Formats: - HTML - YAML - JSON - 2026-07-24T08:10:26.997594
  • gps_subgroup_search • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '7776.cw', 'ambient_counter': 75, 'ambient_order': 7776, 'ambient_tex': 'C_3^4:(S_3\\times \\SD_{16})', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 648, 'counter': 26, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '7776.cw.6._.F', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '6.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': 6, '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': '1296.3522', 'subgroup_hash': 3522, 'subgroup_order': 1296, 'subgroup_tex': 'C_3^4:\\SD_{16}', 'supersolvable': False, 'sylow': 0}
  • gps_subgroup_data • Show schema
    {'ambient': '7776.cw', 'aut_centralizer_order': None, 'aut_label': None, 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '7776.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2._.F', '3.a1.a1'], 'contains': ['12._.B', '12._.I', '12._.K', '54._.K', '54._.M'], 'core': '12._.B', 'coset_action_label': None, 'count': 3, 'diagramx': [8815, -1, 9028, -1, 8693, -1, 8489, -1], 'generators': [2017, 4352, 96, 2, 200, 2592, 6048, 6212], 'label': '7776.cw.6._.F', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2._.F', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '6.f1.a1', 'projective_image': '7776.cw', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6._.F', 'subgroup_fusion': None, 'weyl_group': '2592.cv'}
  • gps_groups • Show schema
    {'Agroup': False, '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': 24, 'aut_gen_orders': [2, 2, 8, 2], 'aut_gens': [[10461591, 23411674, 42212730, 42494788, 37173307, 4521260, 29213543, 14659948], [7666975, 3644059, 10409068, 34858262, 24998138, 23020464, 29213543, 14659948], [42212730, 26212087, 10461591, 34910677, 7285539, 15883908, 1713592, 7505410], [23556038, 7666975, 3644059, 3447199, 41391625, 18109023, 13339061, 30515490], [34910677, 26067759, 7666975, 7949150, 20945103, 23323415, 2107652, 10411571]], 'aut_group': '5184.cc', 'aut_hash': 9016091067240435718, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5184, 'aut_permdeg': 360, 'aut_perms': [2293588064937398692118534326523967685067419042365624720946919804179416712791892402115444922647426784514604846793056313029924527606107061583996262604238318820222454730896213496650652005136144492995599002571092461094232583827356986182783084297479623878124608719225648754195411598649427077292573579400418198303000179523684165780432865404384595204743372642000547242985315095106459307803982502144006410229544469949387962481398194499428493182537694572178942314654628840971437886572382403159090462762464874449083758671944252534293037596749694245839014675573158942127117335335315279343782961340447300430423387723583245418531551757326824755028008704952760391563770255431737249131516840195579474064225128873558687513982145286869927118088094658251780421934204851995496750661392, 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False, 'aut_tex': 'C_3^4:D_8: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': 24, 'autcentquo_group': '5184.cc', 'autcentquo_hash': 9016091067240435718, 'autcentquo_nilpotent': False, 'autcentquo_order': 5184, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^4:D_8:C_2^2', 'cc_stats': [[1, 1, 1], [2, 36, 1], [2, 81, 1], [3, 8, 4], [3, 16, 3], [4, 162, 1], [4, 324, 1], [6, 72, 4], [8, 162, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1296.3522', 'commutator_count': 1, 'commutator_label': '324.164', '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': 3522, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 36, 1, 1], [2, 81, 1, 1], [3, 8, 1, 4], [3, 16, 1, 3], [4, 162, 1, 1], [4, 324, 1, 1], [6, 72, 1, 4], [8, 162, 2, 1]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 96, 'exponent': 24, 'exponents_of_order': [4, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 1, 4], [16, 1, 3]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1296.3522', 'hash': 3522, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [3, 3, 3, 3, 8, 6, 4, 2], 'inner_gens': [[10461591, 23411674, 42212730, 42494788, 2483953, 41475925, 17858912, 34906619], [10461591, 23411674, 42212730, 42494788, 25142463, 34369274, 24503393, 