Formats: - HTML - YAML - JSON - 2026-07-19T02:55:33.644929
  • gps_subgroup_search • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '944784.rx', 'ambient_counter': 466, 'ambient_order': 944784, 'ambient_tex': 'C_3^6.S_3^2:S_3^2', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': True, 'core_order': 26244, 'counter': 126, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '944784.rx.36.A', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '36.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '36.10', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 10, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 36, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'S_3^2', 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': None, 'subgroup_hash': 7803675823749077841, 'subgroup_order': 26244, 'subgroup_tex': 'C_3^5.C_3^2.D_6', 'supersolvable': True, 'sylow': 0}
  • gps_subgroup_data • Show schema
    {'ambient': '944784.rx', 'aut_centralizer_order': None, 'aut_label': '36.A', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': None, 'complements': None, 'conjugacy_class_count': 1, 'contained_in': None, 'contains': None, 'core': '36.A', 'coset_action_label': None, 'count': 1, 'diagramx': [2052, 1423, 2139, 7135], 'generators': [5423661851800953859178358486848134772, 17684330570401657831907612774644, 191472051032343538037001321677135365749125, 4865804190709907763, 271353676394454377792702129150099764, 127651814341427291341507040516023521515254, 418802671169936714973312003, 3259463770706534265128308294164584130, 26978572725239511210388, 10628907036246685625578073782240332522387], 'label': '944784.rx.36.A', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '36.A', 'normal_contained_in': [], 'normal_contains': [], 'normalizer': '1.a1', 'old_label': '36.a1', 'projective_image': '944784.rx', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '36.A', 'subgroup_fusion': None, 'weyl_group': None}
  • label None does not appear in gps_groups
  • gps_groups • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 36, 'aut_gen_orders': [18, 12, 36, 12], 'aut_gens': [[244050880369023955532401898366136741200668, 297836716282674354632265147391380864733439, 371989532658154783760856509160051209468852], [116698076512707024490563436850920521244226, 308170127269697832572559729010210166959173, 361659907452150912008539939729998801997467], [159563821392093273817149243553580007074399, 95739815439710224363804009597159013809228, 106366566326727483880687395889058934529917], [371991433817613907939595806834347768048242, 297535516318980987423612988807932266701174, 308168786151247430076429023192930721488514], [307872441842612155838820698641199481749612, 361658548982365794838147758440567939193000, 244050065439601433060822742973121355624814]], 'aut_group': None, 'aut_hash': 1591828275226491776, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 51018336, 'aut_permdeg': 1026, 'aut_perms': 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'aut_phi_ratio': 162.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 54, 1, 1], [2, 486, 2, 1], [2, 729, 1, 2], [2, 4374, 1, 1], [2, 6561, 1, 1], [3, 2, 2, 1], [3, 4, 1, 3], [3, 8, 1, 2], [3, 8, 2, 2], [3, 16, 1, 1], [3, 36, 1, 1], [3, 72, 1, 2], [3, 72, 9, 1], [3, 144, 1, 1], [3, 144, 9, 1], [3, 324, 1, 1], [3, 648, 6, 1], [3, 1296, 3, 1], [3, 1296, 6, 1], [4, 13122, 2, 1], [6, 108, 1, 3], [6, 108, 2, 1], [6, 216, 1, 2], [6, 216, 2, 1], [6, 972, 1, 1], [6, 972, 2, 2], [6, 1458, 2, 1], [6, 1944, 1, 1], [6, 1944, 2, 1], [6, 1944, 9, 1], [6, 2916, 1, 3], [6, 8748, 1, 2], [6, 8748, 2, 1], [6, 17496, 1, 1], [6, 17496, 6, 1], [6, 26244, 1, 2], [9, 36, 3, 1], [9, 36, 9, 1], [9, 72, 3, 2], [9, 72, 9, 1], [9, 144, 3, 1], [9, 162, 6, 1], [9, 324, 3, 1], [9, 324, 6, 1], [9, 648, 3, 3], [9, 648, 9, 1], [9, 648, 18, 1], [9, 1296, 9, 1], [12, 26244, 2, 1], [18, 108, 9, 1], [18, 216, 9, 1], [18, 972, 3, 1], [18, 972, 6, 3], [18, 972, 9, 1], [18, 1458, 6, 1], [18, 1944, 3, 2], [18, 1944, 9, 1], [18, 1944, 18, 1], [18, 2916, 3, 2], [18, 2916, 6, 1], [18, 2916, 9, 1], [18, 5832, 3, 2], [18, 8748, 3, 1], [18, 8748, 6, 2], [18, 17496, 3, 1], [36, 26244, 6, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^6.C_3^4.C_6^3.