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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '2834352.ot', 'ambient_counter': 384, 'ambient_order': 2834352, 'ambient_tex': 'C_3^7.(C_3^2\\wr C_2.D_4)', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': True, 'core_order': 236196, 'counter': 12, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '2834352.ot.12.A', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '12.a1.N', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '12.1', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 12, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_3:C_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': None, 'subgroup_hash': 4796398716679184265, 'subgroup_order': 236196, 'subgroup_tex': 'C_3^6.C_3^3.D_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '2834352.ot', 'aut_centralizer_order': None, 'aut_label': '12.A', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '2834352.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': None, 'contains': None, 'core': '12.A', 'coset_action_label': None, 'count': 1, 'diagramx': [-1, 2560, -1, 4624], 'generators': [271353676006674129020684387317943043, 9147456921273669403019911296244, 51704033854380251097744, 418802671169936795924582403, 295241022183162525929235278507022801987, 2554547282942135044, 8222838669689166026487225087350850, 109891728871050621252759776875563, 255296065142822556182124257284229418573508, 10628907044783947079682850239896283847108, 109900536862373786112239422044986, 3793956811293730067355006271457072068], 'label': '2834352.ot.12.A', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '12.A', 'normal_contained_in': [], 'normal_contains': [], 'normalizer': '1.a1', 'old_label': '12.a1.N', 'projective_image': '2834352.ot', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '12.A', 'subgroup_fusion': None, 'weyl_group': '2834352.ot'}
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label None does not appear in gps_groups
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': None, 'aut_cyclic': None, 'aut_derived_length': None, 'aut_exponent': None, 'aut_gen_orders': None, 'aut_gens': [[212142124449419219710734249030901005417721, 169891273666237684944772110217727360666874], [340082126171418722947104875026770124094155, 170186498233600647368353812710445721359211], [212433842738998018693653674688343977333815, 180519654239351320278048684040665975277314], [319410126205133667777156248671083008049213, 42540651047644819113058591539783294893490], [329453219153457352152745353758157162324025, 42546906789415323412549307189142586303318], [202104209282072291141666687198870172784586, 180519654431663434066992724564807163582010], [329453745296407391752300647430298574497521, 169893718135546263785981386168102356401874], [319413393790070298888297740204046094588325, 169893718071760121575029745054445357114852], [329447515028083181188391576194214730151557, 308164151891364663546751261771699642967015], [74452666394728918614583458910354884618958, 159855803078390033379544212465550307373483], [202104209291219329287242649584729029737625, 169891273666236878413551335602178529476970], [212141869541420552414878370810817305577721, 180519654423126560417859871001419931243373], [202103954383220661939677871706167890416829, 180519654423125753810212155991385305178638], [212141869541420552414873505006720001072685, 180519383087133689926236021121163767934014], [212141598187745771126347639078513433658362, 169891273675383528700201273198018125338734], [202104209291219329235538615730348894761865, 169891002304025223424188582298236985021914], [191475302264119712829670599372036437140105, 159852832228679596588548846148329678481678], [202104209299755815104415520214802411971229, 180519128179135410410633537432730841259374], [191767546915805422876294187729033369557680, 159563295048305602057110277074326567417970], [329451309049934110921492779753245016307677, 371695929674038204296962485754192972735520]], 'aut_group': None, 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 102036672, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': 108.0, 'aut_solvable': None, 'aut_stats': [[1, 1, 1, 1], [2, 162, 1, 1], [2, 729, 2, 1], [2, 1458, 1, 1], [2, 6561, 1, 1], [3, 4, 2, 2], [3, 8, 1, 2], [3, 16, 1, 1], [3, 16, 2, 1], [3, 36, 1, 1], [3, 36, 2, 1], [3, 72, 