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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '768.1084932', 'ambient_counter': 1084932, 'ambient_order': 768, 'ambient_tex': 'C_2^4:C_{48}', 'central': False, 'central_factor': False, 'centralizer_order': 256, 'characteristic': False, 'core_order': 16, 'counter': 53, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '768.1084932.24.h1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '24.h1', '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': 24, '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': '32.36', 'subgroup_hash': 36, 'subgroup_order': 32, 'subgroup_tex': 'C_2^2\\times C_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '768.1084932', 'aut_centralizer_order': 256, 'aut_label': '24.h1', 'aut_quo_index': None, 'aut_stab_index': 9, 'aut_weyl_group': '8.5', 'aut_weyl_index': 2304, 'centralizer': '3.a1', 'complements': None, 'conjugacy_class_count': 3, 'contained_in': ['12.a1', '12.f1', '12.g1', '12.h1', '12.l1'], 'contains': ['48.b1', '48.i1', '48.q1', '48.x1', '48.y1', '48.bb1', '48.bc1'], 'core': '48.b1', 'coset_action_label': None, 'count': 9, 'diagramx': [8074, -1, 8077, -1], 'generators': [49681, 557585, 697117], 'label': '768.1084932.24.h1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '12.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1', 'old_label': '24.h1', 'projective_image': '48.49', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '24.h1', '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': '32.36', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 3, 2, 2, 2, 2, 2], 'aut_gens': [[1, 2, 4], [3, 2, 4], [1, 2, 28], [2, 3, 4], [1, 2, 6], [1, 2, 7], [1, 18, 4], [17, 2, 4], [1, 2, 20]], 'aut_group': '384.20089', 'aut_hash': 20089, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 384, 'aut_permdeg': 10, 'aut_perms': [374407, 1, 823080, 2810574, 2003886, 367920, 5160, 368046], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [4, 1, 2, 1], [4, 1, 6, 1], [8, 1, 16, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4:S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '384.20089', 'autcent_hash': 20089, 'autcent_nilpotent': False, 'autcent_order': 384, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^4:S_4', '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], [4, 1, 8], [8, 1, 16]], 'center_label': '32.36', 'center_order': 32, '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', '2.1', '2.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 36, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['8.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [4, 1, 2, 4], [8, 1, 4, 4]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 28, 'exponent': 8, 'exponents_of_order': [5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '8.5', 'hash': 36, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 4]], '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, 32]], 'label': '32.36', 'linC_count': 1792, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 48, 'linQ_dim': 6, 'linQ_dim_count': 48, 'linR_count': 96, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2*C8', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 32, 'number_divisions': 16, 'number_normal_subgroups': 38, 'number_subgroup_autclasses': 14, 'number_subgroup_classes': 38, 'number_subgroups': 38, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7], [4, 8], [8, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[3, 2, 4], [1, 2, 28], [2, 3, 4], [1, 2, 6], [1, 2, 7], [1, 18, 4], [17, 2, 4], [1, 2, 20]], 'outer_group': '384.20089', 'outer_hash': 20089, 'outer_nilpotent': False, 'outer_order': 384, 'outer_permdeg': 10, 'outer_perms': [374407, 1, 823080, 2810574, 2003886, 367920, 5160, 368046], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^4:S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 12, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 8], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 4]], 'representations': {'PC': {'code': 557068, 'gens': [1, 2, 3], 'pres': [5, -2, 2, 2, -2, -2, 42, 58]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [25038659807093174, 125101739941717066, 24992906222183252]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [34876415, 3397304, 28816400, 3054512, 28233362]}, 'Perm': {'d': 12, 'gens': [40176, 362880, 39916800, 16582, 5167]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 8], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times C_8', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '192.935', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 12, 'aut_gen_orders': [12, 4, 4, 4, 4], 'aut_gens': [[648556, 1032719, 573457, 49665, 453063], [974004, 49681, 49665, 557585, 907345], [140653, 573953, 49665, 573457, 236109], [957620, 1032719, 573457, 49665, 175055], [664957, 1032719, 557585, 49665, 907345], [664957, 1032719, 557585, 49665, 43979]], 'aut_group': None, 'aut_hash': 