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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '5184.rw', 'ambient_counter': 465, 'ambient_order': 5184, 'ambient_tex': 'C_3^4:(C_4\\times \\SD_{16})', 'central': False, 'central_factor': False, 'centralizer_order': 432, 'characteristic': False, 'core_order': 1, 'counter': 512, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '5184.rw.2592.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '2592.c1.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': 2592, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': True, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '5184.rw', 'aut_centralizer_order': None, 'aut_label': '2592.c1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '12.d1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['864.e1.a1', '864.e1.b1', '864.f1.a1', '864.g1.a1', '864.g1.b1', '864.h1.a1', '1296.b1.a1', '1296.f1.a1', '1296.l1.a1'], 'contains': ['5184.a1.a1'], 'core': '5184.a1.a1', 'coset_action_label': None, 'count': 12, 'diagramx': [6958, -1, 2513, -1, 6313, -1, 2115, -1], 'generators': [12], 'label': '5184.rw.2592.c1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '72.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '12.d1.a1', 'old_label': '2592.c1.a1', 'projective_image': '5184.rw', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '2592.c1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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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}
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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': 24, 'aut_gen_orders': [12, 24, 24, 6, 8], 'aut_gens': [[1, 4, 24, 576, 1728], [2891, 148, 2000, 2304, 4032], [4817, 3092, 4832, 576, 1728], [4001, 2124, 2008, 3456, 1152], [2267, 3748, 2000, 576, 3456], [3331, 3996, 3680, 2880, 4608]], 'aut_group': None, 'aut_hash': 2723700116397070647, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20736, 'aut_permdeg': 48, 'aut_perms': [8545806002176253348541307703934972840849634289735899916149555, 1755590269203518878033982760996030867957556088655489655748382, 6436331217157943305973349168628224338516571645134391774196193, 8720835638387874913217388784361353829467023220179161521972137, 224592308406934243837329933950156223535334242455262795309359], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 2], [2, 12, 1, 1], [2, 81, 1, 1], [2, 108, 1, 1], [3, 4, 2, 1], [3, 8, 1, 1], [3, 32, 2, 1], [4, 9, 2, 1], [4, 18, 1, 1], [4, 36, 1, 1], [4, 81, 2, 1], [4, 108, 2, 1], [4, 162, 1, 1], [4, 162, 2, 1], [4, 324, 1, 1], [4, 324, 2, 1], [6, 24, 1, 1], [6, 36, 2, 1], [6, 48, 2, 1], [6, 72, 1, 1], [6, 96, 2, 1], [6, 216, 1, 1], [8, 18, 1, 2], [8, 162, 1, 2], [8, 162, 2, 2], [12, 72, 2, 2], [12, 144, 2, 1], [12, 216, 2, 1], [24, 72, 2, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4.C_8^2.C_2^2', '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': 24, 'autcentquo_group': None, 'autcentquo_hash': 2723700116397070647, 'autcentquo_nilpotent': False, 'autcentquo_order': 20736, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^4.C_8^2.C_2^2', 'cc_stats': [[1, 1, 1], [2, 9, 2], [2, 12, 1], [2, 81, 1], [2, 108, 1], [3, 4, 2], [3, 8, 1], [3, 32, 2], [4, 9, 2], [4, 18, 1], [4, 36, 1], [4, 81, 2], [4, 108, 2], [4, 162, 3], [4, 324, 3], [6, 24, 1], [6, 36, 2], [6, 48, 2], [6, 72, 1], [6, 96, 2], [6, 216, 1], [8, 18, 2], [8, 162, 6], [12, 72, 4], [12, 144, 2], [12, 216, 2], [24, 72, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '5184.rw', 'commutator_count': 1, 'commutator_label': '324.161', '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', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 465, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['144.182', 1], ['36.9', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 2], [2, 12, 1, 1], [2, 81, 1, 1], [2, 108, 1, 1], [3, 4, 1, 2], [3, 8, 1, 1], [3, 32, 1, 2], [4, 9, 2, 1], [4, 18, 1, 1], [4, 36, 1, 1], [4, 81, 2, 1], [4, 108, 2, 1], [4, 162, 1, 1], [4, 162, 2, 1], [4, 324, 1, 1], [4, 324, 2, 1], [6, 24, 1, 1], [6, 36, 1, 2], [6, 48, 1, 2], [6, 72, 1, 1], [6, 96, 1, 2], [6, 216, 1, 1], [8, 18, 2, 1], [8, 162, 2, 1], [8, 162, 4, 1], [12, 72, 1, 2], [12, 72, 2, 1], [12, 144, 1, 2], [12, 216, 2, 1], [24, 72, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2150400, 'exponent': 24, 'exponents_of_order': [6, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[32, 1, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '5184.rw', 'hash': 6808996672383237400, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [4, 6, 24, 3, 3], 'inner_gens': [[1, 1412, 3352, 2880, 2304], [4377, 4, 3912, 3456, 1152], [4185, 724, 24, 2880, 4608], [4609, 2308, 4632, 576, 1728], [1153, 2308, 4056, 576, 1728]], 'inner_hash': 6808996672383237400, 'inner_nilpotent': False, 'inner_order': 5184, 'inner_split': False, 'inner_tex': 'C_3^4:(C_4\\times \\SD_{16})', 'inner_used': [1, 2, 3], 'irrC_degree': 32, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 32, 'irrep_stats': [[1, 16], [2, 12], [4, 8], [8, 14], [32, 4]], 'label': '5184.rw', 'linC_count': 64, 'linC_degree': 12, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 32, 'linQ_dim': 12, 'linQ_dim_count': 32, 'linR_count': 32, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^4:(C4*SD16)', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 36, 'number_characteristic_subgroups': 48, 'number_conjugacy_classes': 54, 'number_divisions': 40, 'number_normal_subgroups': 48, 'number_subgroup_autclasses': 408, 'number_subgroup_classes': 515, 'number_subgroups': 21052, 'old_label': None, 'order': 5184, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 219], [3, 80], [4, 1908], [6, 672], [8, 1008], [12, 1008], [24, 288]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [901, 0], 'outer_gens': [[825, 1972, 3688, 4608, 4032], [2595, 2900, 1768, 576, 1728]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 11], [8, 7], [16, 4], [32, 4]], 'representations': {'PC': {'code': '36709702435868393907459507662120400104865762227664223015993700347224634260191519511908766762153902701298982412836563514420974322840412166806426695', 'gens': [1, 3, 5, 9, 10], 'pres': [10, 2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 20, 42362, 69432, 82, 100483, 333, 167604, 53414, 48924, 7384, 144, 228005, 41775, 22345, 26315, 175, 209446, 178096, 52106, 36156, 206, 338567, 15377, 259208, 51858, 77788, 10848, 3298, 1148, 230409, 172819, 28829, 19249, 2459, 3669]}, 'Perm': {'d': 15, 'gens': [1520637524, 224189, 45507, 24951910345, 338677, 63975, 355419268534, 219574, 87737376696, 85949]}}, 'schur_multiplier': [2, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:(C_4\\times \\SD_{16})', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
