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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '450.20', 'ambient_counter': 20, 'ambient_order': 450, 'ambient_tex': 'C_3\\times C_5^2:S_3', 'central': False, 'central_factor': False, 'centralizer_order': 3, 'characteristic': True, 'core_order': 225, 'counter': 2, 'cyclic': False, 'direct': False, 'hall': 15, 'label': '450.20.2.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '2.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '2.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 2, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2', 'simple': False, 'solvable': True, 'special_labels': ['C1'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '225.5', 'subgroup_hash': 5, 'subgroup_order': 225, 'subgroup_tex': 'C_5^2:C_3^2', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '450.20', 'aut_centralizer_order': 1, 'aut_label': '2.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '1200.951', 'aut_weyl_index': 1, 'centralizer': '150.a1.a1', 'complements': ['225.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['6.a1.a1', '6.b1.a1', '6.c1.a1', '50.a1.a1'], 'core': '2.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [2724, 6888, 1853, 3445, 1085, 6986, 1085, 3014], 'generators': [2, 378, 90, 60], 'label': '450.20.2.a1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['6.a1.a1', '6.b1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2.a1.a1', 'projective_image': '150.5', 'quotient_action_image': '2.1', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '2.a1.a1', 'subgroup_fusion': None, 'weyl_group': '150.5'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '9.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 120, 'aut_gen_orders': [2, 24, 10, 15], 'aut_gens': [[1, 3, 15], [2, 96, 201], [1, 138, 21], [25, 3, 165], [85, 3, 15]], 'aut_group': '7200.ed', 'aut_hash': 6220936114514737460, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 7200, 'aut_permdeg': 28, 'aut_perms': [822257375533289975245973065, 2661482157070296743019733704, 159493940519245865975079964226, 251895467090468234355162646468], 'aut_phi_ratio': 60.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 25, 6, 1], [5, 3, 8, 1], [15, 3, 16, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_{25}:D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, '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': '1200.946', 'autcentquo_hash': 946, 'autcentquo_nilpotent': False, 'autcentquo_order': 1200, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_{25}:C_2', 'cc_stats': [[1, 1, 1], [3, 1, 2], [3, 25, 6], [5, 3, 8], [15, 3, 16]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '75.2', 'commutator_count': 1, 'commutator_label': '25.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '3.1', '5.1', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['75.2', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 25, 2, 3], [5, 3, 4, 2], [15, 3, 8, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 4, 'exponent': 15, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [3, 5], 'faithful_reps': [[3, 0, 16]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '225.5', 'hash': 5, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 15, 'inner_gen_orders': [3, 5, 5], 'inner_gens': [[1, 141, 111], [103, 3, 15], [145, 3, 15]], 'inner_hash': 2, 'inner_nilpotent': False, 'inner_order': 75, 'inner_split': True, 'inner_tex': 'C_5^2:C_3', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 6, 'irrep_stats': [[1, 9], [3, 24]], 'label': '225.5', 'linC_count': 16, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 6, 'linQ_dim': 14, 'linQ_dim_count': 6, 'linR_count': 8, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C5^2:C3^2', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 33, 'number_divisions': 9, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 10, 'number_subgroup_classes': 16, 'number_subgroups': 120, 'old_label': None, 'order': 225, 'order_factorization_type': 22, 'order_stats': [[1, 1], [3, 152], [5, 24], [15, 48]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 2, 24], 'outer_gen_pows': [0, 2, 0], 'outer_gens': [[2, 12, 165], [1, 99, 129], [152, 186, 132]], 'outer_group': '96.113', 'outer_hash': 113, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 11, 'outer_perms': [40344, 