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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1728.6120', 'ambient_counter': 6120, 'ambient_order': 1728, 'ambient_tex': 'C_6^2.(C_4\\times D_6)', 'central': False, 'central_factor': False, 'centralizer_order': 864, 'characteristic': True, 'core_order': 6, 'counter': 727, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '1728.6120.288.e1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '288.e1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '288.531', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 531, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 288, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_6^2.C_2^3', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '6.2', 'subgroup_hash': 2, 'subgroup_order': 6, 'subgroup_tex': 'C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1728.6120', 'aut_centralizer_order': None, 'aut_label': '288.e1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '2.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['96.i1.a1', '96.j1.a1', '96.s1.a1', '144.b1.a1', '144.e1.a1', '144.g1.a1', '144.bi1.a1', '144.bi1.b1', '144.bi1.c1', '144.bi1.d1'], 'contains': ['576.a1.a1', '864.e1.a1'], 'core': '288.e1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [3445, 7007, 7539, 3789, 4979, 6386, 8322, 5596], 'generators': [74, 576], 'label': '1728.6120.288.e1.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '288.e1.a1', 'normal_contained_in': ['96.j1.a1', '96.i1.a1', '144.g1.a1', '144.b1.a1', '144.e1.a1'], 'normal_contains': ['576.a1.a1', '864.e1.a1'], 'normalizer': '1.a1.a1', 'old_label': '288.e1.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '288.e1.a1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '6.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [5]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', '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': 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], [3, 1, 2], [6, 1, 2]], 'center_label': '6.2', 'center_order': 6, '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', '3.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 6, 'eulerian_function': 1, 'exponent': 6, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '6.2', 'hash': 2, 'hyperelementary': 6, '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 6]], 'label': '6.2', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C6', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 6, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 6, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [3, 2], [6, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[5]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 5, 'pgroup': 0, 'primary_abelian_invariants': [2, 3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2]], 'representations': {'PC': {'code': 21, 'gens': [1], 'pres': [2, -2, -3, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [73]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [31, 56]}, 'Perm': {'d': 5, 'gens': [24, 4]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6', 'transitive_degree': 6, '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': '32.21', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': None, 'aut_cyclic': None, 'aut_derived_length': None, 'aut_exponent': None, 'aut_gen_orders': None, 'aut_gens': [[1, 4, 24, 144], [81, 4, 600, 144], [1, 628, 24, 144], [1, 20, 24, 720], [1, 4, 120, 720], [3, 4, 120, 720], [73, 20, 120, 144], [865, 20, 24, 720], [1, 6, 24, 144], [1, 76, 120, 720], [1, 884, 120, 144], [1, 4, 120, 722], [1, 4, 120, 792], [1, 20, 120, 1008], [577, 580, 24, 144]], 'aut_group': None, 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 55296, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': 96.0, 'aut_solvable': None, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 2, 1, 1], [3, 6, 1, 2], [3, 12, 1, 1], [4, 2, 4, 1], [4, 9, 8, 1], [4, 18, 4, 5], [6, 2, 1, 7], [6, 6, 1, 14], [6, 12, 1, 7], [12, 2, 8, 1], [12, 12, 4, 2], [12, 12, 8, 1], [12, 18, 8, 2], [12, 36, 4, 4]], 'aut_supersolvable': None, 'aut_tex': None, 'autcent_abelian': None, 'autcent_cyclic': None, 'autcent_exponent': None, 'autcent_group': None, 'autcent_hash': None, 'autcent_nilpotent': None, 'autcent_order': None, 'autcent_solvable': None, 'autcent_split': None, 'autcent_supersolvable': None, 'autcent_tex': None, 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': None, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 1, 7], [3, 2, 1], [3, 6, 2], [3, 12, 1], [4, 2, 4], [4, 9, 8], [4, 18, 20], [6, 2, 7], [6, 6, 14], [6, 12, 7], [12, 2, 8], [12, 12, 16], [12, 18, 16], [12, 36, 16]], 'center_label': '8.5', 'center_order': 8, 'central_product': False, 'central_quotient': '216.102', 'commutator_count': 1, 'commutator_label': '54.10', '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': 6120, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 2, 1, 1], [3, 6, 1, 2], [3, 12, 1, 1], [4, 2, 2, 2], [4, 9, 2, 4], [4, 18, 1, 4], [4, 18, 2, 8], [6, 2, 1, 7], [6, 6, 1, 14], [6, 12, 1, 7], [12, 2, 4, 2], [12, 12, 2, 8], [12, 18, 2, 4], [12, 18, 4, 2], [12, 36, 1, 4], [12, 36, 2, 6]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24192, 'exponent': 12, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '24.15', 'frattini_quotient': '72.46', 'hash': 6120, 'hyperelementary': 1, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': [2, 6, 3, 6], 'inner_gens': [[1, 20, 24, 792], [81, 4, 1272, 792], [1, 628, 24, 144], [1225, 1228, 24, 144]], 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': 216, 'inner_split': None, 'inner_tex': 'C_6.S_3^2', 'inner_used': None, 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 40], [4, 24], [6, 32]], 'label': '1728.6120', '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': 'C6^2.