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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1728.32842', 'ambient_counter': 32842, 'ambient_order': 1728, 'ambient_tex': 'C_6^2.(C_6\\times D_4)', 'central': False, 'central_factor': False, 'centralizer_order': 12, 'characteristic': False, 'core_order': 12, 'counter': 228, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1728.32842.36.a1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '36.a1.b1', 'outer_equivalence': False, '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': '48.45', 'subgroup_hash': 45, 'subgroup_order': 48, 'subgroup_tex': 'C_6\\times D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1728.32842', 'aut_centralizer_order': 8, 'aut_label': '36.a1', 'aut_quo_index': None, 'aut_stab_index': 18, 'aut_weyl_group': '32.46', 'aut_weyl_index': 144, 'centralizer': '144.j1.b1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['12.g1.b1', '18.a1.b1', '18.b1.b1', '18.c1.a1'], 'contains': ['72.b1.b1', '72.f1.a1', '72.x1.b1', '72.be1.b1', '72.bq1.b1', '108.a1.b1'], 'core': '144.a1.a1', 'coset_action_label': None, 'count': 9, 'diagramx': [5229, -1, 3028, -1, 4765, -1, 4228, -1], 'generators': [439, 864, 876, 936, 8], 'label': '1728.32842.36.a1.b1', 'mobius_quo': None, 'mobius_sub': -2, 'normal_closure': '4.a1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '9.a1.a1', 'old_label': '36.a1.b1', 'projective_image': '288.889', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '36.a1.b1', 'subgroup_fusion': None, 'weyl_group': '16.11'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '24.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 2, 2, 2, 2, 2, 2], 'aut_gens': [[1, 2, 4], [1, 2, 5], [25, 14, 44], [1, 38, 28], [1, 2, 20], [1, 27, 28], [1, 3, 4], [1, 26, 4]], 'aut_group': '128.1755', 'aut_hash': 1755, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 128, 'aut_permdeg': 10, 'aut_perms': [21025, 2810575, 374407, 1, 183120, 21288, 1270440], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 4, 1], [3, 1, 2, 1], [4, 2, 2, 1], [6, 1, 2, 1], [6, 1, 4, 1], [6, 2, 8, 1], [12, 2, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^4:D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '64.202', 'autcent_hash': 202, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3:D_4', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 4], [3, 1, 2], [4, 2, 2], [6, 1, 6], [6, 2, 8], [12, 2, 4]], 'center_label': '12.5', 'center_order': 12, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 45, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 4], [3, 1, 2, 1], [4, 2, 1, 2], [6, 1, 2, 3], [6, 2, 2, 4], [12, 2, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 273, 'exponent': 12, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '24.15', 'hash': 45, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [1, 2, 2], 'inner_gens': [[1, 2, 4], [1, 2, 28], [1, 26, 4]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 24], [2, 6]], 'label': '48.45', 'linC_count': 64, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 8, 'linQ_dim': 4, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6*D4', 'ngens': 5, 'nilpotency_class': 2, 'nilpotent': True, '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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 30, 'number_divisions': 20, 'number_normal_subgroups': 38, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 54, 'number_subgroups': 70, 'old_label': None, 'order': 48, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 11], [3, 2], [4, 4], [6, 22], [12, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [0, 4, 0, 0], 'outer_gens': [[1, 2, 20], [25, 27, 20], [25, 39, 20], [25, 14, 45]], 'outer_group': '32.46', 'outer_hash': 46, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [7, 23, 5183, 15136], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [4, 2]], 'representations': {'PC': {'code': 4858585473, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -2, -3, 217, 42, 58]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [35931072, 16563488, 7233423]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [34649835, 11532102, 34603218, 34591355, 34649828]}, 'Perm': {'d': 9, 'gens': [41040, 10824, 90744, 4, 90720]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6\\times D_4', 'transitive_degree': 24, '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': '24.