2, 2], [3, 72, 3, 1], [3, 72, 4, 1], [3, 108, 2, 1], [3, 144, 2, 1], [3, 144, 6, 2], [3, 162, 1, 1], [3, 216, 2, 1], [3, 216, 12, 1], [3, 324, 1, 1], [3, 432, 12, 1], [3, 648, 6, 2], [3, 1296, 3, 1], [3, 1296, 6, 2], [3, 1296, 12, 1], [4, 39366, 4, 1], [6, 324, 1, 1], [6, 324, 2, 1], [6, 648, 1, 1], [6, 648, 2, 2], [6, 972, 2, 1], [6, 972, 4, 1], [6, 1458, 2, 1], [6, 1944, 2, 1], [6, 1944, 4, 1], [6, 2916, 1, 1], [6, 2916, 2, 5], [6, 5832, 1, 1], [6, 5832, 3, 1], [6, 5832, 6, 2], [6, 5832, 12, 1], [6, 13122, 1, 3], [6, 13122, 2, 1], [6, 26244, 1, 2], [6, 26244, 2, 2], [9, 108, 2, 1], [9, 216, 2, 2], [9, 432, 2, 1], [9, 648, 2, 2], [9, 972, 2, 2], [9, 972, 6, 1], [9, 1296, 2, 2], [9, 1944, 2, 1], [9, 1944, 3, 1], [9, 3888, 6, 3], [9, 5832, 1, 2], [9, 5832, 2, 1], [12, 78732, 4, 1], [18, 8748, 2, 5], [18, 8748, 6, 1], [18, 17496, 2, 5], [18, 17496, 3, 1], [18, 26244, 2, 4], [18, 52488, 1, 2], [18, 52488, 2, 1], [36, 78732, 12, 1]], 'aut_supersolvable': None, 'aut_tex': None, 'autcent_abelian': None, 'autcent_cyclic': None, 'autcent_exponent': None, 'autcent_group': None, 'autcent_hash': None, 'autcent_nilpotent': None, 'autcent_order': None, 'autcent_solvable': None, 'autcent_split': None, 'autcent_supersolvable': None, 'autcent_tex': None, 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': None, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 162, 1], [2, 729, 2], [2, 1458, 1], [2, 6561, 1], [3, 4, 4], [3, 8, 2], [3, 16, 3], [3, 36, 3], [3, 72, 11], [3, 108, 2], [3, 144, 14], [3, 162, 1], [3, 216, 14], [3, 324, 1], [3, 432, 12], [3, 648, 12], [3, 1296, 27], [4, 39366, 4], [6, 324, 3], [6, 648, 5], [6, 972, 6], [6, 1458, 2], [6, 1944, 6], [6, 2916, 11], [6, 5832, 28], [6, 13122, 5], [6, 26244, 6], [9, 108, 2], [9, 216, 4], [9, 432, 2], [9, 648, 4], [9, 972, 10], [9, 1296, 4], [9, 1944, 5], [9, 3888, 18], [9, 5832, 4], [12, 78732, 4], [18, 8748, 16], [18, 17496, 13], [18, 26244, 8], [18, 52488, 4], [36, 78732, 12]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '2834352.ot', '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', '3.1'], 'composition_length': 15, 'conjugacy_classes_known': True, 'counter': 384, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 162, 1, 1], [2, 729, 1, 2], [2, 1458, 1, 1], [2, 6561, 1, 1], [3, 4, 1, 4], [3, 8, 1, 2], [3, 16, 1, 3], [3, 36, 1, 1], [3, 36, 2, 1], [3, 72, 1, 5], [3, 72, 2, 3], [3, 108, 2, 1], [3, 144, 1, 12], [3, 144, 2, 1], [3, 162, 1, 1], [3, 216, 2, 7], [3, 324, 1, 1], [3, 432, 2, 6], [3, 648, 2, 6], [3, 1296, 1, 9], [3, 1296, 2, 9], [4, 39366, 2, 2], [6, 324, 1, 3], [6, 648, 1, 3], [6, 648, 2, 1], [6, 972, 2, 3], [6, 1458, 2, 1], [6, 1944, 2, 3], [6, 2916, 1, 9], [6, 2916, 2, 1], [6, 5832, 1, 4], [6, 5832, 2, 12], [6, 13122, 1, 3], [6, 13122, 2, 1], [6, 26244, 1, 2], [6, 26244, 2, 2], [9, 108, 1, 2], [9, 216, 1, 4], [9, 432, 1, 2], [9, 648, 2, 2], [9, 972, 2, 2], [9, 972, 3, 2], [9, 1296, 2, 2], [9, 1944, 1, 5], [9, 3888, 1, 6], [9, 3888, 2, 6], [9, 5832, 1, 2], [9, 5832, 2, 1], [12, 78732, 2, 2], [18, 8748, 1, 6], [18, 8748, 2, 2], [18, 8748, 3, 2], [18, 17496, 1, 9], [18, 17496, 2, 2], [18, 26244, 2, 4], [18, 52488, 1, 2], [18, 52488, 2, 1], [36, 78732, 6, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 36, 'exponents_of_order': [11, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[24, 0, 12], [24, 1, 6], [48, 0, 24], [48, 1, 12], [72, 0, 36], [144, 0, 18], [144, 1, 9], [216, 1, 6], [432, 1, 6]], 'familial': False, 'frattini_label': '729.504', 'frattini_quotient': '3888.dk', 'hash': 6268035797580749796, 'hyperelementary': 1, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': [18, 36], 'inner_gens': [[212142124449419219710734249030901005417721, 42251682323107812518608579189324080364652], [201805190610750031386356674882469932533811, 169891273666237684944772110217727360666874]], 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': 2834352, 'inner_split': None, 'inner_tex': 'C_3^7.(C_3^2\\wr C_2.D_4)', 'inner_used': None, 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 8], [2, 10], [4, 32], [8, 16], [12, 12], [16, 9], [24, 42], [36, 20], [48, 36], [72, 44], [108, 24], [144, 29], [216, 10], [432, 6]], 'label': '2834352.ot', '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': 'C3^7.