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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1728.46787', 'ambient_counter': 46787, 'ambient_order': 1728, 'ambient_tex': 'C_2\\times C_6^2:D_{12}', 'central': False, 'central_factor': False, 'centralizer_order': 36, 'characteristic': False, 'core_order': 4, 'counter': 996, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1728.46787.144.cy1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '144.cy1.a1', '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': 144, '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': '12.3', 'subgroup_hash': 3, 'subgroup_order': 12, 'subgroup_tex': 'A_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1728.46787', 'aut_centralizer_order': 96, 'aut_label': '144.cy1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '24.12', 'aut_weyl_index': 384, 'centralizer': '48.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['48.bu1.a1', '48.bv1.a1', '48.bw1.a1', '48.bx1.a1', '72.bc1.a1', '72.bc1.b1', '72.cw1.a1'], 'contains': ['432.b1.a1', '576.g1.a1'], 'core': '432.b1.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [489, -1, 412, -1, 625, -1, 672, -1], 'generators': [345944, 329687, 33321], 'label': '1728.46787.144.cy1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '16.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.a1.a1', 'old_label': '144.cy1.a1', 'projective_image': '1728.46787', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '144.cy1.a1', 'subgroup_fusion': None, 'weyl_group': '12.3'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '3.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 3, 2, 2], 'aut_gens': [[4, 16, 7], [11, 16, 23], [19, 23, 16], [8, 16, 7], [15, 16, 7]], 'aut_group': '24.12', 'aut_hash': 12, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24, 'aut_permdeg': 4, 'aut_perms': [2, 4, 16, 7], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 4, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_4', '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': 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, 3, 1], [3, 4, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, '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', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 4, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 4, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[3, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '12.3', 'hash': 3, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 2, 2], 'inner_gens': [[4, 7, 23], [19, 16, 7], [15, 16, 7]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'A_4', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 3, 'irrQ_dim': 3, 'irrR_degree': 3, 'irrep_stats': [[1, 3], [3, 1]], 'label': '12.3', 'linC_count': 1, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 1, 'linQ_dim': 3, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'A4', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 5, 'number_subgroup_classes': 5, 'number_subgroups': 10, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [3, 8]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3, 7, 16]], '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': 3, 'perfect': False, 'permutation_degree': 4, 'pgroup': 0, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [3, 1]], 'representations': {'PC': {'code': 9572, 'gens': [1, 2, 3], 'pres': [3, -3, -2, 2, 55, 29]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [4828, 4708]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [31, 21], 'family': 'PSL'}, {'d': 2, 'q': 3, 'gens': [362, 766], 'family': 'PSU'}, {'d': 3, 'q': 3, 'gens': [10315, 1638], 'family': 'Omega'}, {'d': 3, 'q': 3, 'gens': [10315, 1638], 'family': 'POmega'}, {'d': 2, 'q': 3, 'gens': [31, 21], 'family': 'PSigmaL'}, {'d': 1, 'q': 4, 'gens': [3, 7], 'family': 'AGL'}], 'GLFp': {'d': 3, 'p': 2, 'gens': [458, 314, 124]}, 'Perm': {'d': 4, 'gens': [4, 16, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'A_4', 'transitive_degree': 4, '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': [4, 2, 12, 6, 12, 6, 12], 'aut_gens': [[329295, 192688, 173796, 252459, 208961], [592731, 329489, 317555, 252459, 241887], [329295, 137418, 158298, 252459, 560177], [592731, 609169, 469756, 505311, 515883], [329295, 598193, 166194, 241875, 252471], [329295, 593293, 12145, 252459, 505299], [329295, 285587, 481124, 560193, 505299], [329295, 137418, 328923, 241875, 208961]], 'aut_group': None, 'aut_hash': 8765063099206293740, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 9216, 'aut_permdeg': 38, 'aut_perms': [356687815650180491291197543550721770602250486, 98631104855017234839743416254758466751149614, 281779835186876897114942576196803940700595496, 326461513717763429449058281300523838397215444, 