-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '648.295', 'ambient_counter': 295, 'ambient_order': 648, 'ambient_tex': 'C_2\\times C_9^2:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 162, 'characteristic': False, 'core_order': 18, 'counter': 12, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '648.295.12.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '12.a1.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': 12, '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': '54.9', 'subgroup_hash': 9, 'subgroup_order': 54, 'subgroup_tex': 'C_3\\times C_{18}', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '648.295', 'aut_centralizer_order': 162, 'aut_label': '12.a1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': '36.12', 'aut_weyl_index': 648, 'centralizer': '4.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '6.a1.a1'], 'contains': ['24.a1.a1', '36.a1.a1', '36.b1.a1', '36.b1.b1', '36.b1.c1'], 'core': '36.a1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [4392, -1, 6329, -1, 2825, -1, 2798, -1], 'generators': [324, 288, 444, 216], 'label': '648.295.12.a1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.a1', 'old_label': '12.a1.a1', 'projective_image': '324.35', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '12.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '54.9', '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, 22], [1, 39], [2, 51], [2, 5], [20, 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], [2, 1, 1, 1], [3, 1, 2, 1], [3, 1, 6, 1], [6, 1, 2, 1], [6, 1, 6, 1], [9, 1, 18, 1], [18, 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], [2, 1, 1], [3, 1, 8], [6, 1, 8], [9, 1, 18], [18, 1, 18]], 'center_label': '54.9', 'center_order': 54, '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', '3.1', '3.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 9, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 4], [6, 1, 2, 4], [9, 1, 6, 3], [18, 1, 6, 3]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 12, 'exponent': 18, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '18.5', 'hash': 9, '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, 54]], 'label': '54.9', 'linC_count': 648, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 27, 'linQ_dim': 8, 'linQ_dim_count': 27, 'linR_count': 162, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3*C18', 'ngens': 4, 'nilpotency_class': 1, '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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 54, 'number_divisions': 16, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 20, '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': [3, 3, 2, 2, 6], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 22], [1, 39], [2, 51], [2, 5], [20, 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': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 9], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 8], [6, 6]], 'representations': {'PC': {'code': 467051, 'gens': [1, 2], 'pres': [4, -3, -2, -3, -3, 21, 46]}, 'GLFp': {'d': 2, 'p': 19, 'gens': [13719, 75456]}, 'Perm': {'d': 14, 'gens': [6227020800, 357120, 79833600, 80884]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 18], '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_{18}', 'transitive_degree': 54, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 72, 'aut_gen_orders': [18, 12, 12, 4], 'aut_gens': [[1, 4, 36], [451, 4, 612], [391, 296, 260], [407, 508, 644], [223, 592, 196]], 'aut_group': None, 'aut_hash': 7935501427042882995, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 23328, 'aut_permdeg': 324, 'aut_perms': [16081590991859316671764281960037551388639712094954080252881782089122134475168740963818641749798934941859714944121755227733785436302391794760031015162131503681612681964026490734743977969739548693465108276639105723673813870206168107474873468578291093965983716313205930933268394125595203030498275371245057573925015567499016934499799929793021300533562205909156538644185168585820956280431675739598832701884877111687557465245197188825638515925221043747081877783693812604156239212772784938834675911749712690784963366312889236778398669938620151805661019464468698529892777830665932423445293501551375663742003779241577417329666259174306223077975281881939203245241526001706682132477413, 222168850902440016820153752470441291698438133111232355341720384998357673216923548732317685636983249296222019033921291722168731128650781813258879792913209612541142041612286498254076095781167832715597252500556296802467566006896618306610761642602424390594858393581989260388181777665097576031886034311082045075066828917686943045161632789867492103599242122826856332373947047879786440923189664771732737140673909921630370179127251462157843742760335039666940463565197233834810678911943997565418113234715663209831793121028290901350686163350776982190846077293075277170687489409610765712358693447205062389420229311265244434000733317091783697569026629176123550306948992797566966488774658, 