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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1458.421', 'ambient_counter': 421, 'ambient_order': 1458, 'ambient_tex': 'C_9.C_3^2\\times C_{18}', 'central': False, 'central_factor': True, 'centralizer_order': 162, 'characteristic': False, 'core_order': 54, 'counter': 39, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1458.421.27.g1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '27.g1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '27.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 27, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3\\times C_9', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '54.11', 'subgroup_hash': 11, 'subgroup_order': 54, 'subgroup_tex': 'C_9:C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1458.421', 'aut_centralizer_order': 486, 'aut_label': '27.g1', 'aut_quo_index': 1, 'aut_stab_index': 72, 'aut_weyl_group': '54.8', 'aut_weyl_index': 34992, 'centralizer': '9.f1', 'complements': ['54.i1'], 'conjugacy_class_count': 72, 'contained_in': ['9.c1', '9.j1'], 'contains': ['54.g1', '81.b1', '81.d1'], 'core': '27.g1', 'coset_action_label': None, 'count': 72, 'diagramx': [1772, 1687, 1387, 1399], 'generators': [511784, 153730, 317188, 20170], 'label': '1458.421.27.g1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '27.g1', 'normal_contained_in': ['9.c1', '9.j1'], 'normal_contains': ['54.g1', '81.b1', '81.d1'], 'normalizer': '1.a1', 'old_label': '27.g1', 'projective_image': '243.61', 'quotient_action_image': '3.1', 'quotient_action_kernel': '9.1', 'quotient_action_kernel_order': 9, 'quotient_fusion': None, 'short_label': '27.g1', 'subgroup_fusion': None, 'weyl_group': '9.2'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '18.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 6, 'aut_gen_orders': [3, 3, 3, 2], 'aut_gens': [[1, 3], [1, 22], [37, 21], [1, 39], [37, 51]], 'aut_group': '54.8', 'aut_hash': 8, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 54, 'aut_permdeg': 9, 'aut_perms': [165480, 126081, 146378, 219768], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [3, 3, 1, 2], [6, 1, 2, 1], [6, 3, 1, 2], [9, 3, 6, 1], [18, 3, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^2:S_3', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 3, 'autcent_group': '9.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 9, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 1, 2], [3, 3, 2], [6, 1, 2], [6, 3, 2], [9, 3, 6], [18, 3, 6]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '9.2', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 11, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['27.4', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [3, 3, 2, 1], [6, 1, 2, 1], [6, 3, 2, 1], [9, 3, 2, 3], [18, 3, 2, 3]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 24, 'exponent': 18, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[3, 0, 2]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '18.5', 'hash': 11, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [3, 3], 'inner_gens': [[1, 21], [37, 3]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 9, 'inner_split': True, 'inner_tex': 'C_3^2', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 6, 'irrep_stats': [[1, 18], [3, 4]], 'label': '54.11', 'linC_count': 2, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C9:C6', 'ngens': 4, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 10, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 22, 'number_divisions': 12, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 16, 'number_subgroups': 20, 'old_label': None, 'order': 54, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 1], [3, 8], [6, 8], [9, 18], [18, 18]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [2, 0], 'outer_gens': [[1, 33], [1, 22]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 8], [6, 2]], 'representations': {'PC': {'code': 2604245483, 'gens': [1, 2], 'pres': [4, -3, -2, -3, -3, 169, 21, 506, 46]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101732552035792, 73186672638480016]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [28596988, 10475581, 28836015, 23068812]}, 'Perm': {'d': 11, 'gens': [3628800, 357120, 243, 80884]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 6], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_9:C_6', 'transitive_degree': 18, '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': '486.250', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 72, 'aut_gen_orders': [6, 72, 6], 'aut_gens': [[153730, 85954, 330382, 164690], [374464, 505537, 378952, 341603], [201805, 144763, 507772, 347822], [508015, 91957, 378727, 348065]], 'aut_group': None, 'aut_hash': 8386236818640945532, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1889568, 'aut_permdeg': 126, 'aut_perms': [6908350340959909343525159772729327199524821285232916859886764856943037756742765105312328831646496217797809674433265597588929697330389394087434045099014884383614451604085066538107938367755850445992325894345265184, 19656366423910296183416163060403654217091579396574663985355977476181692330061232641379256163539674290580610021913496160063545400585384940518294824981770206108366354321814125154165078735397141354121120079734591366, 23213493815810268783293309151209574095410788697482115693401894113329329478858231111285148622478719397434293776059242161776457985935008930399901843819627661133115554968440280620165184916529346456011310174444702368], 'aut_phi_ratio': 3888.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [3, 1, 6, 1], [3, 3, 24, 1], [6, 1, 2, 1], [6, 1, 6, 1], [6, 3, 24, 1], [9, 1, 18, 1], [9, 1, 54, 1], [9, 3, 48, 1], [9, 3, 144, 1], [18, 1, 18, 1], [18, 1, 54, 1], [18, 3, 48, 1], [18, 3, 144, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4.