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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '54000.c', 'ambient_counter': 3, 'ambient_order': 54000, 'ambient_tex': 'D_5^3:C_3^2:S_3', 'central': False, 'central_factor': False, 'centralizer_order': 3, 'characteristic': False, 'core_order': 750, 'counter': 69, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '54000.c.24.h1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, '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': '2250.g', 'subgroup_hash': 3638640786514561450, 'subgroup_order': 2250, 'subgroup_tex': 'C_3\\times C_5^3:C_6', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '54000.c', 'aut_centralizer_order': None, 'aut_label': '24.h1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '18000.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['6.j1', '8.b1'], 'contains': ['48.c1', '72.a1', '72.k1', '120.u1', '600.u1'], 'core': '72.a1', 'coset_action_label': None, 'count': 8, 'diagramx': None, 'generators': [5406, 10800, 7440, 10872, 7202, 45360], 'label': '54000.c.24.h1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '8.b1', 'old_label': '24.h1', 'projective_image': '54000.c', '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': '2250.g'}
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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': 3, 'aut_exponent': 120, 'aut_gen_orders': [4, 4, 12, 8, 8], 'aut_gens': [[1, 6, 30, 150], [311, 1038, 60, 1050], [575, 1842, 1890, 300], [1121, 126, 132, 300], [1871, 966, 588, 300], [107, 1398, 534, 2100]], 'aut_group': None, 'aut_hash': 4315712504607614885, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 144000, 'aut_permdeg': 275, 'aut_perms': 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'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': '24000.bf', 'autcentquo_hash': 5630595401936538143, 'autcentquo_nilpotent': False, 'autcentquo_order': 24000, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\times F_{25}:C_2', 'cc_stats': [[1, 1, 1], [2, 125, 1], [3, 1, 2], [3, 25, 6], [5, 2, 2], [5, 6, 20], [6, 125, 8], [15, 2, 4], [15, 6, 40], [15, 50, 12]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '750.30', 'commutator_count': 1, 'commutator_label': '125.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '5.1', '5.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 7, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['750.30', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 125, 1, 1], [3, 1, 2, 1], [3, 25, 2, 3], [5, 2, 2, 1], [5, 6, 2, 10], [6, 125, 2, 4], [15, 2, 4, 1], [15, 6, 4, 10], [15, 50, 4, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 30, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[6, 0, 32]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '2250.g', 'hash': 3638640786514561450, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [6, 5, 5, 5], 'inner_gens': [[1, 1434, 1416, 600], [1003, 6, 30, 150], [1045, 6, 30, 150], [1801, 6, 30, 150]], 'inner_hash': 30, 'inner_nilpotent': False, 'inner_order': 750, 'inner_split': False, 'inner_tex': 'C_5^3:C_6', 'inner_used': [1, 2], 'irrC_degree': 6, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 12, 'irrep_stats': [[1, 18], [2, 18], [6, 60]], 'label': '2250.g', '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*C5^3:C6', 'ngens': 6, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 13, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 96, 'number_divisions': 35, 'number_normal_subgroups': 22, 'number_subgroup_autclasses': 48, 'number_subgroup_classes': 128, 'number_subgroups': 3072, 'old_label': None, 'order': 2250, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 125], [3, 152], [5, 124], [6, 1000], [15, 848]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [2, 2, 8, 6], 'outer_gen_pows': [0, 4, 2, 3], 'outer_gens': [[5, 102, 114, 2100], [751, 1452, 594, 1650], [1, 1044, 12, 1950], [1501, 468, 1890, 1050]], 'outer_group': '192.1302', 'outer_hash': 1302, 'outer_nilpotent': False, 'outer_order': 192, 'outer_permdeg': 13, 'outer_perms': [18593280, 1654863846, 2482911360, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6\\times \\OD_{16}', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 8], [4, 1], [8, 4], [12, 10], [24, 10]], 'representations': {'PC': {'code': '95391130413901881391604041793737207', 'gens': [1, 3, 4, 5], 'pres': [6, -2, -3, -5, 5, -3, -5, 12, 25814, 710, 33987, 5841, 18004, 118, 64805]}, 'Perm': {'d': 18, 'gens': [757494103970403, 400411340584080]}}, 'schur_multiplier': [15], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 6], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times C_5^3:C_6', 'transitive_degree': 45, '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': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 60, 'aut_gen_orders': [12, 60, 12, 30], 'aut_gens': [[1, 6, 12, 360, 10800], [30689, 5442, 2064, 39780, 4320], [40031, 1260, 17580, 9594, 288], [28075, 14220, 20640, 36114, 144], [5681, 6702, 6042, 41472, 43200]], 'aut_group': None, 'aut_hash': 8717420453910764587, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 648000, 'aut_permdeg': 624, 'aut_perms': 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'aut_phi_ratio': 45.