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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1344.4058', 'ambient_counter': 4058, 'ambient_order': 1344, 'ambient_tex': '(C_2\\times C_{12}).D_{28}', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 4, 'counter': 241, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1344.4058.168.e1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '168.e1.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': 168, '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': '8.2', 'subgroup_hash': 2, 'subgroup_order': 8, 'subgroup_tex': 'C_2\\times C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.4058', 'aut_centralizer_order': 192, 'aut_label': '168.e1', 'aut_quo_index': None, 'aut_stab_index': 42, 'aut_weyl_group': '8.3', 'aut_weyl_index': 8064, 'centralizer': '168.e1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['24.g1.a1', '56.e1.a1', '84.e1.a1', '84.g1.a1', '84.g1.b1'], 'contains': ['336.a1.a1', '336.f1.a1'], 'core': '336.a1.a1', 'coset_action_label': None, 'count': 42, 'diagramx': [8133, -1, 9052, -1, 7268, -1, 1229, -1], 'generators': [3, 56], 'label': '1344.4058.168.e1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.d1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '42.d1.a1', 'old_label': '168.e1.a1', 'projective_image': '672.708', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '168.e1.a1', 'subgroup_fusion': None, 'weyl_group': '4.2'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 3], [5, 3]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [1, 22], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.3', 'autcent_hash': 3, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': '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], [4, 1, 4]], 'center_label': '8.2', 'center_order': 8, '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'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 2, '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, 8]], 'label': '8.2', 'linC_count': 12, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 4, 'linQ_dim': 3, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C4', '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': 8, 'number_divisions': 6, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 4]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 3], [5, 3]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [1, 22], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2]], 'representations': {'PC': {'code': 34, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16426]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [251, 504]}, 'Perm': {'d': 6, 'gens': [22, 120, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], '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_4', 'transitive_degree': 8, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 168, 'aut_gen_orders': [6, 6, 12, 12, 6, 42, 6], 'aut_gens': [[1, 2, 8, 112], [741, 282, 104, 788], [737, 506, 24, 112], [17, 62, 72, 844], [1029, 1010, 760, 168], [49, 902, 24, 620], [721, 1126, 8, 172], [753, 230, 744, 560]], 'aut_group': None, 'aut_hash': 729292253715603644, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 64512, 'aut_permdeg': 104, 'aut_perms': [7919405973325071166685661721464111317504644370919786530813733850620921677222414781468787075488558258028806340779669458314497867376870552840602740742252333558151695868, 3150968893710943898833824227707619750453999908141092994325746658235458265674568527313302726167576652980405224678073665252709812776358558156340148880917192501632487522, 94383511570546850379591896408530337365781622349201240394270317325455610872239844575387986684979147326513818279174063029625517350522947820960053743045588045238807706, 2745824442970007323113262357843660381585643188219212329674861077512689803398251888124120191831323673978801321979949971967241042999635711157028993145833315778162763786, 8755807803915520558737402980290580882583344904934713991818671875918883388721463940070324378743045387466933331612275554465496619441010740807259936731042819923397372669, 3154810574109369366273219675796625303324094422756222774020592840875333889497215813009324752325720604173667858834993562458981785224084476391602701585580584201003746875, 8649141006095481753105913672213252765301024524416275356312042332929121255164013724722228086856757196091845664832451094967494002803066563507999056965313202553440133458], 'aut_phi_ratio': 168.