(C_3^2\\times Q_8).C_3^4.C_2^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 18, 'autcent_group': None, 'autcent_hash': 9045648380693578052, 'autcent_nilpotent': False, 'autcent_order': 39366, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^5.C_3^4.C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '48.29', 'autcentquo_hash': 29, 'autcentquo_nilpotent': False, 'autcentquo_order': 48, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 1, 8], [3, 3, 24], [6, 1, 8], [6, 3, 24], [9, 1, 72], [9, 3, 192], [18, 1, 72], [18, 3, 192]], 'center_label': '162.23', 'center_order': 162, 'central_product': True, 'central_quotient': '9.2', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 421, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['81.14', 1], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 4], [3, 3, 2, 12], [6, 1, 2, 4], [6, 3, 2, 12], [9, 1, 6, 12], [9, 3, 2, 24], [9, 3, 6, 24], [18, 1, 6, 12], [18, 3, 2, 24], [18, 3, 6, 24]], 'element_repr_type': 'GLZq', 'elementary': 3, 'eulerian_function': 1263600, 'exponent': 18, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '162.55', 'hash': 421, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [3, 1, 3, 1], 'inner_gens': [[153730, 85954, 330868, 164690], [153730, 85954, 330382, 164690], [153244, 85954, 330382, 164690], [153730, 85954, 330382, 164690]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 9, 'inner_split': False, 'inner_tex': 'C_3^2', 'inner_used': [1, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': None, 'irrep_stats': [[1, 486], [3, 108]], 'label': '1458.421', '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': 'C9.C3^2*C18', 'ngens': 7, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 16, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 594, 'number_divisions': 154, 'number_normal_subgroups': 818, 'number_subgroup_autclasses': 72, 'number_subgroup_classes': 986, 'number_subgroups': 1322, 'old_label': None, 'order': 1458, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 1], [3, 80], [6, 80], [9, 648], [18, 648]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 72, 'outer_gen_orders': [24, 24, 24], 'outer_gen_pows': [19684, 19684, 19684], 'outer_gens': [[201562, 256135, 197335, 348065], [196849, 151009, 508015, 171179], [153226, 269329, 201562, 524978]], 'outer_group': None, 'outer_hash': 7187075652081419979, 'outer_nilpotent': False, 'outer_order': 209952, 'outer_permdeg': 21, 'outer_perms': [12934219117314229014, 24939298464629, 2581667054525610506], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_3:S_3.C_6^2.C_3^4.C_2^2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 38, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3, 3, 9], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 80], [6, 54], [18, 18]], 'representations': {'PC': {'code': 53283262981667330427, 'gens': [1, 2, 4, 5], 'pres': [7, -3, -3, -3, -3, -2, -3, -3, 50, 14367, 102, 166]}, 'GLZN': {'d': 2, 'p': 54, 'gens': [158437, 3956593, 8268343, 8345645, 3936643, 6412903, 2991835]}, 'GLZq': {'d': 2, 'q': 27, 'gens': [212521, 511784, 196840, 78736, 19693, 87946, 19927]}, 'Perm': {'d': 38, 'gens': [12201962650488652486714904273541269120823, 22805379974763020892031935207425554274356, 33466329784485185960623305484321400730423, 44103971634799185063965867437183453096756, 54741592634909846309090591073678000806400, 65379716172264364790130485734348063221556, 13763753091226345046315979581580902400000000]}}, 'schur_multiplier': [3, 3, 3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3, 3, 18], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_9.C_3^2\\times C_{18}', 'transitive_degree': 486, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '27.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 6, 'aut_gen_orders': [3, 3, 2, 2, 6], 'aut_gens': [[1, 3], [1, 23], [1, 12], [2, 24], [2, 4], [11, 15]], 'aut_group': '108.28', 'aut_hash': 28, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 108, 'aut_permdeg': 11, 'aut_perms': [4556304, 11088384, 1, 169567, 26386807], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 1, 6, 1], [9, 1, 18, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^2:D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '108.28', 'autcent_hash': 28, 'autcent_nilpotent': False, 'autcent_order': 108, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^2:D_6', '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, 8], [9, 1, 18]], 'center_label': '27.2', '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': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 4], [9, 1, 6, 3]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 4, 'exponent': 9, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '9.2', 'hash': 2, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 3], [1, 3]], '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': '27.2', 'linC_count': 216, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 9, 'linQ_dim': 8, 'linQ_dim_count': 9, 'linR_count': 54, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*C9', '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': 27, 'number_divisions': 8, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 10, 'number_subgroups': 10, 'old_label': None, 'order': 27, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 8], [9, 18]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [3, 3, 2, 2, 6], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 23], [1, 12], [2, 24], [2, 4], [11, 15]], 'outer_group': '108.28', 'outer_hash': 28, 'outer_nilpotent': False, 'outer_order': 108, 'outer_permdeg': 11, 'outer_perms': [4556304, 11088384, 1, 169567, 26386807], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_3^2:D_6', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 12, 'pgroup': 3, 'primary_abelian_invariants': [3, 9], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4], [6, 3]], 'representations': {'PC': {'code': 34, 'gens': [1, 2], 'pres': [3, -3, 3, -3, 22]}, 'GLFp': {'d': 2, 'p': 19, 'gens': [34311, 75456]}, 'Perm': {'d': 12, 'gens': [357120, 79833600, 80884]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 9], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times C_9', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}
