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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1440.2534', 'ambient_counter': 2534, 'ambient_order': 1440, 'ambient_tex': 'C_{60}.S_4', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 60, 'counter': 8, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1440.2534.4.b1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '4.b1.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': 4, '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': '360.37', 'subgroup_hash': 37, 'subgroup_order': 360, 'subgroup_tex': 'C_{12}.D_{15}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1440.2534', 'aut_centralizer_order': 1, 'aut_label': '4.b1', 'aut_quo_index': None, 'aut_stab_index': 4, 'aut_weyl_group': None, 'aut_weyl_index': 4, 'centralizer': '360.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['8.b1.a1', '12.e1.a1', '12.g1.a1', '12.g1.b1', '12.g1.c1', '20.b1.a1'], 'core': '24.a1.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [6805, -1, 9333, -1, 6835, -1, 5720, -1], 'generators': [1, 24, 4, 80, 2, 480], 'label': '1440.2534.4.b1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '4.b1.a1', 'old_label': '4.b1.a1', 'projective_image': '360.141', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.b1.a1', 'subgroup_fusion': None, 'weyl_group': '90.9'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 120, 'aut_gen_orders': [12, 12, 8, 12, 8], 'aut_gens': [[1, 8, 24], [195, 120, 176], [179, 16, 200], [51, 136, 232], [231, 136, 168], [249, 120, 280]], 'aut_group': None, 'aut_hash': 8203041546889010050, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 34560, 'aut_permdeg': 180, 'aut_perms': [94694331161519753388273225975618193492965841754971514966310725405083509433598673097671754893304665792988055637142604575044285537003693204770539427678982929502996489908543510377343561282462839078979513704743332115307646390406466946374416236400611042981990039204428730710390474490415695601326538555743535405972644790544116110120258, 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67274753856259925503720692622036485811052438580278044591717404418885087772769118027127042252471240407809653123868785876929274785626465977916071141953044167242619048945327268862582531745629320153922799372082979907202896140955097027980296361512019339702498142872577957513195738755798800576385926030792935163448259843922161915476213], 'aut_phi_ratio': 360.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 4, 1], [4, 1, 2, 1], [5, 2, 2, 1], [6, 2, 4, 1], [8, 45, 4, 1], [10, 2, 2, 1], [12, 2, 8, 1], [15, 2, 16, 1], [20, 2, 4, 1], [30, 2, 16, 1], [60, 2, 32, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2\\times \\AGL(2,3)\\times F_5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': '8640.be', 'autcentquo_hash': 9203803121985669655, 'autcentquo_nilpotent': False, 'autcentquo_order': 8640, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\times C_3^2:\\GL(2,3)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 2, 4], [4, 1, 2], [5, 2, 2], [6, 2, 4], [8, 45, 4], [10, 2, 2], [12, 2, 8], [15, 2, 16], [20, 2, 4], [30, 2, 16], [60, 2, 32]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, 'central_quotient': '90.9', 'commutator_count': 1, 'commutator_label': '45.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 37, '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, 4], [4, 1, 2, 1], [5, 2, 2, 1], [6, 2, 1, 4], [8, 45, 4, 1], [10, 2, 2, 1], [12, 2, 2, 4], [15, 2, 4, 4], [20, 2, 4, 1], [30, 2, 4, 4], [60, 2, 8, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 672, 'exponent': 120, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '90.9', 'hash': 37, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [2, 3, 15], 'inner_gens': [[1, 16, 336], [17, 8, 24], [49, 8, 24]], 'inner_hash': 9, 'inner_nilpotent': False, 'inner_order': 90, 'inner_split': False, 'inner_tex': 'C_3:D_{15}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 88]], 'label': '360.37', 'linC_count': 1728, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 48, 'linQ_dim': 10, 'linQ_dim_count': 12, 'linR_count': 576, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C12.D15', 'ngens': 6, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 13, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 96, 'number_divisions': 31, 'number_normal_subgroups': 37, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 48, 'number_subgroups': 168, 'old_label': None, 'order': 360, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 1], [3, 8], [4, 2], [5, 4], [6, 8], [8, 180], [10, 4], [12, 16], [15, 32], [20, 8], [30, 32], [60, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 24, 'outer_gen_orders': [2, 2, 6, 4, 2, 4], 'outer_gen_pows': [0, 0, 0, 0, 1, 0], 'outer_gens': [[1, 16, 40], [5, 8, 96], [3, 120, 272], [1, 136, 40], [1, 136, 328], [1, 120, 152]], 'outer_group': '384.20061', 'outer_hash': 20061, 'outer_nilpotent': False, 'outer_order': 384, 'outer_permdeg': 20, 'outer_perms': [51598353365764567, 16, 45193881432430103, 1002542195319841560, 1919109629252447407, 608403346242652800], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,3):C_2^3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [8], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 9], [4, 7], [8, 9], [16, 4]], 'representations': {'PC': {'code': 78562331567844694241303, 'gens': [1, 4, 5], 'pres': [6, -2, -2, -2, -3, -3, -5, 12, 31, 387, 10084, 118, 10373]}, 'GLZN': {'d': 2, 'p': 50, 'gens': [2574180, 188776, 6125049, 2284481, 5375043, 3976021]}, 'Perm': {'d': 19, 'gens': [356995585548865, 8443680, 12501360, 3, 6706022403, 7115056236288000]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [8], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}.D_{15}', 'transitive_degree': 360, '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': '8.