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [12, 4, 2, 6, 2, 4, 2], 'aut_gens': [[1, 2, 24, 144], [1273, 422, 120, 1128], [37, 250, 408, 1128], [673, 490, 120, 1584], [1537, 118, 120, 1128], [1465, 1234, 120, 1584], [937, 1606, 120, 264], [109, 118, 408, 144]], 'aut_group': None, 'aut_hash': 2073078227855066378, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 4608, 'aut_permdeg': 50, 'aut_perms': [16625156418087568116571113096962271568410242872641939985508653955, 25354831028660692467477430922603328935642959535313837626889852264, 11313893963498666990674570472854408658870567325611904697308234956, 30209684710868022248851542306332935299165423069829158306798987831, 20151192419157081322251807553217262791327199386455905085314059667, 28334654872941916024197467381681396102055957034509160104062329896, 25064246211176107856084088165665050100854385379187559222282492364], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 12, 2, 1], [2, 18, 2, 1], [2, 36, 1, 1], [3, 1, 2, 1], [3, 4, 1, 2], [3, 4, 2, 2], [4, 4, 1, 1], [4, 12, 2, 2], [4, 24, 1, 1], [4, 72, 2, 1], [6, 1, 2, 1], [6, 2, 2, 1], [6, 4, 1, 4], [6, 4, 2, 4], [6, 8, 1, 1], [6, 8, 2, 1], [6, 12, 4, 1], [6, 18, 4, 1], [6, 24, 2, 1], [6, 24, 4, 1], [6, 36, 2, 1], [12, 4, 2, 2], [12, 4, 4, 1], [12, 8, 1, 1], [12, 8, 2, 2], [12, 8, 4, 1], [12, 12, 4, 2], [12, 24, 2, 4], [12, 24, 4, 3], [12, 72, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_2\\times D_6:D_6).C_2^4', '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': 12, 'autcentquo_group': '288.889', 'autcentquo_hash': 889, 'autcentquo_nilpotent': False, 'autcentquo_order': 288, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'D_6\\wr C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 12, 2], [2, 18, 2], [2, 36, 1], [3, 1, 2], [3, 4, 6], [4, 4, 1], [4, 12, 4], [4, 24, 1], [4, 72, 2], [6, 1, 2], [6, 2, 2], [6, 4, 12], [6, 8, 3], [6, 12, 4], [6, 18, 4], [6, 24, 6], [6, 36, 2], [12, 4, 8], [12, 8, 9], [12, 12, 8], [12, 24, 20], [12, 72, 4]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '288.889', 'commutator_count': 1, 'commutator_label': '72.49', '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'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 32842, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['576.5297', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 12, 1, 2], [2, 18, 1, 2], [2, 36, 1, 1], [3, 1, 2, 1], [3, 4, 1, 2], [3, 4, 2, 2], [4, 4, 1, 1], [4, 12, 1, 2], [4, 12, 2, 1], [4, 24, 1, 1], [4, 72, 1, 2], [6, 1, 2, 1], [6, 2, 2, 1], [6, 4, 1, 4], [6, 4, 2, 4], [6, 8, 1, 1], [6, 8, 2, 1], [6, 12, 2, 2], [6, 18, 2, 2], [6, 24, 1, 2], [6, 24, 2, 2], [6, 36, 2, 1], [12, 4, 2, 2], [12, 4, 4, 1], [12, 8, 1, 1], [12, 8, 2, 2], [12, 8, 4, 1], [12, 12, 2, 2], [12, 12, 4, 1], [12, 24, 1, 2], [12, 24, 2, 7], [12, 24, 4, 1], [12, 72, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 349440, 'exponent': 12, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 0, 8], [8, 0, 6]], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '432.754', 'hash': 32842, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 4, 6, 6], 'inner_gens': [[1, 878, 24, 1608], [877, 2, 408, 1128], [1, 1490, 24, 144], [409, 890, 24, 144]], 'inner_hash': 889, 'inner_nilpotent': False, 'inner_order': 288, 'inner_split': True, 'inner_tex': 'D_6\\wr C_2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 4, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 8, 'irrep_stats': [[1, 24], [2, 18], [4, 54], [8, 12]], 'label': '1728.32842', 'linC_count': 8, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 8, 'linQ_dim': 14, 'linQ_dim_count': 96, 'linR_count': 4, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C6^2.(C6*D4)', '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, 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': 50, 'number_characteristic_subgroups': 34, 'number_conjugacy_classes': 108, 'number_divisions': 63, 'number_normal_subgroups': 62, 'number_subgroup_autclasses': 450, 'number_subgroup_classes': 630, 'number_subgroups': 4756, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 99], [3, 26], [4, 220], [6, 414], [12, 968]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 936], 'outer_gens': [[865, 2, 24, 144], [1, 2, 24, 1008], [1, 10, 24, 144], [937, 506, 120, 720]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 25, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 14], [4, 18], [8, 15], [16, 7], [32, 1]], 'representations': {'PC': {'code': 294051830508081354170695613644852833158281399511968576684060643656038206385705, 'gens': [1, 2, 5, 7], 'pres': [9, 2, 2, 2, 3, 2, 3, 2, 2, 3, 7776, 15805, 46, 74, 9193, 1372, 130, 18158, 1319, 101310, 35547, 24972, 186, 107143, 13840, 25945, 214, 101096, 25289, 23354]}, 'Perm': {'d': 25, 'gens': [628421542361789143161736, 1374977352166435658995221, 1948059198073120809600000, 45360, 2569517761302938246169623, 2647329283975505718758400, 3294774614019626683545600, 311, 56]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_6^2.(C_6\\times D_4)', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
