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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '17496.ln', 'ambient_counter': 300, 'ambient_order': 17496, 'ambient_tex': '(C_3^3\\times \\He_3):D_{12}', 'central': False, 'central_factor': False, 'centralizer_order': 729, 'characteristic': False, 'core_order': 3, 'counter': 1526, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '17496.ln.648.bf1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '648.bf1', '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': 648, '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': '27.5', 'subgroup_hash': 5, 'subgroup_order': 27, 'subgroup_tex': 'C_3^3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '17496.ln', 'aut_centralizer_order': None, 'aut_label': '648.bf1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '24.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['216.l1', '216.ek1', '216.em1', '216.er1', '216.ew1', '216.fc1'], 'contains': ['1944.e1', '1944.m1', '1944.q1', '1944.dh1', '1944.eo1'], 'core': '5832.a1', 'coset_action_label': None, 'count': 24, 'diagramx': None, 'generators': [456727548537813895695936, 576, 20206644910181452859], 'label': '17496.ln.648.bf1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '24.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '24.a1', 'old_label': '648.bf1', 'projective_image': '17496.ln', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '648.bf1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '27.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 312, 'aut_gen_orders': [2, 13], 'aut_gens': [[1, 3, 9], [6, 2, 9], [5, 17, 11]], 'aut_group': '11232.a', 'aut_hash': 778507202365856770, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 11232, 'aut_permdeg': 15, 'aut_perms': [6452863345, 297839648544], 'aut_phi_ratio': 624.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [3, 1, 26, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(3,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 312, 'autcent_group': '11232.a', 'autcent_hash': 778507202365856770, 'autcent_nilpotent': False, 'autcent_order': 11232, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(3,3)', '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], [3, 1, 26]], 'center_label': '27.5', 'center_order': 27, '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': ['3.1', '3.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 5, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 3]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 13]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3, 13], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '27.5', 'hash': 5, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 3, 9], [1, 3, 9], [1, 3, 9]], '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, 27]], 'label': '27.5', 'linC_count': 1872, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 234, 'linQ_dim': 6, 'linQ_dim_count': 234, 'linR_count': 234, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3^3', 'ngens': 3, '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': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 27, 'number_divisions': 14, 'number_normal_subgroups': 28, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 28, 'number_subgroups': 28, 'old_label': None, 'order': 27, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 26]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 312, 'outer_gen_orders': [2, 13], 'outer_gen_pows': [0, 0], 'outer_gens': [[6, 2, 9], [5, 17, 11]], 'outer_group': '11232.a', 'outer_hash': 778507202365856770, 'outer_nilpotent': False, 'outer_order': 11232, 'outer_permdeg': 15, 'outer_perms': [6452863345, 297839648544], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\GL(3,3)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 3, 'primary_abelian_invariants': [3, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 13]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3], 'pres': [3, -3, 3, 3]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [41624334336939899, 125101725607167719, 125101750005091125]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [10475581, 9534373, 23068812]}, 'Perm': {'d': 9, 'gens': [80640, 240, 4]}}, 'schur_multiplier': [3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 3], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3', 'transitive_degree': 27, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [6, 6, 6, 6, 12, 6], 'aut_gens': [[53959604401617793769671, 25959450103576957175193], [141790946212024439904714, 192694406491194511282500], [372203718064533633242367, 399594708987382336337433], [366796595534309375057447, 399594708987382336337493], [188861823676500929745991, 372572957935895378094925], [25981476741363360233671, 423117246816449032746180], [134076137490820473137894, 36505858734773231288060]], 'aut_group': None, 'aut_hash': 2049735307842522356, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1259712, 'aut_permdeg': 648, 'aut_perms': 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'aut_phi_ratio': 216.