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 15, 1, 1], [2, 75, 1, 1], [2, 125, 1, 1], [2, 135, 1, 1], [2, 675, 1, 1], [2, 1125, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 300, 2, 1], [3, 600, 2, 1], [5, 6, 2, 1], [5, 8, 2, 1], [5, 12, 2, 2], [5, 24, 2, 1], [6, 30, 1, 1], [6, 30, 3, 1], [6, 150, 1, 1], [6, 150, 3, 1], [6, 250, 1, 1], [6, 750, 1, 1], [6, 900, 2, 1], [6, 1500, 2, 1], [6, 3000, 2, 1], [6, 4500, 2, 1], [10, 30, 4, 1], [10, 54, 2, 1], [10, 60, 2, 2], [10, 72, 2, 1], [10, 108, 2, 2], [10, 150, 2, 1], [10, 216, 2, 1], [10, 270, 4, 1], [10, 540, 2, 2], [10, 1350, 2, 1], [15, 12, 2, 1], [15, 12, 6, 1], [15, 16, 2, 1], [15, 24, 2, 2], [15, 24, 6, 2], [15, 48, 2, 2], [15, 48, 6, 1], [15, 600, 4, 1], [15, 1200, 4, 1], [30, 60, 4, 1], [30, 60, 12, 1], [30, 120, 2, 2], [30, 120, 6, 2], [30, 300, 2, 1], [30, 300, 6, 1], [30, 1800, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_5^3.C_6^2.(C_4\\times S_3^2)', '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': 60, 'autcentquo_group': None, 'autcentquo_hash': 8717420453910764587, 'autcentquo_nilpotent': False, 'autcentquo_order': 648000, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_5^3.C_6^2.(C_4\\times S_3^2)', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 15, 1], [2, 75, 1], [2, 125, 1], [2, 135, 1], [2, 675, 1], [2, 1125, 1], [3, 2, 1], [3, 6, 1], [3, 300, 2], [3, 600, 2], [5, 6, 2], [5, 8, 2], [5, 12, 4], [5, 24, 2], [6, 30, 4], [6, 150, 4], [6, 250, 1], [6, 750, 1], [6, 900, 2], [6, 1500, 2], [6, 3000, 2], [6, 4500, 2], [10, 30, 4], [10, 54, 2], [10, 60, 4], [10, 72, 2], [10, 108, 4], [10, 150, 2], [10, 216, 2], [10, 270, 4], [10, 540, 4], [10, 1350, 2], [15, 12, 8], [15, 16, 2], [15, 24, 16], [15, 48, 10], [15, 600, 4], [15, 1200, 4], [30, 60, 16], [30, 120, 16], [30, 300, 8], [30, 1800, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '54000.c', 'commutator_count': 1, 'commutator_label': None, 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '5.1', '5.1', '5.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 15, 1, 1], [2, 75, 1, 1], [2, 125, 1, 1], [2, 135, 1, 1], [2, 675, 1, 1], [2, 1125, 1, 1], [3, 2, 1, 1], [3, 6, 1, 1], [3, 300, 2, 1], [3, 600, 2, 1], [5, 6, 2, 1], [5, 8, 2, 1], [5, 12, 2, 2], [5, 24, 2, 1], [6, 30, 1, 4], [6, 150, 1, 4], [6, 250, 1, 1], [6, 750, 1, 1], [6, 900, 2, 1], [6, 1500, 2, 1], [6, 3000, 2, 1], [6, 4500, 2, 1], [10, 30, 2, 2], [10, 54, 2, 1], [10, 60, 2, 2], [10, 72, 2, 1], [10, 108, 2, 2], [10, 150, 2, 1], [10, 216, 2, 1], [10, 270, 2, 2], [10, 540, 2, 2], [10, 1350, 2, 1], [15, 12, 2, 4], [15, 16, 2, 1], [15, 24, 2, 8], [15, 48, 2, 5], [15, 600, 4, 1], [15, 1200, 4, 1], [30, 60, 2, 8], [30, 120, 2, 8], [30, 300, 2, 4], [30, 1800, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 744, 'exponent': 30, 'exponents_of_order': [4, 3, 3], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[12, 1, 24], [24, 1, 24], [48, 1, 8]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '18000.d', 'hash': 7387495986454361018, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [6, 2, 30, 30, 5], 'inner_gens': [[1, 3420, 10560, 43506, 288], [7747, 6, 228, 6840, 10800], [7093, 150, 12, 6840, 10800], [43627, 4326, 4332, 360, 43200], [10873, 6, 12, 21960, 10800]], 'inner_hash': 7387495986454361018, 'inner_nilpotent': False, 'inner_order': 54000, 'inner_split': False, 'inner_tex': 'D_5^3:C_3^2:S_3', 'inner_used': [1, 2, 3], 'irrC_degree': 12, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 12, 'irrep_stats': [[1, 12], [2, 6], [3, 4], [6, 26], [8, 12], [12, 48], [16, 6], [24, 36], [48, 10]], 'label': '54000.c', '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': True, 'name': 'D5^3:C3^2:S3', '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, 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, 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': 61, 'number_characteristic_subgroups': 29, 'number_conjugacy_classes': 160, 'number_divisions': 87, 'number_normal_subgroups': 29, 'number_subgroup_autclasses': 1050, 'number_subgroup_classes': 1710, 'number_subgroups': 291198, 'old_label': None, 'order': 54000, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 2159], [3, 1808], [5, 124], [6, 21520], [10, 7716], [15, 8192], [30, 12480]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 3534], 'outer_gens': [[5, 6, 9666, 1872, 10800], [39097, 2238, 6084, 2760, 21600]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [6, 49], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [3, 4], [4, 2], [6, 10], [12, 8], [16, 2], [24, 24], [32, 3], [48, 18], [64, 1], [96, 5]], 'representations': {'PC': {'code': '531121569954694569393498682229141570344046412924164294722812186829621118540796289700978124348343086521592382600401284825875769852777034758980151804902254663370344223', 'gens': [1, 3, 4, 7, 10], 'pres': [10, 2, 3, 2, 2, 3, 5, 2, 3, 5, 5, 20, 102602, 245802, 422403, 363733, 1543, 113, 516004, 630014, 824, 194, 518405, 324015, 2905, 3045426, 176836, 79826, 39936, 206, 2611207, 144977, 38427, 19237, 317, 3888008, 3258, 129628, 64838, 28809, 108019, 12069]}, 'Perm': {'d': 24, 'gens': [52984131216757424104634, 28207328802774788672943]}}, 'schur_multiplier': [2, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'D_5^3:C_3^2:S_3', 'transitive_degree': 45, 'wreath_data': None, 'wreath_product': False}