(C3^2wrC2.D4)', 'ngens': 2, '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, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 102, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 298, 'number_divisions': 196, 'number_normal_subgroups': 34, 'number_subgroup_autclasses': None, 'number_subgroup_classes': None, 'number_subgroups': None, 'old_label': None, 'order': 2834352, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 9639], [3, 54674], [4, 157464], [6, 443070], [9, 122472], [12, 314928], [18, 787320], [36, 944784]], 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': True, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': '36.12', 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 36, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': 'C_6\\times S_3', 'pc_rank': None, 'perfect': False, 'permutation_degree': 36, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 4], [2, 8], [4, 14], [8, 18], [12, 4], [16, 7], [24, 14], [32, 3], [36, 12], [48, 28], [72, 12], [96, 12], [144, 29], [216, 10], [288, 9], [324, 4], [432, 6], [648, 2]], 'representations': {'PC': {'code': '44918282124218148440910889954223519698376341216865009458561345802379434294881046859830909858178417202388652109310842588683505860004285344184440795456044674946855516434982585965192967861912304642715764713102842939972005661186494503090769681760349287069581760186830215819281037412844639523438335308811816833397921193268563293702425871210094168165009345999680433388489197705326330337086946417225674355414741544003073051601820003121471571279900738977001750000534906402540063743', 'gens': [1, 3, 5, 7, 9, 10, 12, 13, 14, 15], 'pres': [15, 2, 2, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 30, 5387056, 114755222, 53921267, 122, 140668803, 71383938, 23391904, 89801119, 28255084, 19796674, 214, 24533285, 69852260, 22591475, 22448930, 3739850, 51065286, 75781461, 70746516, 1544181, 3019026, 1686276, 411, 226748167, 25957, 56907368, 157347383, 48507698, 16161983, 13469558, 5386178, 1007333, 299196009, 86378424, 18662439, 510354, 3623469, 2967384, 1134099, 113979, 594, 190531450, 1924585, 962320, 160450, 243622091, 7348346, 87130121, 7348391, 2434961, 270671, 45506, 2442972, 204703227, 10235202, 8529372, 117754573, 287854588, 127983283, 2865313, 1379788, 799573, 361123, 68188, 194414, 106288244, 17714774]}, 'Perm': {'d': 36, 'gens': [169891273666237684944772110217727360666874, 212142124449419219710734249030901005417721]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 1, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^7.(C_3^2\\wr C_2.D_4)', 'transitive_degree': 36, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 4], [1, 8], [11, 4]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 49], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 1], [4, 3, 2, 1], [6, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', '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': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 2, 1], [4, 3, 2], [6, 2, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 1], [4, 3, 2, 1], [6, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 12, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, -1, 1]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '6.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 8], [9, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 2]], 'label': '12.1', 'linC_count': 1, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3:C4', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 6, 'number_divisions': 5, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 8, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 6], [6, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3, 4]], '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': 2, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3]], 'representations': {'PC': {'code': 3913, 'gens': [1, 3], 'pres': [3, -2, -2, -3, 6, 74]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [20468488, 16858410]}, 'Lie': [{'d': 2, 'q': 7, 'gens': [2064, 1718, 690, 273], 'family': 'CSOPlus'}, {'d': 2, 'q': 5, 'gens': [537, 504, 427], 'family': 'CSOMinus'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [169, 458]}, 'Perm': {'d': 7, 'gens': [129, 16, 840]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3:C_4', 'transitive_degree': 12, 'wreath_data': None, 'wreath_product': False}