26267005], [10461591, 23411674, 42212730, 42494788, 14121684, 4521260, 40811623, 7472627], [10461591, 23411674, 42212730, 42494788, 20996273, 34369274, 40778804, 7899040], [3447199, 34858262, 26212087, 23556038, 37173307, 29358724, 38018060, 42240645], [34858262, 30749221, 42212730, 7666975, 31649316, 4521260, 6233438, 30462967], [42212730, 3447199, 34858262, 34910677, 18536325, 25730138, 29213543, 14659948], [34858262, 26212087, 7949150, 7666975, 40997914, 31824011, 29213543, 14659948]], 'inner_hash': 3522, 'inner_nilpotent': False, 'inner_order': 1296, 'inner_split': True, 'inner_tex': 'C_3^4:\\SD_{16}', 'inner_used': [1, 2, 3, 4, 5, 6], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 4], [2, 3], [8, 8], [16, 3]], 'label': '1296.3522', 'linC_count': 4, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 4, 'linQ_dim': 8, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^4:SD16', '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': 12, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 18, 'number_divisions': 17, 'number_normal_subgroups': 10, 'number_subgroup_autclasses': 106, 'number_subgroup_classes': 173, 'number_subgroups': 5112, 'old_label': None, 'order': 1296, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 117], [3, 80], [4, 486], [6, 288], [8, 324]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [3553890, 14408200], 'outer_gens': [[42494788, 3644059, 34910677, 34858262, 14040317, 41672785, 40673893, 7702270], [42494788, 30552451, 34910677, 10461591, 18142686, 25535104, 24536212, 26549180]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 1], [8, 8], [16, 3]], 'representations': {'PC': {'code': 723737900592486266733552335904908019302814541802475658469847959055, 'gens': [1, 2, 5, 6, 7, 8], 'pres': [8, -2, -2, -2, -2, -3, 3, 3, 3, 97, 41, 290, 66, 1932, 660, 188, 5381, 3085, 1365, 605, 40326, 17486, 11446, 2046, 8207, 10263, 6943]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [10461591, 23411674, 42212730, 42494788, 37173307, 4521260, 29213543, 14659948]}, 'Perm': {'d': 12, 'gens': [399081600, 3629275, 135354714, 3633865, 79914244, 43626384, 79879104, 80788]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:\\SD_{16}', 'transitive_degree': 12, 'wreath_data': None, 'wreath_product': False}
  • gps_groups • Show 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': 24, 'aut_gen_orders': [2, 8, 6, 3, 3, 3, 3], 'aut_gens': [[1, 2, 16, 96, 288, 864, 2592], [1097, 6374, 6288, 6080, 5760, 4320, 2592], [7277, 6754, 6512, 1952, 2304, 2592, 6048], [5377, 3778, 6928, 1824, 7488, 1728, 5184], [2689, 7106, 2608, 96, 2880, 864, 2592], [3745, 3746, 7216, 1824, 288, 864, 2592], [6049, 6914, 5200, 96, 288, 864, 2592], [4321, 1730, 4336, 96, 288, 864, 2592]], 'aut_group': '7776.cw', 'aut_hash': 8212187940285761847, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 7776, 'aut_permdeg': 225, 'aut_perms': [11231268714318088878867744082404562360510990226514671598596467852112105308964526497545494917507570819614888216266281021383412910480090444199656726890735562817848266391167807026829329218021463950310425797760350239940095047612049778933921736396702327347419707361994649417221333711014234975279434550226154902335999269557510670199033126742173541646154396175709579011930117732152429908326556194085736976454255994395052238842347447493575891, 271488471834983269251046137259492763515915003995221494409717119951333197286100364036583659752021311128779764125120279156946945205540495446662436510353065328138944162389991798972755511793960772457380697811405572817014850016201186111969749740903678315688645256938866510402925918126491118337594428275942698973508801470662403635347663373648639380918765342617885808996318545785064680948853159290733391847567884710391620582840464911809613, 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3302816221221346167156239570433023206807198695911642394148280340914675319075148046167865021574676711696534232922260963903526143177151461033136319250086932922362131365492203886266752913381077024331008938579790627142836879708503323697457810436676660229198424190801765952193471637272914891703511418952589999848488134513895951846689129715050975465644604538067691783267654912945124854336555689378797042407906406051827279503106181467985778], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 