7474513708491177873, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 18432, 'aut_permdeg': 40, 'aut_perms': [809423928496687167271487085757255527994810877675, 295345899726570506854301515652037996798773058441, 180323696592935524080812416574114644051812565223, 75645435193711474216406918532443387415632573939, 67180695819489304252814337793113993042085955482], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [2, 3, 1, 2], [2, 3, 6, 1], [3, 4, 2, 1], [4, 1, 2, 1], [4, 1, 6, 1], [4, 3, 2, 1], [4, 3, 6, 1], [6, 4, 2, 1], [6, 4, 12, 1], [8, 1, 4, 1], [8, 1, 12, 1], [8, 3, 4, 1], [8, 3, 12, 1], [12, 4, 4, 1], [12, 4, 12, 1], [16, 1, 32, 1], [16, 3, 32, 1], [24, 4, 8, 1], [24, 4, 24, 1], [48, 4, 64, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2.C_2^6.D_6^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '768.1090215', 'autcent_hash': 1090215, 'autcent_nilpotent': False, 'autcent_order': 768, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '(C_2^3\\times C_4):S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 3, 8], [3, 4, 2], [4, 1, 8], [4, 3, 8], [6, 4, 14], [8, 1, 16], [8, 3, 16], [12, 4, 16], [16, 1, 32], [16, 3, 32], [24, 4, 32], [48, 4, 64]], 'center_label': '64.183', 'center_order': 64, 'central_product': True, 'central_quotient': '12.3', 'commutator_count': 1, 'commutator_label': '4.2', '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', '2.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 1084932, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['16.1', 1], ['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 3, 1, 8], [3, 4, 2, 1], [4, 1, 2, 4], [4, 3, 2, 4], [6, 4, 2, 7], [8, 1, 4, 4], [8, 3, 4, 4], [12, 4, 4, 4], [16, 1, 8, 4], [16, 3, 8, 4], [24, 4, 8, 4], [48, 4, 16, 4]], 'element_repr_type': 'GLZq', 'elementary': 1, 'eulerian_function': 7280, 'exponent': 48, 'exponents_of_order': [8, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.1', 'frattini_quotient': '96.228', 'hash': 1084932, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 1, 2, 2, 1], 'inner_gens': [[648556, 1032719, 557585, 573457, 453063], [648556, 1032719, 573457, 49665, 453063], [107900, 1032719, 573457, 49665, 453063], [632684, 1032719, 573457, 49665, 453063], [648556, 1032719, 573457, 49665, 453063]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'A_4', 'inner_used': [1, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 192], [3, 64]], 'label': '768.1084932', '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': 'C2^4:C48', 'ngens': 9, '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': 24, 'number_characteristic_subgroups': 24, 'number_conjugacy_classes': 256, 'number_divisions': 60, 'number_normal_subgroups': 147, 'number_subgroup_autclasses': 176, 'number_subgroup_classes': 595, 'number_subgroups': 1540, 'old_label': None, 'order': 768, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 31], [3, 8], [4, 32], [6, 56], [8, 64], [12, 64], [16, 128], [24, 128], [48, 256]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 3, 2, 2, 4, 2, 2, 2, 2, 2], 'outer_gen_pows': [32769, 32769, 32769, 32769, 32769, 32769, 32769, 32769, 32769, 32769], 'outer_gens': [[157053, 1032719, 49665, 573457, 453063], [433844, 1015839, 573457, 49665, 437207], [648556, 1032719, 573457, 49665, 321987], [416931, 1032719, 49665, 573457, 453063], [416931, 1032719, 49665, 573457, 190943], [648556, 1032719, 573457, 49665, 437207], [648556, 1032719, 573457, 49665, 645193], [108396, 508447, 573457, 49665, 629337], [108396, 1032719, 573457, 49665, 120921], [648556, 1032719, 573457, 49665, 977367]], 'outer_group': '1536.408641001', 'outer_hash': 5942603984477037536, 'outer_nilpotent': False, 'outer_order': 1536, 'outer_permdeg': 20, 'outer_perms': [122007708013328040, 11212469452800, 23, 122002101778601656, 377744146144662983, 513696337505971200, 1541100224635776000, 1149404675007012480, 1663101012472709160, 122002101778601640], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '(C_2^4\\times C_4):S_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 16, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [3, 8], [4, 8], [6, 4], [8, 8], [12, 4], [16, 4], [24, 4]], 'representations': {'PC': {'code': '45511043742870652668070063131177108096', 'gens': [1, 3, 4, 5, 6], 'pres': [9, -2, -3, -2, -2, 2, -2, -2, -2, -2, 18, 1299, 444, 544, 823, 158, 186, 214]}, 'GLZN': {'d': 2, 'p': 28, 'gens': [340649, 214622, 375039, 22345, 26725, 124067, 33321, 33335, 285389]}, 'GLZq': {'d': 2, 'q': 32, 'gens': [32785, 294921, 124284, 33281, 245767, 557073, 573953, 713485, 453063]}, 'Perm': {'d': 24, 'gens': [181, 28260974647749700644480, 82, 56212811007499316626109, 56212811007499316625920, 56212811007499316625840, 83490668053409500934400, 110527028898728529830400, 56212811007499316620800]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 48], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^4:C_{48}', 'transitive_degree': 192, 'wreath_data': None, 'wreath_product': False}