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': 36, 'autcentquo_group': None, 'autcentquo_hash': 1591828275226491776, 'autcentquo_nilpotent': False, 'autcentquo_order': 51018336, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^6.C_3^5.C_6^2.C_2^3', 'cc_stats': [[1, 1, 1], [2, 54, 1], [2, 486, 2], [2, 729, 2], [2, 4374, 1], [2, 6561, 1], [3, 2, 2], [3, 4, 3], [3, 8, 6], [3, 16, 1], [3, 36, 1], [3, 72, 11], [3, 144, 10], [3, 324, 1], [3, 648, 6], [3, 1296, 9], [4, 13122, 2], [6, 108, 5], [6, 216, 4], [6, 972, 5], [6, 1458, 2], [6, 1944, 12], [6, 2916, 3], [6, 8748, 4], [6, 17496, 7], [6, 26244, 2], [9, 36, 12], [9, 72, 15], [9, 144, 3], [9, 162, 6], [9, 324, 9], [9, 648, 36], [9, 1296, 9], [12, 26244, 2], [18, 108, 9], [18, 216, 9], [18, 972, 30], [18, 1458, 6], [18, 1944, 33], [18, 2916, 21], [18, 5832, 6], [18, 8748, 15], [18, 17496, 3], [36, 26244, 6]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '944784.rx', 'commutator_count': 1, 'commutator_label': None, 'complements_known': False, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 14, 'conjugacy_classes_known': True, 'counter': 466, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 54, 1, 1], [2, 486, 1, 2], [2, 729, 1, 2], [2, 4374, 1, 1], [2, 6561, 1, 1], [3, 2, 1, 2], [3, 4, 1, 3], [3, 8, 1, 6], [3, 16, 1, 1], [3, 36, 1, 1], [3, 72, 1, 11], [3, 144, 1, 10], [3, 324, 1, 1], [3, 648, 1, 6], [3, 1296, 1, 9], [4, 13122, 1, 2], [6, 108, 1, 5], [6, 216, 1, 4], [6, 972, 1, 5], [6, 1458, 1, 2], [6, 1944, 1, 12], [6, 2916, 1, 3], [6, 8748, 1, 4], [6, 17496, 1, 7], [6, 26244, 1, 2], [9, 36, 3, 4], [9, 72, 3, 5], [9, 144, 3, 1], [9, 162, 3, 2], [9, 324, 3, 3], [9, 648, 3, 12], [9, 1296, 3, 3], [12, 26244, 1, 2], [18, 108, 3, 3], [18, 216, 3, 3], [18, 972, 3, 10], [18, 1458, 3, 2], [18, 1944, 3, 11], [18, 2916, 3, 7], [18, 5832, 3, 2], [18, 8748, 3, 5], [18, 17496, 3, 1], [36, 26244, 3, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 36, 'exponents_of_order': [10, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[24, 1, 18], [48, 1, 9], [72, 1, 18], [144, 1, 12]], 'familial': False, 'frattini_label': '729.504', 'frattini_quotient': '1296.3528', 'hash': 4214627173336113295, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [36, 4, 6], 'inner_gens': [[244050880369023955532401898366136741200668, 180229591054657757954706805809947997895946, 307867552619483051164819294722402277621709], [234009994187128676174962741623002502470158, 297836716282674354632265147391380864733439, 287204516031430505936748878732044555615436], [52873544325509173749300093954575351296707, 361363053052184771814282767790243654249845, 371989532658154783760856509160051209468852]], 'inner_hash': 4214627173336113295, 'inner_nilpotent': False, 'inner_order': 944784, 'inner_split': None, 'inner_tex': 'C_3^6.S_3^2:S_3^2', 'inner_used': [1, 2, 3], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 8], [2, 10], [4, 14], [8, 12], [12, 52], [16, 1], [18, 24], [24, 52], [36, 46], [48, 13], [72, 84], [144, 18]], 'label': '944784.rx', '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': None, 'name': 'C3^6.S3^2:S3^2', 'ngens': 3, '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, 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': 90, 'number_characteristic_subgroups': 21, 'number_conjugacy_classes': 334, 'number_divisions': 182, 'number_normal_subgroups': 47, 'number_subgroup_autclasses': None, 'number_subgroup_classes': None, 'number_subgroups': None, 'old_label': None, 'order': 944784, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 13419], [3, 18224], [4, 26244], [6, 251208], [9, 40824], [12, 52488], [18, 384912], [36, 157464]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [3, 6, 3], 'outer_gen_pows': [20666822203593226439125122521227417770387, 298500476807218563126693594451883025211, 191468239887785193731351893401426515626803], 'outer_gens': [[244050608832821975934962137751646970679163, 42546906890768935849334701416122070137545, 96033155211515977039322372550327217062212], [307872441842612155838820698641199481749612, 361658548982365794838147758440567939193000, 244050065439601433060822742973121355624814], [297539030727396874833311957404280989356964, 244343660283603373721722495989509740539025, 159555961309457492391578870081030219217115]], 'outer_group': '54.12', 'outer_hash': 12, 'outer_nilpotent': False, 'outer_order': 54, 'outer_permdeg': 18, 'outer_perms': [1989816157143936, 1535593698060819, 5947020276487111], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3\\times C_3^2', 'pc_rank': None, 'perfect': False, 'permutation_degree': 36, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 8], [2, 10], [4, 14], [8, 12], [12, 4], [16, 1], [24, 22], [36, 20], [48, 10], [54, 8], [72, 22], [108, 14], [144, 10], [216, 24], [432, 3]], 