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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '768.1088504', 'ambient_counter': 1088504, 'ambient_order': 768, 'ambient_tex': 'C_2\\times A_4:\\OD_{32}', 'central': False, 'central_factor': False, 'centralizer_order': 96, 'characteristic': False, 'core_order': 16, 'counter': 119, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '768.1088504.16.j1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '16.j1.b1', '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': 16, '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': '48.23', 'subgroup_hash': 23, 'subgroup_order': 48, 'subgroup_tex': 'C_2\\times C_{24}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '768.1088504', 'aut_centralizer_order': 48, 'aut_label': '16.j1', 'aut_quo_index': None, 'aut_stab_index': 8, 'aut_weyl_group': '16.14', 'aut_weyl_index': 384, 'centralizer': '8.i1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.e1.b1', '8.i1.a1'], 'contains': ['32.f1.b1', '32.i1.a1', '32.j1.a1', '48.g1.b1'], 'core': '48.g1.b1', 'coset_action_label': None, 'count': 4, 'diagramx': [5162, -1, 4486, -1, 5758, -1, 3744, -1], 'generators': [508447, 557073, 622611, 417459, 294921], 'label': '768.1088504.16.j1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.e1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.g1.a1', 'old_label': '16.j1.b1', 'projective_image': '96.194', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '16.j1.b1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '48.23', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 2, 2, 4], 'aut_gens': [[1, 2], [1, 14], [1, 22], [25, 22], [25, 15]], 'aut_group': '32.46', 'aut_hash': 46, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 32, 'aut_permdeg': 8, 'aut_perms': [16, 23, 11536, 16576], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [3, 1, 2, 1], [4, 1, 2, 2], [6, 1, 2, 1], [6, 1, 4, 1], [8, 1, 8, 1], [12, 1, 4, 2], [24, 1, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.46', 'autcent_hash': 46, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times D_4', '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, 3], [3, 1, 2], [4, 1, 4], [6, 1, 6], [8, 1, 8], [12, 1, 8], [24, 1, 16]], 'center_label': '48.23', 'center_order': 48, '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', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 23, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['8.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [4, 1, 2, 2], [6, 1, 2, 3], [8, 1, 4, 2], [12, 1, 4, 2], [24, 1, 8, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 24, 'exponent': 24, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '12.5', 'hash': 23, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 48]], 'label': '48.23', 'linC_count': 384, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 4, 'linQ_dim': 6, 'linQ_dim_count': 4, 'linR_count': 16, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C24', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 14, 'number_conjugacy_classes': 48, 'number_divisions': 16, 'number_normal_subgroups': 22, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 22, 'number_subgroups': 22, 'old_label': None, 'order': 48, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 4], [6, 6], [8, 8], [12, 8], [24, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 14], [1, 22], [25, 22], [25, 15]], 'outer_group': '32.46', 'outer_hash': 46, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [16, 23, 11536, 16576], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 8, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 4], [8, 2]], 'representations': {'PC': {'code': 1286603201, 'gens': [1, 2], 'pres': [5, -2, -2, -2, -2, -3, 26, 42, 58]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [24992906222183252, 41624324273564377]}, 'GLFp': {'d': 2, 'p': 73, 'gens': [17894809, 20228936]}, 'Perm': {'d': 13, 'gens': [27760320, 479001600, 4, 11614560, 3669864]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 24], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_{24}', 'transitive_degree': 48, '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': '32.36', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [12, 4, 12, 12, 12, 4, 4], 'aut_gens': [[49681, 299447, 957620, 573457, 557585], [508447, 18336, 433332, 557585, 49665], [1032719, 168763, 141165, 49665, 557585], [508447, 915077, 974516, 49665, 573457], [49681, 20192, 450228, 557585, 49665], [1032719, 581519, 973988, 49665, 573457], [508447, 111101, 664941, 49665, 557585], [1032719, 897765, 681837, 49665, 557585]], 'aut_group': None, 'aut_hash': 3625057183303488335, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6144, 'aut_permdeg': 192, 'aut_perms': 