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [12, 4, 12, 12, 12, 30], 'aut_gens': [[1, 8, 120, 240], [549, 1264, 840, 1200], [79, 1024, 120, 360], [581, 824, 120, 240], [881, 296, 840, 600], [383, 464, 840, 1080], [501, 488, 840, 1080]], 'aut_group': None, 'aut_hash': 2220674056071093365, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 34560, 'aut_permdeg': 360, 'aut_perms': 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1409444480070792693853267250514774229796170658417984554819865762212335415963359754161400656882640622793441667697385777424901227383918001582709061749592027466179601692954935192397093065268684684514779796445521623814708870759802082795582964686643216183674759673751161716970779601996814286369565366200189703144113705392050352584539148010763563638399580426281127699671594360161737167620699148437108428204617861967869592838942833639308124893128754249232888407316208664752736543679477837467328306181295710779572568961496699503693541604847885822083734617223987498624377713194464451582180030996043626871806128848703368053167802906616539927713727741374973197125072466597788431906594799225392241087622158519238167456727730314561856634213080456017172700868026843280598799564021], 'aut_phi_ratio': 90.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 2, 1, 1], [3, 8, 3, 1], [4, 1, 2, 1], [4, 3, 2, 1], [5, 2, 2, 1], [6, 2, 1, 1], [6, 6, 1, 2], [6, 8, 3, 1], [8, 90, 4, 2], [10, 2, 2, 1], [10, 6, 2, 2], [12, 2, 2, 1], [12, 6, 2, 1], [12, 8, 6, 1], [15, 2, 4, 1], [15, 8, 12, 1], [20, 2, 4, 1], [20, 6, 4, 1], [30, 2, 4, 1], [30, 6, 4, 2], [30, 8, 12, 1], [60, 2, 8, 1], [60, 6, 8, 1], [60, 8, 24, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2\\times F_5\\times C_3:S_3:S_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': None, 'autcentquo_hash': 928336707834802966, 'autcentquo_nilpotent': False, 'autcentquo_order': 8640, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\times C_3:S_3:S_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [3, 2, 1], [3, 8, 3], [4, 1, 2], [4, 3, 2], [5, 2, 2], [6, 2, 1], [6, 6, 2], [6, 8, 3], [8, 90, 8], [10, 2, 2], [10, 6, 4], [12, 2, 2], [12, 6, 2], [12, 8, 6], [15, 2, 4], [15, 8, 12], [20, 2, 4], [20, 6, 4], [30, 2, 4], [30, 6, 8], [30, 8, 12], [60, 2, 8], [60, 6, 8], [60, 8, 24]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, 'central_quotient': '360.141', 'commutator_count': 1, 'commutator_label': '180.31', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 2534, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 2, 1, 1], [3, 8, 1, 3], [4, 1, 2, 1], [4, 3, 2, 1], [5, 2, 2, 1], [6, 2, 1, 1], [6, 6, 1, 2], [6, 8, 1, 3], [8, 90, 4, 2], [10, 2, 2, 1], [10, 6, 2, 2], [12, 2, 2, 1], [12, 6, 2, 1], [12, 8, 2, 3], [15, 2, 4, 1], [15, 8, 4, 3], [20, 2, 4, 1], [20, 6, 4, 1], [30, 2, 4, 1], [30, 6, 4, 2], [30, 8, 4, 3], [60, 2, 8, 1], [60, 6, 8, 1], [60, 8, 8, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 40320, 'exponent': 120, 'exponents_of_order': [5, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[6, 0, 8]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '360.141', 'hash': 2534, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 15, 2, 6], 'inner_gens': [[1, 112, 120, 1320], [17, 8, 840, 1080], [1, 728, 120, 240], [601, 848, 120, 240]], 'inner_hash': 141, 'inner_nilpotent': False, 'inner_order': 360, 'inner_split': True, 'inner_tex': 'C_{15}:S_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 6, 'irrQ_degree': 48, 'irrQ_dim': 48, 'irrR_degree': 12, 'irrep_stats': [[1, 8], [2, 88], [3, 8], [6, 28]], 'label': '1440.2534', 'linC_count': 288, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 11, 'linQ_degree_count': 12, 'linQ_dim': 11, 'linQ_dim_count': 6, 'linR_count': 72, 'linR_degree': 7, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C60.S4', '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, 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': 32, 'number_characteristic_subgroups': 31, 'number_conjugacy_classes': 132, 'number_divisions': 44, 'number_normal_subgroups': 49, 'number_subgroup_autclasses': 104, 'number_subgroup_classes': 136, 'number_subgroups': 1032, 'old_label': None, 'order': 1440, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 7], [3, 26], [4, 8], [5, 4], [6, 38], [8, 720], [10, 28], [12, 64], [15, 104], [20, 32], [30, 152], [60, 256]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 4, 2, 6], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 88, 120, 360], [1, 104, 120, 240], [5, 8, 120, 240], [7, 1072, 120, 1320]], 'outer_group': '96.206', 'outer_hash': 206, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 11, 'outer_perms': [362881, 5190, 289, 3996720], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{12}:C_2^3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [8], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 9], [3, 2], [4, 7], [6, 3], [8, 9], [12, 4], [16, 4], [24, 3], [48, 1]], 'representations': {'PC': {'code': 599670727224064573816768091738821250788295105, 'gens': [1, 4, 6, 7], 'pres': [8, -2, -2, -2, -3, -5, -2, 2, -3, 16, 41, 3587, 123, 3844, 5069, 73926, 7590, 166, 61447]}, 'GLZN': {'d': 2, 'p': 100, 'gens': [21402081, 49000049, 51752576, 57000057, 1505001, 61414689, 51500051, 73329601]}, 'Perm': {'d': 20, 'gens': [13160522058128937, 12568, 18498, 6759455986483200, 1394852659200, 559198080, 122000787836928000, 256094948229120000]}}, 'schur_multiplier': [6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [8], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{60}.S_4', 'transitive_degree': 360, 'wreath_data': None, 'wreath_product': False}