'representations': {'PC': {'code': '110935153570703862790360226702015346895914113911266049559080125977082826926620622182845277371577384672368192817900802256549532295051771683686873446863630629212575071048278934753370089582406263544618747623377413779479160410461699435256787083547875926776680703420739142744466849743152729230734418935936602178551222120903951815570587676194842363019044166193511379109378248307402373843711', 'gens': [1, 2, 4, 6, 9, 11, 12, 13, 14], 'pres': [14, 2, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 11229624, 9342425, 71, 27344858, 9609168, 32192163, 445217, 4727719, 157, 34324084, 7894338, 12943172, 1837140, 61396277, 9645067, 9106809, 3480167, 1483585, 243, 36070278, 31116980, 7341802, 2447304, 2509058, 384, 870919, 217749, 24227, 8113, 96090632, 58586998, 2898540, 1007042, 313048, 18222, 498, 60489, 60503, 272197, 90771, 15199, 79767082, 39850296, 3891926, 349324, 99858, 124820, 5666, 64665227, 55847257, 84992556, 46427498, 38584768, 3249846, 828896, 2083618, 619260, 19780, 61725915]}, 'Perm': {'d': 36, 'gens': [371989532658154783760856509160051209468852, 244050880369023955532401898366136741200668, 297836716282674354632265147391380864733439]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': None, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 243, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^6.S_3^2:S_3^2', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
  • gps_groups • Show schema
    {'Agroup': True, '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': [2, 2, 2, 3, 3], 'aut_gens': [[475, 450, 148, 244], [450, 475, 147, 243], [572, 450, 148, 147], [572, 693, 243, 147], [719, 693, 148, 244], [572, 693, 148, 244]], 'aut_group': '72.40', 'aut_hash': 40, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 72, 'aut_permdeg': 6, 'aut_perms': [450, 1, 25, 243, 244], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 2, 1], [2, 9, 1, 1], [3, 2, 2, 1], [3, 4, 1, 1], [6, 6, 2, 1]], 'aut_supersolvable': False, 'aut_tex': '\\SOPlus(4,2)', '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': '72.40', 'autcentquo_hash': 40, 'autcentquo_nilpotent': False, 'autcentquo_order': 72, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\SOPlus(4,2)', 'cc_stats': [[1, 1, 1], [2, 3, 2], [2, 9, 1], [3, 2, 2], [3, 4, 1], [6, 6, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '36.10', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 10, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['6.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 2], [2, 9, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [6, 6, 1, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [2, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '36.10', 'hash': 10, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 3, 3], 'inner_gens': [[475, 450, 148, 147], [475, 450, 243, 244], [475, 598, 148, 244], [572, 450, 148, 244]], 'inner_hash': 10, 'inner_nilpotent': False, 'inner_order': 36, 'inner_split': True, 'inner_tex': 'S_3^2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 4, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 4], [4, 1]], 'label': '36.10', 'linC_count': 5, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 5, 'linQ_dim': 4, 'linQ_dim_count': 5, 'linR_count': 5, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'S3^2', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 9, 'number_divisions': 9, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 15, 'number_subgroup_classes': 22, 'number_subgroups': 60, 'old_label': None, 'order': 36, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 15], [3, 8], [6, 12]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [25], 'outer_gens': [[450, 475, 147, 148]], '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': 3, 'perfect': False, 'permutation_degree': 6, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 1]], 'representations': {'PC': {'code': 415852963, 'gens': [1, 2, 4], 'pres': [4, -2, -2, -3, -3, 81, 21, 98, 199]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [11780519, 20465590, 35990610]}, 'Lie': [{'d': 4, 'q': 2, 'gens': [51450, 44940], 'family': 'OmegaPlus'}, {'d': 4, 'q': 2, 'gens': [51450, 44940], 'family': 'POmegaPlus'}, {'d': 4, 'q': 2, 'gens': [33827, 33841, 35873, 50209], 'family': 'SpinPlus'}], 'GLFp': {'d': 3, 'p': 3, 'gens': [11017, 14041, 8101, 8183]}, 'Perm': {'d': 6, 'gens': [475, 450, 148, 244]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'S_3^2', 'transitive_degree': 6, 'wreath_data': None, 'wreath_product': False}