-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '960.10096', 'ambient_counter': 10096, 'ambient_order': 960, 'ambient_tex': 'C_5\\times C_4^2.D_6', 'central': False, 'central_factor': False, 'centralizer_order': 480, 'characteristic': False, 'core_order': 1, 'counter': 167, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '960.10096.480.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '480.c1', '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': 480, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': True, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '960.10096', 'aut_centralizer_order': None, 'aut_label': '480.c1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '2.b1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['96.c1', '160.c1', '240.b1', '240.f1'], 'contains': ['960.a1'], 'core': '960.a1', 'coset_action_label': None, 'count': 4, 'diagramx': None, 'generators': [2], 'label': '960.10096.480.c1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '240.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.b1', 'old_label': '480.c1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '480.c1', 'subgroup_fusion': None, 'weyl_group': None}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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]], 'center_label': '2.1', 'center_order': 2, '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'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', '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': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '160.228', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [12, 4, 12, 12, 12, 12, 6, 12], 'aut_gens': [[1, 2, 4, 16], [163, 490, 732, 216], [491, 10, 724, 952], [881, 482, 486, 216], [561, 482, 12, 112], [803, 482, 246, 16], [321, 482, 492, 696], [89, 490, 4, 304], [403, 2, 246, 696]], 'aut_group': None, 'aut_hash': 4060798118887912709, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 49152, 'aut_permdeg': 68, 'aut_perms': [847201541533104018696478311909290380682502285839325057919204609867297591404040914012791357488326, 779628553323091807565952466608094000900468651282029856175987853547901414999066500907291811459324, 782367364451634378668318363094600921375276940993073272945916419864074107526412771526156042162804, 1301878823270683752113933249989016151243555833688776359038803051460419544919997981455379403510506, 203345269196047420473203565704165421202047643287861596274354572826781708555990968258235296628208, 1656891666919595558991701472323247703143625215153119888153658575532993450277234728081207329068952, 205596030793298087450163603946109252893283245540244347013529249979073655724881188980760479040332, 632152282530761706268169956833123640239934686938136817518493996770456397711152865527897406636094], 'aut_phi_ratio': 192.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [3, 2, 1, 1], [4, 2, 2, 2], [4, 2, 8, 1], [4, 6, 8, 2], [5, 1, 4, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 4, 2, 1], [10, 1, 4, 1], [10, 1, 8, 1], [10, 2, 8, 1], [12, 2, 4, 1], [12, 4, 2, 1], [12, 4, 8, 1], [15, 2, 4, 1], [20, 2, 8, 2], [20, 2, 32, 1], [20, 6, 32, 2], [30, 2, 4, 1], [30, 2, 8, 1], [30, 4, 8, 1], [60, 2, 16, 1], [60, 4, 8, 1], [60, 4, 32, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3:(C_2\\times C_4\\times C_2^6.C_2^5)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': None, 'autcent_hash': 1718285292446712972, 'autcent_nilpotent': True, 'autcent_order': 1024, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '48.51', 'autcentquo_hash': 51, 'autcentquo_nilpotent': False, 'autcentquo_order': 48, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times D_6', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [3, 2, 1], [4, 2, 12], [4, 6, 16], [5, 1, 4], [6, 2, 3], [6, 4, 2], [10, 1, 12], [10, 2, 8], [12, 2, 4], [12, 4, 10], [15, 2, 4], [20, 2, 48], [20, 6, 64], [30, 2, 12], [30, 4, 8], [60, 2, 16], [60, 4, 40]], 'center_label': '20.5', 'center_order': 20, 'central_product': True, 'central_quotient': '48.51', 'commutator_count': 1, 'commutator_label': '6.2', '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', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 