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 28, 2, 1], [3, 2, 1, 1], [4, 2, 2, 1], [4, 4, 2, 1], [4, 28, 2, 1], [4, 84, 2, 1], [4, 168, 1, 1], [6, 2, 1, 1], [6, 4, 1, 1], [6, 28, 4, 1], [7, 2, 3, 1], [8, 24, 2, 1], [12, 4, 2, 1], [12, 4, 4, 1], [12, 28, 4, 1], [14, 2, 3, 1], [14, 4, 3, 1], [21, 4, 3, 1], [28, 4, 6, 1], [28, 8, 6, 1], [42, 4, 3, 1], [42, 4, 6, 1], [56, 24, 12, 1], [84, 8, 6, 1], [84, 8, 12, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_{42}\\times D_4).C_6.C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 84, 'autcentquo_group': None, 'autcentquo_hash': 4773224379489722456, 'autcentquo_nilpotent': False, 'autcentquo_order': 8064, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{42}.(C_2^4\\times C_6).C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 28, 2], [3, 2, 1], [4, 2, 2], [4, 4, 2], [4, 28, 2], [4, 84, 2], [4, 168, 1], [6, 2, 1], [6, 4, 1], [6, 28, 4], [7, 2, 3], [8, 24, 2], [12, 4, 6], [12, 28, 4], [14, 2, 3], [14, 4, 3], [21, 4, 3], [28, 4, 6], [28, 8, 6], [42, 4, 9], [56, 24, 12], [84, 8, 18]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '672.708', 'commutator_count': 1, 'commutator_label': '168.39', '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', '7.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 4058, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 28, 1, 2], [3, 2, 1, 1], [4, 2, 1, 2], [4, 4, 1, 2], [4, 28, 1, 2], [4, 84, 1, 2], [4, 168, 1, 1], [6, 2, 1, 1], [6, 4, 1, 1], [6, 28, 2, 2], [7, 2, 3, 1], [8, 24, 1, 2], [12, 4, 1, 2], [12, 4, 2, 2], [12, 28, 2, 2], [14, 2, 3, 1], [14, 4, 3, 1], [21, 4, 3, 1], [28, 4, 3, 2], [28, 8, 3, 2], [42, 4, 3, 1], [42, 4, 6, 1], [56, 24, 6, 2], [84, 8, 3, 2], [84, 8, 6, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 10752, 'exponent': 168, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[8, -1, 6]], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '168.50', 'hash': 4058, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 4, 14, 6], 'inner_gens': [[1, 6, 776, 112], [677, 2, 680, 616], [689, 674, 8, 112], [1, 954, 8, 112]], 'inner_hash': 708, 'inner_nilpotent': False, 'inner_order': 672, 'inner_split': True, 'inner_tex': '(C_2\\times C_6):D_{28}', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 48, 'irrQ_dim': 96, 'irrR_degree': 16, 'irrep_stats': [[1, 8], [2, 46], [4, 32], [8, 10]], 'label': '1344.4058', 'linC_count': 822, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 32, 'linQ_dim': 16, 'linQ_dim_count': 32, 'linR_count': 216, 'linR_degree': 12, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': '(C2*C12).D28', '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, 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': 28, 'number_characteristic_subgroups': 28, 'number_conjugacy_classes': 96, 'number_divisions': 43, 'number_normal_subgroups': 64, 'number_subgroup_autclasses': 168, 'number_subgroup_classes': 284, 'number_subgroups': 2228, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 59], [3, 2], [4, 404], [6, 118], [7, 6], [8, 48], [12, 136], [14, 18], [21, 12], [28, 72], [42, 36], [56, 288], [84, 144]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 12], 'outer_gen_pows': [0, 0, 56, 56], 'outer_gens': [[1, 2, 8, 560], [673, 2, 8, 112], [393, 338, 8, 112], [393, 1014, 40, 1236]], 'outer_group': '96.221', 'outer_hash': 221, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 11, 'outer_perms': [1680, 744, 3669864, 10888828], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{12}:C_2^3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [4, 8], [6, 4], [8, 1], [12, 8], [24, 3], [48, 1]], 'representations': {'PC': {'code': 10779316853140107071173631861347443084440471737305981956665, 'gens': [1, 2, 4, 6], 'pres': [8, 2, 2, 2, 2, 7, 2, 2, 3, 97, 41, 16226, 4050, 24835, 10891, 91, 3844, 14797, 9429, 141, 31374, 166, 28687]}, 'Perm': {'d': 26, 'gens': [621574835364378657689852, 358492124931567, 4411036800533, 580161317289, 5875149352558, 7122198257764, 6758061133824000, 16754357281155028254720000]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_2\\times C_{12}).D_{28}', 'transitive_degree': 336, 'wreath_data': None, 'wreath_product': False}