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318299428896529513707849828360789184126542137890191208139354409169047785422065877470648743370021104916125907661463589531747416032710003015108481093200986021447224492787651825835125430993867783091486182418376961593437804822415239428383052461718578741137256343886428066423809284397088581426440357921055221030916941165878646920980405328102972493275686761550623], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 3, 1, 2], [2, 3, 2, 1], [2, 6, 2, 1], [3, 8, 1, 1], [4, 1, 2, 2], [4, 2, 2, 1], [4, 3, 2, 2], [4, 6, 2, 1], [6, 8, 1, 1], [6, 8, 2, 1], [6, 8, 4, 1], [8, 1, 4, 2], [8, 2, 4, 1], [8, 3, 4, 2], [8, 6, 4, 1], [12, 8, 2, 2], [12, 8, 4, 1], [16, 12, 16, 2], [24, 8, 4, 2], [24, 8, 8, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_2^3\\times A_4).C_2^6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.2167', 'autcent_hash': 2167, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4^2:C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '48.48', 'autcentquo_hash': 48, 'autcentquo_nilpotent': False, 'autcentquo_order': 48, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2\\times S_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 3, 4], [2, 6, 2], [3, 8, 1], [4, 1, 4], [4, 2, 2], [4, 3, 4], [4, 6, 2], [6, 8, 7], [8, 1, 8], [8, 2, 4], [8, 3, 8], [8, 6, 4], [12, 8, 8], [16, 12, 32], [24, 8, 16]], 'center_label': '16.5', 'center_order': 16, 'central_product': True, 'central_quotient': '48.48', 'commutator_count': 1, 'commutator_label': '24.13', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 1088504, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['384.5680', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 3, 1, 4], [2, 6, 1, 2], [3, 8, 1, 1], [4, 1, 2, 2], [4, 2, 1, 2], [4, 3, 2, 2], [4, 6, 1, 2], [6, 8, 1, 3], [6, 8, 2, 2], [8, 1, 4, 2], [8, 2, 2, 2], [8, 3, 4, 2], [8, 6, 2, 2], [12, 8, 2, 4], [16, 12, 4, 8], [24, 8, 4, 4]], 'element_repr_type': 'GLZq', 'elementary': 1, 'eulerian_function': 20160, 'exponent': 48, 'exponents_of_order': [8, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.1', 'frattini_quotient': '96.226', 'hash': 1088504, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 3, 2, 2], 'inner_gens': [[49681, 824231, 957620, 573457, 557585], [573953, 299447, 681853, 49665, 557585], [49681, 28640, 957620, 557585, 49665], [49681, 823719, 449700, 573457, 557585], [49681, 299447, 433828, 573457, 557585]], 'inner_hash': 48, 'inner_nilpotent': False, 'inner_order': 48, 'inner_split': False, 'inner_tex': 'C_2\\times S_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 40], [3, 32], [6, 8]], 'label': '768.1088504', 'linC_count': 384, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 11, 'linQ_degree_count': 8, 'linQ_dim': 11, 'linQ_dim_count': 8, 'linR_count': 48, 'linR_degree': 7, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C2*A4:OD32', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 32, 'number_characteristic_subgroups': 43, 'number_conjugacy_classes': 112, 'number_divisions': 50, 'number_normal_subgroups': 101, 'number_subgroup_autclasses': 284, 'number_subgroup_classes': 489, 'number_subgroups': 1626, 'old_label': None, 'order': 768, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 31], [3, 8], [4, 32], [6, 56], [8, 64], [12, 64], [16, 384], [24, 128]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 4], 'outer_gen_pows': [49681, 49681, 32769, 32769, 32769, 32769], 'outer_gens': [[49681, 37055, 957620, 573457, 557585], [49681, 446515, 957620, 573457, 557585], [49681, 766057, 681853, 49665, 557585], [49681, 914149, 664941, 49665, 557585], [49681, 258169, 957620, 573457, 557585], [508447, 840103, 450228, 573457, 557585]], 'outer_group': '128.2163', 'outer_hash': 2163, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 12, 'outer_perms': [87102727, 127387576, 40291223, 11520, 127387560, 303171263], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:D_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 8], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [3, 8], [4, 8], [6, 4], [8, 2], [12, 4], [16, 2], [24, 2]], 'representations': {'PC': {'code': '1482085955519320388227563783068344264788363336708', 'gens': [1, 2, 6, 8, 9], 'pres': [9, -2, -2, -2, -2, -2, -2, -3, -2, 2, 325, 46, 74, 102, 4334, 158, 4047, 20752, 916, 709, 1511, 305]}, 'GLZN': {'d': 2, 'p': 68, 'gens': [8020353, 4087629, 2829897, 316745, 16587479, 5236, 10376289, 314465, 18708763]}, 'GLZq': {'d': 2, 'q': 32, 'gens': [32785, 98307, 299447, 294921, 124284, 442909, 33281, 557073, 573953]}, 'Perm': {'d': 22, 'gens': [51347038622219078823, 51212945926204877284, 355461131765, 4397976680734, 5800817303020, 2966658509284, 107425784095838208000, 51212587272118272000, 122000787836928000]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 8], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times A_4:\\OD_{32}', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
