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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '324000.bm', 'ambient_counter': 39, 'ambient_order': 324000, 'ambient_tex': 'C_{15}^3.(C_4\\times S_4)', 'central': False, 'central_factor': False, 'centralizer_order': 9, 'characteristic': False, 'core_order': 1, 'counter': 546, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '324000.bm.360.bb1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '360.bb1', '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': 360, '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': '900.127', 'subgroup_hash': 127, 'subgroup_order': 900, 'subgroup_tex': 'C_{15}^2:C_2^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '324000.bm', 'aut_centralizer_order': None, 'aut_label': '360.bb1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '36000.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['72.e1', '120.f1', '120.k1', '180.bf1'], 'contains': ['720.v1', '720.cb1', '720.dj1', '1080.q1', '1080.bf1', '1800.t1', '1800.bg1'], 'core': '324000.a1', 'coset_action_label': None, 'count': 60, 'diagramx': None, 'generators': [232585, 69120, 15360, 122400, 12, 82368], 'label': '324000.bm.360.bb1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '2.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '60.h1', 'old_label': '360.bb1', 'projective_image': '324000.bm', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '360.bb1', 'subgroup_fusion': None, 'weyl_group': '600.164'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 120, 'aut_gen_orders': [2, 4, 4, 20, 30], 'aut_gens': [[1, 6, 60], [1, 6, 660], [1, 213, 348], [1, 354, 480], [553, 282, 660], [53, 306, 60]], 'aut_group': '9600.ci', 'aut_hash': 6485580844893215951, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 9600, 'aut_permdeg': 32, 'aut_perms': [112751007213293177005440806740974740, 201189092643295239692475030103994330, 68521610844905257696729830235577114, 231819553106314888639349802396635057, 185772374973083441770924348878021612], 'aut_phi_ratio': 40.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 15, 2, 1], [2, 25, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [5, 2, 4, 1], [5, 4, 4, 1], [6, 15, 4, 1], [6, 25, 2, 1], [6, 50, 1, 1], [6, 50, 2, 1], [10, 30, 4, 1], [15, 2, 8, 1], [15, 4, 4, 1], [15, 4, 8, 3], [15, 4, 16, 1], [30, 30, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'D_6\\times F_5\\wr C_2', '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': 120, 'autcentquo_group': '4800.bn', 'autcentquo_hash': 4310732139677402518, 'autcentquo_nilpotent': False, 'autcentquo_order': 4800, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5^2:D_6', 'cc_stats': [[1, 1, 1], [2, 15, 2], [2, 25, 1], [3, 1, 2], [3, 2, 3], [5, 2, 4], [5, 4, 4], [6, 15, 4], [6, 25, 2], [6, 50, 3], [10, 30, 4], [15, 2, 8], [15, 4, 44], [30, 30, 8]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '300.40', 'commutator_count': 1, 'commutator_label': '75.3', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '5.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 127, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['300.40', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 15, 1, 2], [2, 25, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [5, 2, 2, 2], [5, 4, 2, 2], [6, 15, 2, 2], [6, 25, 2, 1], [6, 50, 1, 1], [6, 50, 2, 1], [10, 30, 2, 2], [15, 2, 4, 2], [15, 4, 2, 2], [15, 4, 4, 10], [30, 30, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 30, 'exponents_of_order': [2, 2, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[4, 0, 16]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '900.127', 'hash': 127, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [2, 10, 15], 'inner_gens': [[1, 54, 240], [13, 6, 840], [721, 126, 60]], 'inner_hash': 40, 'inner_nilpotent': False, 'inner_order': 300, 'inner_split': False, 'inner_tex': 'C_{15}:D_{10}', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 8, 'irrep_stats': [[1, 12], [2, 30], [4, 48]], 'label': '900.127', 'linC_count': 16, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 60, 'linQ_dim': 12, 'linQ_dim_count': 60, 'linR_count': 352, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C15^2:C2^2', '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, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 20, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 90, 'number_divisions': 34, 'number_normal_subgroups': 30, 'number_subgroup_autclasses': 68, 'number_subgroup_classes': 112, 'number_subgroups': 864, 'old_label': None, 'order': 900, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 55], [3, 8], [5, 24], [6, 260], [10, 120], [15, 192], [30, 240]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [1, 0, 1, 0], 'outer_gens': [[5, 42, 780], [5, 213, 612], [1, 42, 120], [1, 42, 840]], 'outer_group': '32.22', 'outer_hash': 22, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 10, 'outer_perms': [806527, 85680, 806400, 408534], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3:C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 6], [8, 8], [16, 10]], 'representations': {'PC': {'code': 5565023042968870248160869260022071, 'gens': [1, 3, 5], 'pres': [6, -2, -3, -2, -5, -3, -5, 12, 974, 50, 1155, 7204, 4216, 118, 25925, 4337]}, 'GLFq': {'d': 4, 'q': 4, 'gens': [3230696963, 2210795970, 2160064514, 27331201, 2149582850, 3856979952]}, 'Perm': {'d': 16, 'gens': [363624, 87657293544, 3, 3991680, 6504, 1482509952000]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{15}^2:C_2^2', 'transitive_degree': 30, '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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 180, 'aut_gen_orders': [30, 12, 2, 30, 30, 15], 'aut_gens': [[1, 2, 24, 48, 1440, 21600], [279793, 220042, 295656, 85008, 290880, 185760], [285801, 248762, 291240, 25464, 257760, 28032], [79201, 107162, 175704, 92208, 20160, 302400], [291361, 273818, 291384, 293712, 1440, 21600], [149809, 244010, 792, 14448, 20160, 33120], [191521, 16898, 24, 10128, 1440, 21600]], 'aut_group': '648000.v', 'aut_hash': 2811864275658790406, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 648000, 'aut_permdeg': 1215, 'aut_perms': 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'aut_phi_ratio': 7.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 90, 1, 1], [2, 125, 1, 1], [2, 135, 1, 1], [2, 450, 1, 1], [2, 675, 1, 1], [3, 6, 1, 1], [3, 8, 1, 1], [3, 12, 1, 1], [3, 1800, 1, 1], [4, 1125, 1, 2], [4, 1350, 1, 2], [4, 3375, 1, 2], [4, 4050, 1, 1], [4, 6750, 1, 2], [4, 20250, 1, 1], [5, 12, 1, 1], [5, 16, 1, 1], [5, 24, 1, 2], [5, 48, 1, 1], [6, 180, 1, 2], [6, 270, 1, 1], [6, 360, 1, 1], [6, 750, 1, 1], [6, 900, 1, 2], [6, 1000, 1, 1], [6, 1350, 1, 1], [6, 1500, 1, 1], [6, 1800, 1, 1], [6, 9000, 1, 1], [9, 3600, 1, 1], [10, 360, 1, 2], [10, 720, 1, 2], [10, 1080, 1, 3], [10, 1800, 1, 1], [10, 2700, 1, 1], [12, 2700, 1, 2], [12, 4500, 1, 4], [12, 13500, 1, 2], [12, 27000, 1, 2], [15, 16, 2, 1], [15, 24, 1, 1], [15, 24, 2, 2], [15, 48, 1, 4], [15, 48, 2, 10], [15, 96, 1, 6], [15, 96, 2, 7], [15, 7200, 1, 1], [18, 18000, 1, 1], [20, 5400, 1, 2], [20, 16200, 1, 1], [30, 360, 2, 2], [30, 720, 1, 2], [30, 720, 2, 10], [30, 1080, 2, 2], [30, 1800, 2, 1], [30, 2160, 1, 1], [30, 3600, 1, 1], [30, 3600, 2, 1], [30, 5400, 1, 1], [45, 7200, 2, 1], [60, 5400, 2, 2]], 'aut_supersolvable': False, 'aut_tex': 'D_{15}\\wr S_3.C_4', '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': 180, 'autcentquo_group': '648000.v', 'autcentquo_hash': 2811864275658790406, 'autcentquo_nilpotent': False, 'autcentquo_order': 648000, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'D_{15}\\wr S_3.C_4', 'cc_stats': [[1, 1, 1], [2, 90, 1], [2, 125, 1], [2, 135, 1], [2, 450, 1], [2, 675, 1], [3, 6, 1], [3, 8, 1], [3, 12, 1], [3, 1800, 1], [4, 1125, 2], [4, 1350, 2], [4, 3375, 2], [4, 4050, 1], [4, 6750, 2], [4, 20250, 1], [5, 12, 1], [5, 16, 1], [5, 24, 2], [5, 48, 1], [6, 180, 2], [6, 270, 1], [6, 360, 1], [6, 750, 1], [6, 900, 2], [6, 1000, 1], [6, 1350, 1], [6, 1500, 1], [6, 1800, 1], [6, 9000, 1], [9, 3600, 1], [10, 360, 2], [10, 720, 2], [10, 1080, 3], [10, 1800, 1], [10, 2700, 1], [12, 2700, 2], [12, 4500, 4], [12, 13500, 2], [12, 27000, 2], [15, 16, 2], [15, 24, 5], [15, 48, 24], [15, 96, 20], [15, 7200, 1], [18, 18000, 1], [20, 5400, 2], [20, 16200, 1], [30, 360, 4], [30, 720, 22], [30, 1080, 4], [30, 1800, 2], [30, 2160, 1], [30, 3600, 3], [30, 5400, 1], [45, 7200, 2], [60, 5400, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '324000.bm', 'commutator_count': 1, 'commutator_label': '40500.j', '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', '3.1', '3.1', '5.1', '5.1', '5.1'], 