(C4*D6)', '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': 57, 'number_characteristic_subgroups': 105, 'number_conjugacy_classes': 128, 'number_divisions': 84, 'number_normal_subgroups': 175, 'number_subgroup_autclasses': 514, 'number_subgroup_classes': 787, 'number_subgroups': 4461, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 7], [3, 26], [4, 440], [6, 182], [12, 1072]], 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': False, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': '256.56092', 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 256, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': 'C_2^8', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 28], [4, 28], [6, 4], [8, 4], [12, 10], [24, 2]], 'representations': {'PC': {'code': 98477231119210916127450760924241623460407976860098527015126084906498601, 'gens': [1, 3, 5, 7], 'pres': [9, 2, 2, 2, 3, 2, 3, 2, 2, 3, 18, 542, 74, 3171, 354, 14332, 6646, 130, 9095, 4244, 49902, 12498, 186, 103687, 25945, 214, 93320, 23354]}, 'Perm': {'d': 21, 'gens': [19646680328720049, 262539173829110529, 87459247, 40279680, 40284856, 16, 372491224308403200, 2689063734976819200, 5251452972974208000]}}, 'schur_multiplier': [2, 2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4, 4], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6^2.(C_4\\times D_6)', 'transitive_degree': 576, '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': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 4, 6, 4, 4, 2, 12], 'aut_gens': [[1, 2, 4, 24], [109, 98, 20, 120], [167, 202, 204, 76], [145, 18, 4, 120], [7, 210, 108, 224], [3, 50, 108, 224], [1, 22, 20, 276], [19, 150, 204, 236]], 'aut_group': None, 'aut_hash': 696738818174563068, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 4608, 'aut_permdeg': 28, 'aut_perms': [234958145418482259275055404496, 104885722843633461227246895015, 301968189164943406956875023311, 64496827101396227570827895961, 34488541435186683321773384039, 8195909242264164471513098280, 222422388440738388685152811246], 'aut_phi_ratio': 48.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 9, 4, 1], [3, 2, 2, 1], [3, 4, 1, 1], [4, 2, 2, 1], [4, 6, 4, 2], [4, 18, 2, 1], [6, 2, 2, 3], [6, 4, 1, 3], [12, 4, 4, 2], [12, 12, 4, 2]], 'aut_supersolvable': False, 'aut_tex': 'D_6^2.C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '72.40', 'autcentquo_hash': 40, 'autcentquo_nilpotent': False, 'autcentquo_order': 72, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\SOPlus(4,2)', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 9, 4], [3, 2, 2], [3, 4, 1], [4, 2, 2], [4, 6, 8], [4, 18, 2], [6, 2, 6], [6, 4, 3], [12, 4, 8], [12, 12, 8]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '72.46', 'commutator_count': 1, 'commutator_label': '18.5', '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'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 531, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 9, 1, 4], [3, 2, 1, 2], [3, 4, 1, 1], [4, 2, 2, 1], [4, 6, 1, 4], [4, 6, 2, 2], [4, 18, 2, 1], [6, 2, 1, 6], [6, 4, 1, 3], [12, 4, 2, 4], [12, 12, 1, 4], [12, 12, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1344, 'exponent': 12, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '72.46', 'hash': 531, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 3, 6], 'inner_gens': [[1, 2, 4, 132], [1, 2, 20, 120], [1, 10, 4, 24], [205, 194, 4, 24]], 'inner_hash': 46, 'inner_nilpotent': False, 'inner_order': 72, 'inner_split': False, 'inner_tex': 'S_3\\times D_6', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 20], [4, 12]], 'label': '288.531', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, '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': 'C6^2.C2^3', 'ngens': 7, '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': 21, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 48, 'number_divisions': 38, 'number_normal_subgroups': 66, 'number_subgroup_autclasses': 103, 'number_subgroup_classes': 211, 'number_subgroups': 786, 'old_label': None, 'order': 288, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 39], [3, 8], [4, 88], [6, 24], [12, 128]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 14, 4, 24], [1, 14, 4, 264], [1, 2, 4, 36], [1, 158, 20, 276], [3, 158, 108, 232]], 'outer_group': '64.261', 'outer_hash': 261, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 10, 'outer_perms': [367920, 1174440, 120, 367921, 17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_2^3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 16], [4, 12], [8, 2]], 'representations': {'PC': {'code': 19559067527306321925744373491885349513, 'gens': [1, 2, 3, 5], 'pres': [7, 2, 2, 2, 3, 2, 2, 3, 84, 219, 58, 234, 4624, 2111, 102, 10085, 5052, 124, 9414, 4717]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101743041088268, 91655872448021897, 125101738765657804]}, 'GLZN': {'d': 2, 'p': 30, 'gens': [360023, 40966, 297011, 27829, 482473, 513019, 32773]}, 'Perm': {'d': 14, 'gens': [6314118487, 6354398167, 11537, 7, 16680, 13971242880, 20122058880]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6^2.C_2^3', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
