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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '384.9328', 'ambient_counter': 9328, 'ambient_order': 384, 'ambient_tex': 'S_3\\times C_2^3:D_4', 'central': False, 'central_factor': False, 'centralizer_order': 128, 'characteristic': False, 'core_order': 8, 'counter': 126, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '384.9328.24.t1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '24.t1', 'outer_equivalence': True, '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': 24, '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': '16.14', 'subgroup_hash': 14, 'subgroup_order': 16, 'subgroup_tex': 'C_2^4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '384.9328', 'aut_centralizer_order': None, 'aut_label': '24.t1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '3.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['8.c1', '12.g1', '12.h1', '12.i1'], 'contains': ['48.c1', '48.d1', '48.o1', '48.w1'], 'core': '48.c1', 'coset_action_label': None, 'count': 3, 'diagramx': None, 'generators': [69133, 14113, 179725, 97063], 'label': '384.9328.24.t1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '8.c1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1', 'old_label': '24.t1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '24.t1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 420, 'aut_gen_orders': [6, 3], 'aut_gens': [[1, 2, 4, 8], [12, 4, 5, 3], [10, 14, 7, 5]], 'aut_group': '20160.a', 'aut_hash': 3764836782182912467, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20160, 'aut_permdeg': 8, 'aut_perms': [5193, 5760], 'aut_phi_ratio': 2520.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 15, 1]], 'aut_supersolvable': False, 'aut_tex': 'A_8', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 420, 'autcent_group': '20160.a', 'autcent_hash': 3764836782182912467, 'autcent_nilpotent': False, 'autcent_order': 20160, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'A_8', '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, 15]], 'center_label': '16.14', 'center_order': 16, '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'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 14, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 4]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [4], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '16.14', 'hash': 14, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1, 1], 'inner_gens': [[1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 8]], '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, 16]], 'label': '16.14', 'linC_count': 840, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 840, 'linQ_dim': 4, 'linQ_dim_count': 840, 'linR_count': 840, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^4', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 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': 16, 'number_divisions': 16, 'number_normal_subgroups': 67, 'number_subgroup_autclasses': 5, 'number_subgroup_classes': 67, 'number_subgroups': 67, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 15]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 420, 'outer_gen_orders': [6, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[12, 4, 5, 3], [10, 14, 7, 5]], 'outer_group': '20160.a', 'outer_hash': 3764836782182912467, 'outer_nilpotent': False, 'outer_order': 20160, 'outer_permdeg': 8, 'outer_perms': [5193, 5760], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'A_8', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3, 4], 'pres': [4, -2, 2, 2, 2]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [7233746, 7115648, 35812976, 7115160]}, 'GLFp': {'d': 4, 'p': 2, 'gens': [33837, 18465, 27183, 18467]}, 'Perm': {'d': 8, 'gens': [5040, 120, 6, 1]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^4', 'transitive_degree': 16, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [12, 6, 6, 6, 2, 12], 'aut_gens': [[13903, 69133, 69557, 266185, 97255], [155813, 69613, 24427, 69557, 180205], [152141, 235297, 24427, 245471, 262867], [318329, 318247, 266473, 79859, 263059], [69347, 152083, 190303, 321929, 262867], [14191, 69229, 152219, 266473, 96871], [79787, 318055, 24715, 235169, 180205]], 'aut_group': None, 'aut_hash': 2776734662927589858, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 147456, 'aut_permdeg': 48, 'aut_perms': [7263970475724668788450749568740431631923535270163496892886729, 4649014806612517493609007474753352709951819569043775879799612, 9799051000951023568212403395912459834463319216588472863649705, 3245078251436274858624688449958924809845053605727463152623732, 8584767042967323047907588219582038038246523002357099612142028, 2190974095308718327480657004066090051525831912395648259236644], 'aut_phi_ratio': 1152.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 3, 2], [2, 3, 8, 1], [2, 4, 6, 1], [2, 12, 6, 1], [3, 2, 1, 1], [4, 4, 2, 1], [4, 4, 6, 1], [4, 12, 2, 1], [4, 12, 6, 1], [6, 2, 1, 1], [6, 2, 3, 2], [6, 8, 6, 1], [12, 8, 2, 1], [12, 8, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^8.C_3.D_6.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4096.bri', 'autcent_hash': 3677823439335518020, 'autcent_nilpotent': True, 'autcent_order': 4096, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^{12}', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '36.10', 'autcentquo_hash': 10, 'autcentquo_nilpotent': False, 'autcentquo_order': 36, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3^2', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 3, 8], [2, 4, 6], [2, 12, 6], [3, 2, 1], [4, 4, 8], [4, 12, 8], [6, 2, 7], [6, 8, 6], [12, 8, 8]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '48.51', 'commutator_count': 1, 'commutator_label': '24.15', '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', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 9328, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['6.1', 1], ['64.73', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 3, 1, 8], [2, 4, 1, 6], [2, 12, 1, 6], [3, 2, 1, 1], [4, 4, 1, 6], [4, 4, 2, 1], [4, 12, 1, 6], [4, 12, 2, 1], [6, 2, 1, 7], [6, 8, 1, 6], [12, 8, 1, 6], [12, 8, 2, 1]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 43680, 'exponent': 12, 'exponents_of_order': [7, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '48.51', 'hash': 9328, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 2, 2, 3], 'inner_gens': [[13903, 69133, 69269, 190303, 97255], [13903, 69133, 69557, 266185, 96871], [14191, 69133, 69557, 100597, 97255], [96985, 69133, 235457, 266185, 97255], [13903, 69517, 69557, 266185, 97255]], 'inner_hash': 51, 'inner_nilpotent': False, 'inner_order': 48, 'inner_split': False, 'inner_tex': 'C_2^2\\times D_6', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 36], [4, 14]], 'label': '384.9328', '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': True, 'metacyclic': False, 'monomial': True, 'name': 'S3*C2^3:D4', 'ngens': 8, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 18, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 66, 'number_divisions': 63, 'number_normal_subgroups': 223, 'number_subgroup_autclasses': 216, 'number_subgroup_classes': 1812, 'number_subgroups': 6998, 'old_label': None, 'order': 384, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 127], [3, 2], [4, 128], [6, 62], [12, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 6, 4, 2], 'outer_gen_pows': [13825, 13825, 13825, 13825], 'outer_gens': [[263173, 69133, 152507, 266185, 97255], [321713, 152083, 79859, 152507, 180205], [152141, 235297, 152507, 245471, 262867], [96985, 152371, 152219, 266473, 97255]], 'outer_group': '3072.bbo', 'outer_hash': 8324277783353456216, 'outer_nilpotent': False, 'outer_order': 3072, 'outer_permdeg': 18, 'outer_perms': [6267312023, 1068369959024648, 1108907864103589, 753233848029496], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^7.S_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 32], [4, 14], [8, 1]], 'representations': {'PC': {'code': 108323344119670297593651950605313, 'gens': [1, 2, 3, 5, 7], 'pres': [8, -2, -2, -2, -2, -2, -2, -2, -3, 290, 66, 8324, 2420, 116, 8974, 166, 8207]}, 'GLZN': {'d': 2, 'p': 24, 'gens': [155665, 13903, 14017, 69269, 155741, 96775, 14113, 179725]}, 'Perm': {'d': 15, 'gens': [87657298129, 186810626309, 479004643, 375196, 12316, 18498, 87657292800, 3991680]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times C_2^3:D_4', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
