-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1296.887', 'ambient_counter': 887, 'ambient_order': 1296, 'ambient_tex': 'C_6^2.D_{18}', 'central': False, 'central_factor': False, 'centralizer_order': 24, 'characteristic': False, 'core_order': 2, 'counter': 347, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '1296.887.324.f1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '324.f1.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': 324, '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': '4.1', 'subgroup_hash': 1, 'subgroup_order': 4, 'subgroup_tex': 'C_4', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '1296.887', 'aut_centralizer_order': 288, 'aut_label': '324.f1', 'aut_quo_index': None, 'aut_stab_index': 54, 'aut_weyl_group': '2.1', 'aut_weyl_index': 15552, 'centralizer': '54.h1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['108.t1.a1', '108.u1.a1', '108.v1.a1', '108.w1.a1', '162.c1.a1', '162.d1.b1', '162.d1.c1'], 'contains': ['648.b1.a1'], 'core': '648.b1.a1', 'coset_action_label': None, 'count': 27, 'diagramx': [8785, -1, 8990, -1, 8495, -1, 8424, -1], 'generators': [3], 'label': '1296.887.324.f1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.d1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '27.c1.a1', 'old_label': '324.f1.a1', 'projective_image': '648.297', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '324.f1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [3]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', '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, 1], [4, 1, 2]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [2], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4]], 'label': '4.1', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 1], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -2, -2, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [56, 15], 'family': 'CSOPlus'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [21]}, 'Perm': {'d': 4, 'gens': [22, 7]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'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': 36, 'aut_gen_orders': [18, 2, 12, 6, 12, 12], 'aut_gens': [[1, 2, 12, 216], [333, 322, 876, 1080], [1089, 538, 1284, 216], [333, 218, 372, 216], [1193, 1202, 924, 1080], [1193, 1082, 420, 1080], [329, 1054, 276, 216]], 'aut_group': None, 'aut_hash': 3470787610861739022, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 31104, 'aut_permdeg': 144, 'aut_perms': [2972100677950393210924002383708636898310707853981653439509689033320407093369147193064300285251401855206583448792170426830137334013371689837186540087319093530566519860987915346721874337577799657339584723965377199229476831223340073604589634452947670731, 3518325857194164296719357624816584974912934612327549059451852556913788167791513756302524880350062103637388881472393605302769451040928787108432005083862754534857196747705670227250717392724492227305550161994732101671380499089089341903342160199328573417, 3430581952144546514299825936747596398557950869368428342853326900210856658630867460579578808680633662086687338337691768671435895833821984559392339052440386885837845671453292287844867387585599990782068771626230881165251751467444014055858019533965865725, 1414525190405698553161038165809619857823357923220642496203534570444382032461860848864523731556234632683080636466227712860201589826415881852580811098047030775618146328735631179798812553904542881587236379087230866064423988428197688599556056396155329208, 1473282440807741789834168886190886135657439073578702831856167005082556353795448684547438120709096881208598788216037986174972876237899729292918236246045262216779060371193472018921227202297761519559062406693178067775692100486608968597787690337292087648, 2567514860269919206288863982494700028723342249231399939471102561077206204766374573077357632025549165677871376423850614606307921679564014662595023366542590565577918430302608180439551649796320713114817616491394548542054667578993216714407521541886876841], 'aut_phi_ratio': 72.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 18, 2, 1], [2, 54, 2, 1], [3, 2, 1, 2], [3, 4, 1, 1], [3, 6, 1, 1], [3, 12, 1, 1], [4, 54, 2, 1], [6, 2, 1, 2], [6, 2, 2, 2], [6, 4, 1, 1], [6, 4, 2, 1], [6, 6, 1, 1], [6, 6, 2, 1], [6, 12, 1, 1], [6, 12, 2, 1], [6, 18, 4, 1], [6, 54, 4, 1], [9, 6, 3, 1], [9, 12, 3, 1], [12, 54, 4, 1], [18, 6, 3, 1], [18, 6, 6, 1], [18, 12, 3, 1], [18, 12, 6, 1], [18, 18, 12, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_3\\times C_9).C_6^2.C_2^5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 18, 'autcentquo_group': '972.429', 'autcentquo_hash': 429, 'autcentquo_nilpotent': False, 'autcentquo_order': 972, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_3^3.S_3^2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 18, 2], [2, 54, 2], [3, 2, 2], [3, 4, 1], [3, 6, 1], [3, 12, 1], [4, 54, 2], [6, 2, 6], [6, 4, 3], [6, 6, 3], [6, 12, 3], [6, 18, 4], [6, 54, 4], [9, 6, 3], [9, 12, 3], [12, 54, 4], [18, 6, 9], [18, 12, 9], [18, 18, 12]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '324.37', 'commutator_count': 1, 'commutator_label': '162.24', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 887, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['648.55', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 18, 1, 2], [2, 54, 1, 2], [3, 2, 1, 2], [3, 4, 1, 1], [3, 6, 1, 1], [3, 12, 1, 1], [4, 54, 1, 2], [6, 2, 1, 6], [6, 4, 1, 3], [6, 6, 1, 3], [6, 12, 1, 3], [6, 18, 2, 2], [6, 54, 2, 2], [9, 6, 3, 1], [9, 12, 3, 1], [12, 54, 2, 2], [18, 6, 3, 3], [18, 12, 3, 3], [18, 18, 6, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 18144, 'exponent': 36, 'exponents_of_order': [4, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '18.5', 'frattini_quotient': '72.46', 'hash': 887, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [2, 6, 9, 3], 'inner_gens': [[1, 658, 12, 1080], [653, 2, 1068, 1080], [1, 458, 12, 216], [433, 434, 12, 216]], 'inner_hash': 37, 'inner_nilpotent': False, 'inner_order': 324, 'inner_split': True, 'inner_tex': 'C_3^2:D_{18}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 42], [4, 16], [6, 8], [12, 4]], 'label': '1296.887', 'linC_count': 72, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 13, 'linQ_degree_count': 8, 'linQ_dim': 13, 'linQ_dim_count': 8, 'linR_count': 24, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C6^2.D18', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 31, 'number_characteristic_subgroups': 31, 'number_conjugacy_classes': 78, 'number_divisions': 46, 'number_normal_subgroups': 63, 'number_subgroup_autclasses': 216, 'number_subgroup_classes': 361, 'number_subgroups': 3883, 'old_label': None, 'order': 1296, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 147], [3, 26], [4, 108], [6, 366], [9, 54], [12, 216], [18, 378]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 12], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[649, 118, 12, 1080], [757, 118, 12, 1080], [649, 10, 12, 1080], [1, 766, 804, 1080]], 'outer_group': '96.167', 'outer_hash': 167, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 11, 'outer_perms': [41040, 90720, 3719544, 8509684], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 24, '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, 8], [12, 12]], 'representations': {'PC': {'code': 332223068011143471352593927154044954241661170551436204589, 'gens': [1, 2, 4, 7], 'pres': [8, -2, -2, -3, -2, -3, -3, -2, -3, 10529, 41, 194, 17099, 7027, 91, 12492, 4580, 156, 3469, 60486, 30254, 166, 55303, 27663]}, 'Perm': {'d': 24, 'gens': [2452749366752054105850, 12936675055, 186814252800, 186810624000, 29377311667228901428676, 16371, 56412204651114504192000, 104197]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6^2.D_{18}', 'transitive_degree': 72, 'wreath_data': None, 'wreath_product': False}