1, 8116084], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3\\times \\OD_{16}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4], [12, 2], [24, 2]], 'representations': {'PC': {'code': 24100739628151, 'gens': [1, 2, 3], 'pres': [4, 3, 5, 3, 5, 1129, 1334, 46, 1491]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [120852205970, 84161084168, 99619348751, 110636744024]}, 'Perm': {'d': 18, 'gens': [401488805019603, 3, 779743974659304, 85475086139640]}}, 'schur_multiplier': [15], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_5^2:C_3^2', 'transitive_degree': 45, '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': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [4, 2, 2, 3, 5, 5, 2], 'aut_gens': [[1, 2, 6, 90], [235, 254, 78, 270], [307, 290, 84, 360], [57, 202, 132, 342], [419, 416, 438, 108], [307, 290, 6, 90], [379, 434, 6, 90], [1, 2, 66, 90]], 'aut_group': '1200.951', 'aut_hash': 951, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1200, 'aut_permdeg': 17, 'aut_perms': [161962148252563, 45849626322441, 92122999329435, 107621682549173, 44281709476155, 235938134016283, 31112], 'aut_phi_ratio': 10.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 15, 1, 1], [3, 1, 2, 1], [3, 50, 1, 1], [3, 50, 2, 1], [5, 3, 4, 1], [5, 6, 2, 1], [6, 15, 2, 1], [10, 15, 4, 1], [15, 3, 8, 1], [15, 6, 4, 1], [30, 15, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_5^2:(C_4\\times D_6)', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '600.151', 'autcentquo_hash': 151, 'autcentquo_nilpotent': False, 'autcentquo_order': 600, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_5^2:(C_4\\times S_3)', 'cc_stats': [[1, 1, 1], [2, 15, 1], [3, 1, 2], [3, 50, 3], [5, 3, 4], [5, 6, 2], [6, 15, 2], [10, 15, 4], [15, 3, 8], [15, 6, 4], [30, 15, 8]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '150.5', 'commutator_count': 1, 'commutator_label': '75.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '5.1', '5.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 20, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['150.5', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 15, 1, 1], [3, 1, 2, 1], [3, 50, 1, 1], [3, 50, 2, 1], [5, 3, 4, 1], [5, 6, 2, 1], [6, 15, 2, 1], [10, 15, 4, 1], [15, 3, 8, 1], [15, 6, 4, 1], [30, 15, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72, 'exponent': 30, 'exponents_of_order': [2, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[3, 0, 16], [6, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '450.20', 'hash': 20, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [2, 3, 5, 5], 'inner_gens': [[1, 4, 384, 90], [5, 2, 438, 108], [163, 110, 6, 90], [1, 74, 6, 90]], 'inner_hash': 5, 'inner_nilpotent': False, 'inner_order': 150, 'inner_split': True, 'inner_tex': 'C_5^2:S_3', 'inner_used': [1, 2, 3], 'irrC_degree': 3, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 6, 'irrep_stats': [[1, 6], [2, 3], [3, 24], [6, 6]], 'label': '450.20', 'linC_count': 16, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 6, 'linQ_dim': 14, 'linQ_dim_count': 6, 'linR_count': 8, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3*C5^2:S3', 'ngens': 5, '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': 8, 'number_conjugacy_classes': 39, 'number_divisions': 12, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 26, 'number_subgroup_classes': 26, 'number_subgroups': 244, 'old_label': None, 'order': 450, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 15], [3, 152], [5, 24], [6, 30], [10, 60], [15, 48], [30, 120]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 2, 66, 90], [1, 2, 42, 180]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 1], [12, 3], [24, 3]], 'representations': {'PC': {'code': 700301926691713929039967, 'gens': [1, 2, 3, 5], 'pres': [5, -2, -3, -3, -5, 5, 41, 5762, 3292, 57, 5043, 2348, 1359]}, 'GLZN': {'d': 2, 'p': 50, 'gens': [125049, 188776, 2702031, 4671337, 3976021]}, 'Perm': {'d': 18, 'gens': [64165879816560, 3, 425258361649560, 798157540354320, 1180064644017840]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times C_5^2:S_3', 'transitive_degree': 45, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], '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, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, 'wreath_data': None, 'wreath_product': False}