10096, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['192.1075', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [3, 2, 1, 1], [4, 2, 1, 4], [4, 2, 2, 4], [4, 6, 1, 8], [4, 6, 2, 4], [5, 1, 4, 1], [6, 2, 1, 3], [6, 4, 1, 2], [10, 1, 4, 3], [10, 2, 4, 2], [12, 2, 2, 2], [12, 4, 1, 2], [12, 4, 2, 4], [15, 2, 4, 1], [20, 2, 4, 4], [20, 2, 8, 4], [20, 6, 4, 8], [20, 6, 8, 4], [30, 2, 4, 3], [30, 4, 4, 2], [60, 2, 8, 2], [60, 4, 4, 2], [60, 4, 8, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 5110560, 'exponent': 60, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '240.206', 'hash': 10096, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 2, 6], 'inner_gens': [[1, 2, 4, 176], [1, 2, 484, 16], [1, 482, 4, 16], [801, 2, 4, 16]], '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], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 160], [2, 80], [4, 30]], 'label': '960.10096', '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': 'C5*C4^2.D6', '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': 32, 'number_characteristic_subgroups': 38, 'number_conjugacy_classes': 270, 'number_divisions': 80, 'number_normal_subgroups': 302, 'number_subgroup_autclasses': 168, 'number_subgroup_classes': 532, 'number_subgroups': 976, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 120], [5, 4], [6, 14], [10, 28], [12, 48], [15, 8], [20, 480], [30, 56], [60, 192]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 4, 2, 4, 4, 4, 4], 'outer_gen_pows': [0, 0, 64, 0, 64, 0, 0], 'outer_gens': [[161, 482, 484, 472], [89, 482, 6, 368], [163, 2, 724, 496], [723, 490, 252, 600], [321, 490, 14, 304], [241, 482, 12, 696], [483, 482, 244, 216]], 'outer_group': None, 'outer_hash': 2254786802738627770, 'outer_nilpotent': True, 'outer_order': 1024, 'outer_permdeg': 128, 'outer_perms': [73766644167315114738804790016625678793082355628921460436681908558089873649705551110884961927108869271802396021012417190029803119182810687349561783760163025038702267360068190545402250177610945299575227635884376068142, 264580010069027461472337864709827586647744781205464948124486508143462994841408133988336985354977704362611480710684259972144203193649250887278122435600840407561228969963118973986608202326819578952079764959293911749408, 166379233996234159580285853520836038237666255685811120667396993534023866134947100280724868408037193385795591473649363456102871973896990128821262658727514773495521578404572834225868418232311668808132986999565679066814, 211946619914509325656142161151465579648574410358596761248950292981698369107014866538445448541784905958083249455206399722887437861660654404608639463319228645177517942707352682599965859728808499590712348466756702803029, 189967100298002644937992510126691201477783781647376635153738245052622584298736532141629091154213429927877491625983353628348142557713815803866551590793876060434014842809617287683453991312148527371115346858209401937941, 22453248187434572357427039501254732824681168430966894740284506153603436910264355103427023033438440462808732509891278029326495518124323799493752631811887721383976162739436240418532647377605636489121892791736296375456, 140596322960092000598651271090273976412579939749904383942927864654910518750376597105104030456618489779635916789055384614634598571147570325216545413855179164342116397898633166654555207723524791903431847467299707489376], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^7.C_2^3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 28, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 4, 5], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 16], [4, 22], [8, 18], [16, 6], [32, 2]], 'representations': {'PC': {'code': 35750904539091277305045550929006858403847, 'gens': [1, 2, 3, 5], 'pres': [8, -2, -2, -2, -2, -2, -2, -3, -5, 3840, 5818, 66, 7044, 116, 16901, 141, 39430, 222]}, 'Perm': {'d': 28, 'gens': [12553020084755943525148416720, 22991339605126724732905804800, 34794315026039879700764774400, 46073266816666934362706052480, 33, 87091200, 57366271753015597261553664000, 5760]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 20], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times C_4^2.D_6', 'transitive_degree': 480, 'wreath_data': None, 'wreath_product': False}
