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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '2592.cm', 'ambient_counter': 65, 'ambient_order': 2592, 'ambient_tex': 'S_3^2\\wr C_2', 'central': False, 'central_factor': False, 'centralizer_order': 9, 'characteristic': False, 'core_order': 81, 'counter': 57, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '2592.cm.16.d1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '16.d1', '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': 16, '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': '162.52', 'subgroup_hash': 52, 'subgroup_order': 162, 'subgroup_tex': 'C_3^2\\wr C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '2592.cm', 'aut_centralizer_order': 9, 'aut_label': '16.d1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '288.889', 'aut_weyl_index': 36, 'centralizer': '288.e1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['8.d1', '8.f1', '8.h1', '8.i1', '8.l1'], 'contains': ['32.a1', '48.k1', '48.l1', '48.m1', '48.n1'], 'core': '32.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [2496, -1, 639, -1], 'generators': [166306490, 15214597, 29606670, 167265, 1498112], 'label': '2592.cm.16.d1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '8.d1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.c1', 'old_label': '16.d1', 'projective_image': '2592.cm', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '16.d1', 'subgroup_fusion': None, 'weyl_group': '144.192'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '18.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [3, 2, 3, 3, 3, 2], 'aut_gens': [[736, 2918, 757, 1027], [736, 3431, 757, 1027], [736, 2918, 784, 973], [736, 2972, 757, 1027], [736, 2918, 973, 1027], [5107, 5111, 757, 1027], [733, 2918, 757, 1027]], 'aut_group': '20736.pd', 'aut_hash': 2986519587559918118, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20736, 'aut_permdeg': 17, 'aut_perms': [176625221068800, 4197015660316, 25113582144000, 5331894180480, 37249, 10855], 'aut_phi_ratio': 384.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 1, 8, 1], [3, 2, 4, 1], [3, 2, 32, 1], [6, 9, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^2:\\GL(2,3)\\times \\GL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '48.29', 'autcent_hash': 29, 'autcent_nilpotent': False, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(2,3)', '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, 9, 1], [3, 1, 8], [3, 2, 36], [6, 9, 8]], 'center_label': '9.2', 'center_order': 9, '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', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 52, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['18.4', 1], ['3.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [3, 1, 2, 4], [3, 2, 1, 4], [3, 2, 2, 16], [6, 9, 2, 4]], 'element_repr_type': 'GLZq', 'elementary': 1, 'eulerian_function': 91, 'exponent': 6, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '162.52', 'hash': 52, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [1, 2, 3, 3], 'inner_gens': [[736, 2918, 757, 1027], [736, 2918, 784, 1243], [736, 2945, 757, 1027], [736, 3458, 757, 1027]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 18, 'inner_split': False, 'inner_tex': 'C_3:S_3', 'inner_used': [2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 18], [2, 36]], 'label': '162.52', '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': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^2wrC2', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 54, 'number_divisions': 30, 'number_normal_subgroups': 42, 'number_subgroup_autclasses': 23, 'number_subgroup_classes': 160, 'number_subgroups': 344, 'old_label': None, 'order': 162, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 9], [3, 80], [6, 72]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [2, 6, 2, 6, 4, 4], 'outer_gen_pows': [730, 730, 2918, 2918, 730, 730], 'outer_gens': [[733, 5108, 1243, 784], [733, 2921, 1243, 1270], [736, 2918, 1216, 1000], [5110, 731, 1243, 757], [5104, 5105, 757, 1027], [5107, 731, 757, 1027]], 'outer_group': '1152.157461', 'outer_hash': 7371672288863537087, 'outer_nilpotent': False, 'outer_order': 1152, 'outer_permdeg': 12, 'outer_perms': [1, 18594004, 23, 7711327, 303055320, 44352840], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'S_4\\times \\GL(2,3)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 12], [4, 16]], 'representations': {'PC': {'code': 14018986071, 'gens': [1, 2, 4, 5], 'pres': [5, -3, -2, -3, -3, -3, 26, 248, 909]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [24577758, 15477644, 24507118, 2939606, 36731395]}, 'GLZq': {'d': 2, 'q': 9, 'gens': [757, 733, 3220, 737, 2920]}, 'Perm': {'d': 12, 'gens': [40723336, 91451125, 325, 435, 130682595]}}, 'schur_multiplier': [3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^2\\wr C_2', 'transitive_degree': 18, 'wreath_data': ['C_3^2', 'C_2', '2T1'], 'wreath_product': True}