508854386204979089342462496212427298863973154, 495060207007763464354985760608081422961362644, 98631104854862216466400769978518956406822322], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 3, 1, 2], [2, 3, 2, 1], [2, 12, 2, 1], [2, 36, 2, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 8, 1, 1], [3, 8, 2, 1], [3, 16, 1, 1], [3, 16, 2, 1], [4, 6, 2, 1], [4, 12, 2, 1], [4, 18, 2, 1], [4, 36, 2, 1], [6, 1, 2, 1], [6, 1, 4, 1], [6, 2, 1, 1], [6, 2, 2, 2], [6, 2, 4, 1], [6, 3, 2, 2], [6, 3, 4, 1], [6, 6, 1, 2], [6, 6, 2, 3], [6, 6, 4, 1], [6, 8, 1, 1], [6, 8, 2, 2], [6, 8, 4, 1], [6, 12, 4, 2], [6, 12, 8, 1], [6, 16, 1, 1], [6, 16, 2, 2], [6, 16, 4, 1], [6, 36, 4, 1], [12, 6, 4, 1], [12, 12, 4, 2], [12, 12, 8, 1], [12, 18, 4, 1], [12, 24, 4, 1], [12, 24, 8, 1], [12, 36, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_6\\times A_4).C_2^6.C_2', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '144.183', 'autcentquo_hash': 183, 'autcentquo_nilpotent': False, 'autcentquo_order': 144, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\times S_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 3, 4], [2, 12, 2], [2, 36, 2], [3, 1, 2], [3, 2, 3], [3, 8, 3], [3, 16, 3], [4, 6, 2], [4, 12, 2], [4, 18, 2], [4, 36, 2], [6, 1, 6], [6, 2, 9], [6, 3, 8], [6, 6, 12], [6, 8, 9], [6, 12, 16], [6, 16, 9], [6, 36, 4], [12, 6, 4], [12, 12, 16], [12, 18, 4], [12, 24, 12], [12, 36, 4]], 'center_label': '12.5', 'center_order': 12, 'central_product': True, 'central_quotient': '144.183', 'commutator_count': 1, 'commutator_label': '72.47', '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': 46787, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['288.855', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 1, 4], [2, 12, 1, 2], [2, 36, 1, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 8, 1, 1], [3, 8, 2, 1], [3, 16, 1, 1], [3, 16, 2, 1], [4, 6, 1, 2], [4, 12, 1, 2], [4, 18, 1, 2], [4, 36, 1, 2], [6, 1, 2, 3], [6, 2, 1, 3], [6, 2, 2, 3], [6, 3, 2, 4], [6, 6, 1, 4], [6, 6, 2, 4], [6, 8, 1, 3], [6, 8, 2, 3], [6, 12, 2, 8], [6, 16, 1, 3], [6, 16, 2, 3], [6, 36, 2, 2], [12, 6, 2, 2], [12, 12, 2, 8], [12, 18, 2, 2], [12, 24, 2, 2], [12, 24, 4, 2], [12, 36, 2, 2]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 131040, 'exponent': 12, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '864.4690', 'hash': 46787, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [1, 2, 6, 2, 6], 'inner_gens': [[329295, 192688, 173796, 252459, 208961], [329295, 192688, 2611, 241875, 208961], [329295, 290879, 173796, 241875, 252471], [329295, 500002, 470162, 252459, 208961], [329295, 192688, 165788, 252459, 208961]], 'inner_hash': 183, 'inner_nilpotent': False, 'inner_order': 144, 'inner_split': True, 'inner_tex': 'S_3\\times S_4', 'inner_used': [2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 24], [2, 54], [3, 24], [4, 12], [6, 30]], 'label': '1728.46787', 'linC_count': 128, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 7, 'linQ_degree_count': 16, 'linQ_dim': 7, 'linQ_dim_count': 16, 'linR_count': 16, 'linR_degree': 7, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C2*C6^2:D12', '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': 54, 'number_characteristic_subgroups': 50, 'number_conjugacy_classes': 144, 'number_divisions': 88, 'number_normal_subgroups': 106, 'number_subgroup_autclasses': 653, 'number_subgroup_classes': 1179, 'number_subgroups': 8324, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 111], [3, 80], [4, 144], [6, 672], [12, 720]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4], 'outer_gen_pows': [21953, 21953, 21953, 21953, 21953], 'outer_gens': [[329295, 181320, 8771, 241875, 208961], [329295, 181320, 20153, 241875, 560177], [329295, 192688, 173796, 515895, 560177], [329295, 192688, 173796, 252459, 560177], [592731, 181320, 20153, 505311, 208961]], 'outer_group': '64.202', 'outer_hash': 202, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 10, 'outer_perms': [806401, 1, 127, 367927, 5057], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3:D_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 18], [3, 8], [4, 22], [6, 14], [8, 6], [12, 12]], 'representations': {'PC': {'code': 533821624729984981153743822106665450463292452506774205227948053, 'gens': [1, 2, 3, 6, 8], 'pres': [9, 2, 2, 2, 2, 3, 2, 3, 2, 3, 605, 74, 732, 102, 733, 24638, 12335, 7160, 3119, 158, 12985, 3922, 5875, 214, 23354]}, 'GLZN': {'d': 2, 'p': 28, 'gens': [214622, 19852, 548825, 340271, 158599, 22345, 21977, 33321, 285389]}, 'Perm': {'d': 16, 'gens': [1321086821167, 85696, 806527, 2889377568000, 806400, 304, 4185076896000, 126, 7]}}, 'schur_multiplier': [2, 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_2\\times C_6^2:D_{12}', 'transitive_degree': 72, 'wreath_data': None, 'wreath_product': False}
