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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '5184.ff', 'ambient_counter': 136, 'ambient_order': 5184, 'ambient_tex': 'C_3^4:C_4^2:C_2^2', 'central': False, 'central_factor': False, 'centralizer_order': 18, 'characteristic': False, 'core_order': 162, 'counter': 38, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '5184.ff.16.g1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '16.g1', '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': 16, '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': '324.156', 'subgroup_hash': 156, 'subgroup_order': 324, 'subgroup_tex': 'C_3^3:C_{12}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '5184.ff', 'aut_centralizer_order': None, 'aut_label': '16.g1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '288.g1', 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['8.c1', '8.i1'], 'contains': ['32.a1', '48.j1', '48.k1'], 'core': '32.a1', 'coset_action_label': None, 'count': 8, 'diagramx': [7879, -1, 8013, -1], 'generators': [13207598294400, 37884712320, 4829956433280, 199992240, 3411889217, 7], 'label': '5184.ff.16.g1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '8.c1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.c1', 'old_label': '16.g1', 'projective_image': '2592.ci', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '16.g1', 'subgroup_fusion': None, 'weyl_group': '144.186'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '36.8', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [8, 6, 24, 8], 'aut_gens': [[1027, 414, 1270, 2923], [5623, 2799, 5107, 5377], [5407, 4548, 3463, 2923], [5110, 2583, 5107, 5647], [3433, 2655, 5647, 3463]], 'aut_group': None, 'aut_hash': 1261382922675480711, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 41472, 'aut_permdeg': 26, 'aut_perms': [173945024224936896517505326, 387382419233989634059047651, 338960101470209218898177577, 347823122465707618714202591], 'aut_phi_ratio': 384.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 8, 1], [3, 2, 4, 1], [3, 2, 32, 1], [4, 9, 2, 1], [6, 1, 8, 1], [6, 2, 4, 1], [6, 2, 32, 1], [12, 9, 16, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times \\GL(2,3)\\times \\AGL(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '96.189', 'autcent_hash': 189, 'autcent_nilpotent': False, 'autcent_order': 96, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2\\times \\GL(2,3)', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '432.734', 'autcentquo_hash': 734, 'autcentquo_nilpotent': False, 'autcentquo_order': 432, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 1, 8], [3, 2, 36], [4, 9, 2], [6, 1, 8], [6, 2, 36], [12, 9, 16]], 'center_label': '18.5', 'center_order': 18, 'central_product': True, 'central_quotient': '18.4', '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', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 156, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 2], ['36.7', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 4], [3, 2, 1, 4], [3, 2, 2, 16], [4, 9, 2, 1], [6, 1, 2, 4], [6, 2, 1, 4], [6, 2, 2, 16], [12, 9, 4, 4]], 'element_repr_type': 'GLZq', 'elementary': 1, 'eulerian_function': 364, 'exponent': 12, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '162.52', 'hash': 156, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [1, 2, 3, 3], 'inner_gens': [[1027, 414, 1270, 2923], [1027, 414, 1000, 5107], [1027, 4791, 1270, 2923], [1027, 225, 1270, 2923]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 18, 'inner_split': False, 'inner_tex': 'C_3:S_3', 'inner_used': [2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 36], [2, 72]], 'label': '324.156', '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': 'C3^3:C12', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 108, 'number_divisions': 55, 'number_normal_subgroups': 78, 'number_subgroup_autclasses': 37, 'number_subgroup_classes': 284, 'number_subgroups': 556, 'old_label': None, 'order': 324, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 1], [3, 80], [4, 18], [6, 80], [12, 144]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [2, 6, 4, 12, 2, 2, 2], 'outer_gen_pows': [730, 730, 730, 730, 414, 414, 730], 'outer_gens': [[1243, 4794, 1270, 2923], [5407, 414, 1000, 2923], [2920, 2388, 1270, 2923], [3433, 4794, 1000, 5377], [1027, 414, 3193, 3463], [1027, 414, 5107, 1270], [1243, 99, 1270, 2923]], 'outer_group': '2304.d', 'outer_hash': 8112084019139883741, 'outer_nilpotent': False, 'outer_order': 2304, 'outer_permdeg': 14, 'outer_perms': [415, 127388152, 16582, 33051508439, 6227383680, 40279680, 6267300480], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^2.S_4^2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [4, 3, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 17], [4, 36]], 'representations': {'PC': {'code': 2303853677248935, 'gens': [1, 2, 5, 6], 'pres': [6, -3, -2, -2, -3, -3, -3, 31, 50, 730, 2603]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [10461591, 23411674, 42212730, 28816400, 33479647, 22075175]}, 'GLZq': {'d': 2, 'q': 9, 'gens': [757, 733, 3220, 198, 5840, 2920]}, 'Perm': {'d': 16, 'gens': [87700843440, 147, 240, 1037836800, 45360, 1394852739840]}}, 'schur_multiplier': [3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 12], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3:C_{12}', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '32.