334], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,3)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 3, 'primary_abelian_invariants': [3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4], [6, 1]], 'representations': {'PC': {'code': 68, 'gens': [1, 2, 3], 'pres': [3, -3, 3, -3, 127]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [74589221202652867, 122615752897794827]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [7426, 11017, 8101]}, 'Perm': {'d': 9, 'gens': [138838, 276480, 80884]}}, 'schur_multiplier': [3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '\\He_3', 'transitive_degree': 9, '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': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [8, 12, 12, 8], 'aut_gens': [[1, 4, 12, 36, 108, 324], [573, 32, 28, 108, 396, 324], [843, 24, 8, 144, 900, 324], [475, 32, 4, 936, 504, 324], [617, 32, 28, 144, 828, 648]], 'aut_group': None, 'aut_hash': 6992198563583357107, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 62208, 'aut_permdeg': 162, 'aut_perms': [70714416648274100620318912991534766029851674240456393636608234767151833213848655600149202743145142303020109997131983213730827484230417899936832668470298138376034258251777158700331119215133554955743528304425631260254260621822665505602810618113102181274146255605157467884582858309524078686, 6109508344564060574560825203326477156750809750975276834828859904349688807216683138042769279130765436717961499116628548274398313984432636134750438644606805874342206307449539515313708944046316039215120199674226803014894951920721227374246720810887035434681478849566932688279626824925298415739, 10370951103015595208506217915940071460731980409840721063283744660366713927197359697502796804492708900381944827715870359519716664772152782903929594049268806014463907082231390951634059128656509475853811938959078196868360008709681851545356988178711666304131687905063529252019280355510963155079, 11437322260263483596542393686352694249752034169350528935684078312338775910613544985420122921812722843533409023287917065230699822550890871283713991392277374063096893865903506309806911251837888496310203196269111823792100002054783811628252437079943821471323386290166632928762660827267938724119], 'aut_phi_ratio': 192.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 1, 2, 1], [3, 2, 4, 1], [3, 2, 8, 1], [3, 12, 2, 1], [3, 12, 16, 1], [4, 81, 2, 1], [6, 9, 2, 1], [6, 18, 4, 1], [6, 18, 8, 1], [12, 81, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3:S_3.C_6^2.C_{12}.C_2^3', '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': 24, 'autcentquo_group': None, 'autcentquo_hash': 6992198563583357107, 'autcentquo_nilpotent': False, 'autcentquo_order': 62208, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3:S_3.C_6^2.C_{12}.C_2^3', 'cc_stats': [[1, 1, 1], [2, 9, 1], [3, 1, 2], [3, 2, 12], [3, 12, 18], [4, 81, 2], [6, 9, 2], [6, 18, 12], [12, 81, 4]], 'center_label': '3.1', 'center_order': 3, 'central_product': False, 'central_quotient': '324.163', 'commutator_count': 2, 'commutator_label': '243.62', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 773, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 1, 2, 1], [3, 2, 1, 4], [3, 2, 2, 4], [3, 12, 1, 2], [3, 12, 2, 8], [4, 81, 2, 1], [6, 9, 2, 1], [6, 18, 1, 4], [6, 18, 2, 4], [12, 81, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 7560, 'exponent': 12, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '324.163', 'hash': 773, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [4, 3, 3, 3, 3, 1], 'inner_gens': [[1, 8, 24, 864, 684, 324], [9, 4, 12, 36, 108, 324], [25, 4, 12, 36, 108, 324], [145, 4, 12, 36, 756, 324], [505, 4, 12, 360, 108, 324], [1, 4, 12, 36, 108, 324]], 'inner_hash': 163, 'inner_nilpotent': False, 'inner_order': 324, 'inner_split': False, 'inner_tex': 'C_3^4:C_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4], [2, 8], [3, 8], [4, 18], [6, 16]], 'label': '972.773', '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': '(C3^2*He3):C4', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 12, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 54, 'number_divisions': 32, 'number_normal_subgroups': 25, 'number_subgroup_autclasses': 55, 'number_subgroup_classes': 258, 'number_subgroups': 1894, 'old_label': None, 'order': 972, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 9], [3, 242], [4, 162], [6, 234], [12, 324]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [4, 3, 4, 2, 4, 2, 4], 'outer_gen_pows': [0, 0, 0, 2, 2, 0, 0], 'outer_gens': [[3, 28, 32, 864, 396, 648], [1, 4, 20, 36, 108, 324], [3, 16, 28, 180, 468, 324], [1, 4, 12, 432, 720, 324], [1, 4, 12, 180, 612, 648], [1, 8, 20, 36, 108, 324], [3, 12, 8, 864, 396, 648]], 'outer_group': '192.1485', 'outer_hash': 1485, 'outer_nilpotent': False, 'outer_order': 192, 'outer_permdeg': 16, 'outer_perms': [15372953209893, 17046952929630, 13492302546092, 8415470267082, 2989740787158, 1056896801802, 4130750993369], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,3):C_2^2', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 9], [4, 2], [6, 2], [8, 8], [12, 9]], 'representations': {'PC': {'code': 28223343566055319069038139916447, 'gens': [1, 3, 4, 5, 6, 7], 'pres': [7, -2, -2, -3, -3, -3, 3, -3, 14, 170, 675, 30244, 1271, 28733, 11352, 915]}, 'Perm': {'d': 21, 'gens': [2474978530955979121, 45199173461268984, 538558147670719680, 5535021080578964424, 3, 8093680118674982907, 10512752162793231864]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '(C_3^2\\times \\He_3):C_4', 'transitive_degree': 54, 'wreath_data': None, 'wreath_product': False}