19218318302118904508780544056870616113834586029051451976610836248606671934747294660180961349560073680632234661042029485029803677848227649141574565457165070254536094739354001566171856931380887078272748267834794326835396982880400313148917070117572897697647319089255717868924024886681226044875836437456570125665608075656975882586754263407970022932105052603919321223672528372728926943474565203915212581696088116594512890280224216066182857636736158606361688539545652806239503878975284228820299883824069725464846320981104885885176316397012365341994924499136946250598774806464832813022336774432413544276480038618458099728418761266657018763462578350199799037557312770577568600157752, 155371710701012845664043747635513138717390229465834291504879574517818036100860869095444771890777017108955011462934672607594790523545107869950740707967791437404760244384864458265707838054165954431153573570510029663152966253788776651808841941098168967527510868544296775156631721438270084651895847870697207447860241908711812375031177158037317327352871696699460565379280553393641494805770469871603354726877694852413515474802220703001330853207376160437604755217086370327642112817623597552675592019821999794371132978902884035901133014158112073811609113937343278716591781092521629217844262858394339833639842864735585969174876204801657823412343586607048266429111509891433128514594611], 'aut_phi_ratio': 108.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 81, 1, 2], [3, 4, 2, 1], [4, 81, 4, 1], [6, 4, 2, 1], [9, 4, 18, 1], [18, 4, 18, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_9^2.C_{12}.C_6.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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 72, 'autcentquo_group': '11664.lk', 'autcentquo_hash': 4509669642411733234, 'autcentquo_nilpotent': False, 'autcentquo_order': 11664, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_9^2:C_{24}:C_6', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 81, 2], [3, 4, 2], [4, 81, 4], [6, 4, 2], [9, 4, 18], [18, 4, 18]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '324.35', 'commutator_count': 1, 'commutator_label': '81.2', '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'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 295, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['324.35', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 81, 1, 2], [3, 4, 1, 2], [4, 81, 2, 2], [6, 4, 1, 2], [9, 4, 3, 6], [18, 4, 3, 6]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 36, 'exponents_of_order': [4, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 1, 18]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '72.45', 'hash': 295, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [4, 9, 9], 'inner_gens': [[1, 72, 340], [581, 4, 36], [381, 4, 36]], 'inner_hash': 35, 'inner_nilpotent': False, 'inner_order': 324, 'inner_split': False, 'inner_tex': 'C_9^2:C_4', 'inner_used': [1, 2], 'irrC_degree': 4, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [4, 40]], 'label': '648.295', 'linC_count': 18, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 6, 'linQ_dim': 12, 'linQ_dim_count': 6, 'linR_count': 18, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*C9^2:C4', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 9, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 48, 'number_divisions': 22, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 36, 'number_subgroup_classes': 74, 'number_subgroups': 1276, 'old_label': None, 'order': 648, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 163], [3, 8], [4, 324], [6, 8], [9, 72], [18, 72]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6, 6], 'outer_gen_pows': [1, 2, 1], 'outer_gens': [[1, 172, 284], [3, 76, 632], [325, 292, 64]], 'outer_group': '72.48', 'outer_hash': 48, 'outer_nilpotent': False, 'outer_order': 72, 'outer_permdeg': 10, 'outer_perms': [720, 40350, 725761], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6\\times D_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [4, 4], [12, 12]], 'representations': {'PC': {'code': 61495217765367759772513961991141978309998402203, 'gens': [1, 3, 5], 'pres': [7, 2, 2, 3, 3, 2, 3, 3, 14, 1514, 345, 79, 6051, 346, 11904, 10721, 102, 1349, 12108, 166, 1182, 10597]}, 'Perm': {'d': 20, 'gens': [123600953843512129, 1, 2335037506744344, 265140936561700206, 393592528344800160, 399987211639843080, 32392094382437184]}}, 'schur_multiplier': [18], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_9^2:C_4', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