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 162, 1, 1], [2, 486, 1, 1], [3, 2, 1, 2], [3, 4, 1, 3], [3, 4, 2, 1], [3, 6, 3, 2], [3, 8, 1, 3], [3, 8, 2, 2], [3, 12, 3, 1], [3, 12, 6, 1], [3, 18, 1, 1], [3, 24, 3, 3], [3, 24, 6, 2], [3, 36, 1, 2], [3, 36, 2, 2], [3, 72, 1, 1], [3, 72, 2, 4], [3, 72, 4, 2], [4, 486, 1, 1], [6, 18, 1, 2], [6, 36, 1, 1], [6, 54, 3, 2], [6, 108, 3, 1], [6, 162, 1, 1], [6, 162, 2, 1], [6, 162, 6, 1], [6, 324, 1, 3], [6, 324, 2, 2], [6, 324, 6, 1], [6, 486, 2, 1], [6, 972, 1, 1], [6, 972, 2, 1], [12, 486, 2, 1], [12, 486, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^6.C_3.D_6^2.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': 12, 'autcentquo_group': None, 'autcentquo_hash': 2049735307842522356, 'autcentquo_nilpotent': False, 'autcentquo_order': 1259712, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^6.C_3^3.C_2^5.C_2', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 162, 1], [2, 486, 1], [3, 2, 2], [3, 4, 5], [3, 6, 6], [3, 8, 7], [3, 12, 9], [3, 18, 1], [3, 24, 21], [3, 36, 6], [3, 72, 17], [4, 486, 1], [6, 18, 2], [6, 36, 1], [6, 54, 6], [6, 108, 3], [6, 162, 9], [6, 324, 13], [6, 486, 2], [6, 972, 3], [12, 486, 8]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '17496.ln', 'commutator_count': 2, 'commutator_label': None, 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 300, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 162, 1, 1], [2, 486, 1, 1], [3, 2, 1, 2], [3, 4, 1, 3], [3, 4, 2, 1], [3, 6, 1, 6], [3, 8, 1, 5], [3, 8, 2, 1], [3, 12, 1, 3], [3, 12, 2, 3], [3, 18, 1, 1], [3, 24, 1, 15], [3, 24, 2, 3], [3, 36, 1, 2], [3, 36, 2, 2], [3, 72, 1, 3], [3, 72, 2, 7], [4, 486, 1, 1], [6, 18, 1, 2], [6, 36, 1, 1], [6, 54, 1, 6], [6, 108, 1, 3], [6, 162, 1, 1], [6, 162, 2, 4], [6, 324, 1, 3], [6, 324, 2, 5], [6, 486, 2, 1], [6, 972, 1, 1], [6, 972, 2, 1], [12, 486, 2, 4]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 24, 'exponent': 12, 'exponents_of_order': [7, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[24, 0, 6], [24, 1, 6]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '1944.3917', 'hash': 5700674277148450841, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [12, 6], 'inner_gens': [[53959604401617793769671, 133915603640045337775380], [133941149792958821524505, 25959450103576957175193]], 'inner_hash': 5700674277148450841, 'inner_nilpotent': False, 'inner_order': 17496, 'inner_split': True, 'inner_tex': '(C_3^3\\times \\He_3):D_{12}', 'inner_used': [1, 2], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 4], [2, 13], [4, 22], [6, 24], [8, 24], [12, 18], [24, 21]], 'label': '17496.ln', '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': False, 'metacyclic': False, 'monomial': True, 'name': '(C3^3*He3):D12', 'ngens': 2, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 57, 'number_characteristic_subgroups': 40, 'number_conjugacy_classes': 126, 'number_divisions': 94, 'number_normal_subgroups': 58, 'number_subgroup_autclasses': 2491, 'number_subgroup_classes': 8638, 'number_subgroups': 483456, 'old_label': None, 'order': 17496, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 657], [3, 2186], [4, 486], [6, 10278], [12, 3888]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 6, 6], 'outer_gen_pows': [265997388004176413455920, 0, 0], 'outer_gens': [[192233858983225494588871, 44018669046503294075673], [189256594518860217753751, 399255300007309410623433], [399254073162826218266311, 426766431027912046807833]], 'outer_group': '72.49', 'outer_hash': 49, 'outer_nilpotent': False, 'outer_order': 72, 'outer_permdeg': 10, 'outer_perms': [362935, 49, 726480], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6:D_6', 'pc_rank': None, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 7], [4, 13], [6, 12], [8, 14], [12, 12], [16, 8], [24, 21], [48, 3]], 'representations': {'PC': {'code': '3932659447255379226645934933240863174816626661878446312288633568040098739448526113778807430656784712418700320163975003899436159', 'gens': [1, 2, 5, 6, 7, 8, 9, 10], 'pres': [10, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 132960, 325401, 51, 43082, 82, 27523, 66004, 17414, 622085, 311055, 2185, 374646, 61336, 15566, 13686, 103687, 1265768, 532458, 62128, 37028, 1418, 1478409, 652819, 425429, 89739, 3369]}, 'Perm': {'d': 24, 'gens': [53959604401617793769671, 25959450103576957175193]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '(C_3^3\\times \\He_3):D_{12}', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