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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': 4, 'aut_exponent': 24, 'aut_gen_orders': [3, 2, 4, 2], 'aut_gens': [[167265, 29606670, 131864130, 245419970], [167265, 15210873, 442321936, 245419970], [14926470, 15210873, 60015573, 245419970], [340708470, 427427072, 383201968, 36946503], [340708470, 427427072, 131864130, 245419970]], 'aut_group': '10368.bd', 'aut_hash': 7501325464046845939, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10368, 'aut_permdeg': 48, 'aut_perms': [6333261748170151247379397101807792107478361906718460509809989, 6425926566075662311587714394739769189392980018070586624886128, 2594874863871791810134145702583520901830131244665749372948078, 332935361180815704243733501997502175618076941300232], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 6, 2, 1], [2, 9, 2, 1], [2, 18, 2, 1], [2, 36, 1, 1], [2, 54, 2, 1], [2, 81, 1, 1], [3, 4, 2, 2], [3, 8, 2, 1], [3, 16, 1, 1], [3, 16, 2, 1], [4, 108, 2, 1], [4, 324, 1, 1], [6, 12, 2, 1], [6, 12, 4, 1], [6, 24, 2, 1], [6, 24, 4, 1], [6, 36, 2, 2], [6, 36, 4, 1], [6, 48, 2, 1], [6, 72, 2, 2], [6, 108, 2, 1], [6, 144, 1, 1], [12, 216, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\wr D_4', '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': '10368.bd', 'autcentquo_hash': 7501325464046845939, 'autcentquo_nilpotent': False, 'autcentquo_order': 10368, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\wr D_4', 'cc_stats': [[1, 1, 1], [2, 6, 2], [2, 9, 2], [2, 18, 2], [2, 36, 1], [2, 54, 2], [2, 81, 1], [3, 4, 4], [3, 8, 2], [3, 16, 3], [4, 108, 2], [4, 324, 1], [6, 12, 6], [6, 24, 6], [6, 36, 8], [6, 48, 2], [6, 72, 4], [6, 108, 2], [6, 144, 1], [12, 216, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '2592.cm', 'commutator_count': 1, 'commutator_label': '324.169', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 65, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 6, 1, 2], [2, 9, 1, 2], [2, 18, 1, 2], [2, 36, 1, 1], [2, 54, 1, 2], [2, 81, 1, 1], [3, 4, 1, 4], [3, 8, 1, 2], [3, 16, 1, 3], [4, 108, 1, 2], [4, 324, 1, 1], [6, 12, 1, 6], [6, 24, 1, 6], [6, 36, 1, 8], [6, 48, 1, 2], [6, 72, 1, 4], [6, 108, 1, 2], [6, 144, 1, 1], [12, 216, 1, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 537600, 'exponent': 12, 'exponents_of_order': [5, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 1, 8], [16, 1, 6]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '2592.cm', 'hash': 6952908971992144288, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [3, 2, 4, 2], 'inner_gens': [[167265, 29439589, 295503684, 245419970], [328449, 29606670, 281164389, 245419970], [1498112, 163699712, 131864130, 22392373], [167265, 29606670, 395176320, 245419970]], 'inner_hash': 6952908971992144288, 'inner_nilpotent': False, 'inner_order': 2592, 'inner_split': False, 'inner_tex': 'S_3^2\\wr C_2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 6], [4, 24], [8, 10], [16, 6]], 'label': '2592.cm', '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': 'S3^2wrC2', 'ngens': 4, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 27, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 54, 'number_divisions': 54, 'number_normal_subgroups': 33, 'number_subgroup_autclasses': 438, 'number_subgroup_classes': 1029, 'number_subgroups': 24844, 'old_label': None, 'order': 2592, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 291], [3, 80], [4, 540], [6, 1248], [12, 432]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[340708470, 427427072, 131864130, 245419970], [167265, 29606670, 120996825, 244304654]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 24], [8, 10], [16, 6]], 'representations': {'PC': {'code': '37432814164892359578174990475578987297335169041003976436514570902348843', 'gens': [1, 2, 4, 5, 7, 9], 'pres': [9, 2, 2, 2, 3, 2, 3, 2, 3, 3, 109, 46, 579, 1740, 165, 48973, 1372, 130, 230, 1319, 65787, 1932, 186, 31120, 1771, 11681, 1034]}, 'Perm': {'d': 12, 'gens': [245419970, 167265, 29606670, 131864130]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'S_3^2\\wr C_2', 'transitive_degree': 12, 'wreath_data': ['72.bg1', '1296.d1', '2592.a1', '2T1'], 'wreath_product': True}