'composition_length': 12, 'conjugacy_classes_known': True, 'counter': 39, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 90, 1, 1], [2, 125, 1, 1], [2, 135, 1, 1], [2, 450, 1, 1], [2, 675, 1, 1], [3, 6, 1, 1], [3, 8, 1, 1], [3, 12, 1, 1], [3, 1800, 1, 1], [4, 1125, 2, 1], [4, 1350, 2, 1], [4, 3375, 2, 1], [4, 4050, 1, 1], [4, 6750, 2, 1], [4, 20250, 1, 1], [5, 12, 1, 1], [5, 16, 1, 1], [5, 24, 1, 2], [5, 48, 1, 1], [6, 180, 1, 2], [6, 270, 1, 1], [6, 360, 1, 1], [6, 750, 1, 1], [6, 900, 1, 2], [6, 1000, 1, 1], [6, 1350, 1, 1], [6, 1500, 1, 1], [6, 1800, 1, 1], [6, 9000, 1, 1], [9, 3600, 1, 1], [10, 360, 1, 2], [10, 720, 1, 2], [10, 1080, 1, 3], [10, 1800, 1, 1], [10, 2700, 1, 1], [12, 2700, 2, 1], [12, 4500, 2, 2], [12, 13500, 2, 1], [12, 27000, 2, 1], [15, 16, 2, 1], [15, 24, 1, 1], [15, 24, 2, 2], [15, 48, 1, 4], [15, 48, 2, 10], [15, 96, 1, 6], [15, 96, 2, 7], [15, 7200, 1, 1], [18, 18000, 1, 1], [20, 5400, 2, 1], [20, 16200, 1, 1], [30, 360, 2, 2], [30, 720, 1, 2], [30, 720, 2, 10], [30, 1080, 2, 2], [30, 1800, 2, 1], [30, 2160, 1, 1], [30, 3600, 1, 1], [30, 3600, 2, 1], [30, 5400, 1, 1], [45, 7200, 2, 1], [60, 5400, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 29016, 'exponent': 180, 'exponents_of_order': [5, 4, 3], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[16, 0, 4], [24, 0, 18], [24, 1, 2], [32, 0, 2], [48, 0, 40], [48, 1, 8], [96, 0, 14], [96, 1, 6]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '324000.bm', 'hash': 6992222009207964957, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 180, 'inner_gen_orders': [4, 12, 2, 30, 15, 15], 'inner_gens': [[1, 158074, 288744, 254832, 54720, 158400], [44969, 2, 143280, 52584, 1344, 20544], [288721, 115946, 24, 259152, 1440, 312480], [92257, 146234, 87960, 48, 20160, 302400], [292321, 1538, 24, 2928, 1440, 21600], [208801, 24098, 54744, 43248, 1440, 21600]], 'inner_hash': 6992222009207964957, 'inner_nilpotent': False, 'inner_order': 324000, 'inner_split': True, 'inner_tex': 'C_{15}^3.(C_4\\times S_4)', 'inner_used': [1, 2], 'irrC_degree': 16, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 8], [2, 4], [3, 8], [6, 8], [8, 4], [12, 14], [16, 8], [24, 29], [32, 3], [48, 50], [96, 20]], 'label': '324000.bm', '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': 'C15^3.(C4*S4)', 'ngens': 12, '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, 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': 117, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 156, 'number_divisions': 106, 'number_normal_subgroups': 18, 'number_subgroup_autclasses': 2508, 'number_subgroup_classes': 2508, 'number_subgroups': 1413136, 'old_label': None, 'order': 324000, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 1475], [3, 1826], [4, 49500], [5, 124], [6, 18190], [9, 3600], [10, 9900], [12, 104400], [15, 10424], [18, 18000], [20, 27000], [30, 43560], [45, 14400], [60, 21600]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [164748], 'outer_gens': [[20737, 254690, 88440, 298704, 18720, 280800]], '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': 6, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [3, 4], [4, 1], [6, 6], [8, 4], [12, 12], [16, 4], [24, 11], [32, 3], [48, 18], [64, 1], [96, 27], [192, 7]], 'representations': {'PC': {'code': '2560975055750535628440959797134618647641952709363283191527680408195495050363903468332791041667081841374214075666627686531575688993413408876793589831431805993790723420743847600035909171053287802079058035871018882835538350542411390272189999679389700815586679083982798993803372323456428654556782238484949698423', 'gens': [1, 2, 5, 6, 9, 11], 'pres': [12, 2, 2, 2, 3, 2, 2, 3, 5, 3, 5, 3, 5, 2635200, 3793777, 61, 3531890, 98, 14246403, 17324644, 4298416, 4525588, 1647040, 18347909, 1893041, 3451277, 78449, 777509, 209, 112902, 7378578, 211710, 922362, 4758, 330, 110599, 9745939, 207391, 3162283, 4663, 5909768, 72596, 4587872, 9116, 45428, 428, 17107209, 69141, 4276833, 8685, 43269, 20908810, 1355926, 1808962, 169534, 1718698, 831670, 526, 18662411, 1327127, 5142563, 165935, 1607099, 777671]}, 'Perm': {'d': 24, 'gens': [27081010061517974326131, 2409067128187421056096]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_{15}^3.(C_4\\times S_4)', 'transitive_degree': 45, 'wreath_data': None, 'wreath_product': False}