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [24, 4, 4, 8, 4], 'aut_gens': [[2879290997287, 4035153024017, 1328391555240, 204387127], [9440766281640, 2966618069302, 15284078587567, 10862374523040], [2879290997287, 3314872930457, 9434463056040, 1322052041647], [8563293098760, 203747177, 1323968220840, 15689520], [37896364800, 878736510142, 15284078587560, 10917090254887], [14665160700840, 7206699993257, 1328391555240, 5699647312687]], 'aut_group': None, 'aut_hash': 6148241670926445892, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 331776, 'aut_permdeg': 144, 'aut_perms': [2956370211950635059086805705137391656757300763195586977198823272888985897601914967359841956494435108397700211590077637496610145435673829767681749413188426992746850395234884366771617721703128917382440716866780342842452019707591804385738919729239879326, 2172174657951613796249203941242676947745406030907635778317887403392707056235357213254336652841747817140898106494368283903103339987828079900650943243696445740077463693699891302461037450595510417930686727986265517720204046552776644940002226285849479475, 4064389229724298502835627306819678504223313255729803351187452133646174811703769007907899448055603710726263437258446159501424683729113444028984583006992031629643333280760523960817262273849347001132419757755503682128248354753738824563801368159335512058, 511336471060292743007116183652805229464609714840378578675611089072825331837320926286394441196471870183757162024296319920339728614020162123891585347046608960320900197552871773419899205176515218130217845568875169445721092731829978205509274822557043103, 2909934532499644610522591420421869937144971129713225927880707014702055129398802079256815324904388313540372223268888788203112837514783077016952968604218303366470283418681669966165148079992261472159852095839355475121395192822493179880176622583926849009], 'aut_phi_ratio': 192.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 18, 2, 1], [2, 18, 8, 1], [2, 81, 1, 2], [3, 8, 2, 1], [3, 8, 4, 1], [3, 32, 1, 1], [4, 18, 8, 1], [4, 162, 4, 1], [4, 162, 8, 1], [6, 8, 2, 1], [6, 8, 4, 1], [6, 32, 1, 1], [6, 72, 4, 1], [6, 72, 8, 2], [12, 72, 16, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4.C_2^3.C_2^5.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '20736.z', 'autcentquo_hash': 687264991867046508, 'autcentquo_nilpotent': False, 'autcentquo_order': 20736, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\wr D_4:C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 18, 10], [2, 81, 2], [3, 8, 6], [3, 32, 1], [4, 18, 8], [4, 162, 12], [6, 8, 6], [6, 32, 1], [6, 72, 20], [12, 72, 16]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '2592.ci', 'commutator_count': 1, 'commutator_label': '162.54', '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', '3.1', '3.1', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 136, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 18, 1, 10], [2, 81, 1, 2], [3, 8, 1, 6], [3, 32, 1, 1], [4, 18, 2, 4], [4, 162, 1, 4], [4, 162, 2, 4], [6, 8, 1, 6], [6, 32, 1, 1], [6, 72, 1, 20], [12, 72, 2, 8]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 669614400, 'exponent': 12, 'exponents_of_order': [6, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 0, 16], [8, 1, 8], [32, 1, 1]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '2592.ci', 'hash': 1302982936702346125, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 2, 2], 'inner_gens': [[2879290997287, 13208557023377, 9967165096320, 881929854847], [10862374523047, 4035153024017, 9434463056040, 38048774407], [8301859912927, 529859594897, 1328391555240, 204387127], [3314985463567, 4047607468817, 1328391555240, 204387127]], 'inner_hash': 1359974926057248800, 'inner_nilpotent': False, 'inner_order': 2592, 'inner_split': True, 'inner_tex': 'C_3:S_3^3:C_2^2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 32], [4, 2], [8, 48], [32, 2]], 'label': '5184.ff', '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': 'C3^4:C4^2:C2^2', 'ngens': 4, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 19, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 84, 'number_divisions': 68, 'number_normal_subgroups': 123, 'number_subgroup_autclasses': 341, 'number_subgroup_classes': 1550, 'number_subgroups': 52150, 'old_label': None, 'order': 5184, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 343], [3, 80], [4, 2088], [6, 1520], [12, 1152]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 8, 'outer_gen_orders': [2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 3412695607, 0, 1322056391160, 0, 0, 0], 'outer_gens': [[5412240980047, 13170711864977, 1328391555240, 37896364807], [1322052041640, 12332146997542, 9973500260400, 11950670970000], [2879290997280, 13208557023382, 1353873754927, 40682887], [5412240980047, 6722070138022, 9967165096320, 9992236116967], [2879290997280, 13208557023377, 1353873754920, 40682887], [2879290997280, 4035153024017, 1328391555240, 204387120], [2879290997287, 13208557023382, 1353873754920, 40682887]], 'outer_group': '128.928', 'outer_hash': 928, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 8, 'outer_perms': [11536, 34990, 5161, 40319, 7, 126, 5167], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\wr D_4', 'pc_rank': 7, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 4], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 8], [4, 2], [8, 32], [16, 8], [32, 2]], 'representations': {'PC': {'code': '13237474955693019835211643851857837929357421579314179653775101737706341762235319316769731580295519999072303437110876489666589', 'gens': [1, 2, 3, 5, 7, 8, 9], 'pres': [10, 2, 2, 2, 2, 2, 3, 3, 3, 2, 3, 10921, 16442, 39072, 82, 79683, 63604, 2014, 52224, 13134, 144, 28805, 1935, 8665, 515, 286726, 26896, 71146, 1716, 3406, 322567, 667, 5797, 69868, 48638, 27588, 268, 18429, 43239, 23449]}, 'Perm': {'d': 16, 'gens': [2879290997287, 4035153024017, 1328391555240, 204387127]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 4], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:C_4^2:C_2^2', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