27, 1, 2], [2, 36, 1, 1], [2, 81, 1, 1], [2, 108, 1, 1], [3, 8, 1, 1], [3, 18, 1, 1], [3, 24, 1, 1], [3, 48, 1, 1], [3, 144, 1, 1], [4, 162, 1, 1], [4, 324, 1, 1], [4, 486, 1, 1], [4, 972, 1, 1], [6, 72, 1, 1], [6, 162, 1, 1], [6, 216, 1, 6], [6, 432, 1, 2], [8, 162, 1, 2], [8, 486, 1, 2], [12, 324, 1, 1], [12, 648, 1, 1], [24, 324, 1, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4:(S_3\\times \\SD_{16})', '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': 24, 'autcentquo_group': '7776.cw', 'autcentquo_hash': 8212187940285761847, 'autcentquo_nilpotent': False, 'autcentquo_order': 7776, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^4:(S_3\\times \\SD_{16})', 'cc_stats': [[1, 1, 1], [2, 27, 2], [2, 36, 1], [2, 81, 1], [2, 108, 1], [3, 8, 1], [3, 18, 1], [3, 24, 1], [3, 48, 1], [3, 144, 1], [4, 162, 1], [4, 324, 1], [4, 486, 1], [4, 972, 1], [6, 72, 1], [6, 162, 1], [6, 216, 6], [6, 432, 2], [8, 162, 2], [8, 486, 2], [12, 324, 1], [12, 648, 1], [24, 324, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '7776.cw', 'commutator_count': 1, 'commutator_label': '972.406', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 75, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 27, 1, 2], [2, 36, 1, 1], [2, 81, 1, 1], [2, 108, 1, 1], [3, 8, 1, 1], [3, 18, 1, 1], [3, 24, 1, 1], [3, 48, 1, 1], [3, 144, 1, 1], [4, 162, 1, 1], [4, 324, 1, 1], [4, 486, 1, 1], [4, 972, 1, 1], [6, 72, 1, 1], [6, 162, 1, 1], [6, 216, 1, 6], [6, 432, 1, 2], [8, 162, 2, 1], [8, 486, 2, 1], [12, 324, 1, 1], [12, 648, 1, 1], [24, 324, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 17418240, 'exponent': 24, 'exponents_of_order': [5, 5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[24, 1, 4], [48, 1, 2]], 'familial': False, 'frattini_label': '9.2', 'frattini_quotient': '864.4662', 'hash': 8212187940285761847, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [2, 8, 6, 3, 3, 3, 3], 'inner_gens': [[1, 3046, 16, 1056, 2304, 864, 6912], [6765, 2, 6800, 4544, 576, 2592, 6048], [1, 7426, 16, 1824, 4896, 1728, 5184], [1921, 6306, 880, 96, 2016, 864, 2592], [1441, 578, 6640, 960, 288, 864, 2592], [1, 6050, 1744, 96, 288, 864, 2592], [6049, 6914, 5200, 96, 288, 864, 2592]], 'inner_hash': 8212187940285761847, 'inner_nilpotent': False, 'inner_order': 7776, 'inner_split': True, 'inner_tex': 'C_3^4:(S_3\\times \\SD_{16})', 'inner_used': [1, 2, 3], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 8], [2, 10], [4, 3], [8, 4], [16, 2], [24, 4], [48, 2]], 'label': '7776.cw', 'linC_count': 4, 'linC_degree': 24, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 24, 'linQ_degree_count': 4, 'linQ_dim': 24, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 24, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': None, 'name': 'C3^4:(S3*SD16)', 'ngens': 10, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 33, 'number_characteristic_subgroups': 33, 'number_conjugacy_classes': 33, 'number_divisions': 30, 'number_normal_subgroups': 33, 'number_subgroup_autclasses': 473, 'number_subgroup_classes': 473, 'number_subgroups': 31930, 'old_label': None, 'order': 7776, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 279], [3, 242], [4, 1944], [6, 2394], [8, 1296], [12, 972], [24, 648]], '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': 7, 'perfect': False, 'permutation_degree': 27, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 3], [8, 5], [16, 2], [24, 4], [48, 2]], 'representations': {'PC': {'code': '58455272015971747385343453536179987028011274495789361867144663252820901170989768250196190849712281178343865739492018193190161126809688095', 'gens': [1, 2, 5, 7, 8, 9, 10], 'pres': [10, -2, -2, -2, -2, -2, -3, 3, -3, -3, 3, 60921, 51, 59882, 82, 38723, 170014, 77424, 5934, 144, 85455, 28825, 515, 73926, 159056, 76186, 1716, 8026, 184327, 23057, 24527, 11577, 1747, 116658, 136108, 19478, 9768, 691209, 302419, 172829, 64839, 32449]}, 'Perm': {'d': 27, 'gens': [9167122780557201012549133103, 6632969369720355530109885969, 3257383389079646387271672220]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:(S_3\\times \\SD_{16})', 'transitive_degree': 27, 'wreath_data': None, 'wreath_product': False}