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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '77760.p', 'ambient_counter': 16, 'ambient_order': 77760, 'ambient_tex': 'A_5\\times S_3\\wr S_3', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 1, 'counter': 833, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '77760.p.360.bc1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '360.bc1', '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': '216.170', 'subgroup_hash': 170, 'subgroup_order': 216, 'subgroup_tex': 'C_6\\times S_3^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '77760.p', 'aut_centralizer_order': None, 'aut_label': '360.bc1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '12960.o1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['90.e1', '120.h1', '180.q1'], 'contains': ['720.k1', '720.l1', '720.o1', '720.dg1', '720.dl1', '720.ef1', '720.ex1', '1080.s1', '1080.cc1', '1080.ce1'], 'core': '77760.a1', 'coset_action_label': None, 'count': 180, 'diagramx': None, 'generators': [38922993360, 84596400, 31658024880, 6318955451, 1134000, 22125520080], 'label': '77760.p.360.bc1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '180.q1', 'old_label': '360.bc1', 'projective_image': '77760.p', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '360.bc1', 'subgroup_fusion': None, 'weyl_group': '72.46'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '24.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 2, 2, 2, 3, 3], 'aut_gens': [[1, 6, 36], [5, 6, 180], [128, 183, 120], [5, 6, 36], [1, 114, 36], [1, 30, 180], [109, 114, 36], [1, 78, 36], [25, 6, 36]], 'aut_group': '576.8418', 'aut_hash': 8418, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 576, 'aut_permdeg': 12, 'aut_perms': [40284726, 1134414, 1, 120, 1174446, 126, 91854000, 214104240], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 4, 1], [2, 9, 2, 1], [3, 1, 2, 1], [3, 2, 2, 1], [3, 2, 4, 1], [3, 4, 1, 1], [3, 4, 2, 1], [6, 1, 2, 1], [6, 2, 2, 1], [6, 2, 4, 1], [6, 3, 8, 1], [6, 4, 1, 1], [6, 4, 2, 1], [6, 6, 4, 1], [6, 6, 8, 1], [6, 9, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'D_6^2:C_2^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '72.40', 'autcentquo_hash': 40, 'autcentquo_nilpotent': False, 'autcentquo_order': 72, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\SOPlus(4,2)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 4], [2, 9, 2], [3, 1, 2], [3, 2, 6], [3, 4, 3], [6, 1, 2], [6, 2, 6], [6, 3, 8], [6, 4, 3], [6, 6, 12], [6, 9, 4]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '36.10', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 170, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['6.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 4], [2, 9, 1, 2], [3, 1, 2, 1], [3, 2, 1, 2], [3, 2, 2, 2], [3, 4, 1, 1], [3, 4, 2, 1], [6, 1, 2, 1], [6, 2, 1, 2], [6, 2, 2, 2], [6, 3, 2, 4], [6, 4, 1, 1], [6, 4, 2, 1], [6, 6, 1, 4], [6, 6, 2, 4], [6, 9, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 4368, 'exponent': 6, 'exponents_of_order': [3, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 0, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '216.170', 'hash': 170, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6, 3], 'inner_gens': [[1, 30, 36], [13, 6, 180], [1, 78, 36]], 'inner_hash': 10, 'inner_nilpotent': False, 'inner_order': 36, 'inner_split': True, 'inner_tex': 'S_3^2', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 24], [2, 24], [4, 6]], 'label': '216.170', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, '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': 'C6*S3^2', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 18, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 54, 'number_divisions': 36, 'number_normal_subgroups': 56, 'number_subgroup_autclasses': 72, 'number_subgroup_classes': 162, 'number_subgroups': 492, 'old_label': None, 'order': 216, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 31], [3, 26], [6, 158]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[5, 114, 180], [128, 183, 120], [113, 114, 36], [109, 114, 36]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [126, 55, 289, 288], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 16], [4, 10], [8, 2]], 'representations': {'PC': {'code': 6879565087467783599445, 'gens': [1, 3, 5], 'pres': [6, -2, -3, -2, -3, -2, -3, 12, 542, 50, 579, 916, 88, 881]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101729074815868, 41624330840588104, 125101738763505778]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [16278328, 8405204, 25633100, 28816400, 29360962, 17443190]}, 'Perm': {'d': 11, 'gens': [3719521, 3760560, 1, 30, 8427600, 11980800]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6\\times S_3^2', 'transitive_degree': 24, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 180, 'aut_gen_orders': [6, 4, 6], 'aut_gens': [[7276964411, 13067354201], [46946188856, 7243534180], [49937494350, 12976674569], [7276964412, 1683803562]], 'aut_group': '155520.m', 'aut_hash': 3820719417594226523, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 155520, 'aut_permdeg': 42, 'aut_perms': [1876262248220482368162687231326036845095429426317, 702414369353014982359618143419748284370182909523513, 1265358009089005506898710751950454472433343447893953], 'aut_phi_ratio': 7.5, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 15, 1, 1], [2, 18, 1, 1], [2, 27, 1, 2], [2, 54, 1, 1], [2, 135, 1, 1], [2, 270, 1, 1], [2, 405, 1, 2], [2, 810, 1, 1], [3, 6, 1, 1], [3, 8, 1, 1], [3, 12, 1, 1], [3, 20, 1, 1], [3, 72, 1, 1], [3, 120, 1, 1], [3, 160, 1, 1], [3, 240, 1, 1], [3, 1440, 1, 1], [4, 54, 1, 1], [4, 162, 1, 1], [4, 810, 1, 1], [4, 2430, 1, 1], [5, 12, 2, 1], [6, 36, 1, 4], [6, 54, 1, 1], [6, 72, 1, 1], [6, 90, 1, 1], [6, 108, 1, 1], [6, 120, 1, 1], [6, 180, 1, 2], [6, 216, 1, 1], [6, 360, 1, 1], [6, 540, 1, 6], [6, 720, 1, 4], [6, 810, 1, 1], [6, 1080, 1, 4], [6, 1440, 1, 1], [6, 1620, 1, 1], [6, 2160, 1, 1], [6, 3240, 1, 1], [6, 4320, 1, 1], [9, 144, 1, 1], [9, 2880, 1, 1], [10, 108, 2, 1], [10, 216, 2, 1], [10, 324, 2, 2], [10, 648, 2, 1], [12, 108, 1, 1], [12, 1080, 1, 1], [12, 1620, 1, 1], [12, 2160, 1, 1], [12, 3240, 1, 1], [15, 72, 2, 1], [15, 96, 2, 1], [15, 144, 2, 1], [15, 864, 2, 1], [18, 2160, 1, 1], [20, 648, 2, 1], [20, 1944, 2, 1], [30, 432, 2, 4], [30, 648, 2, 1], [30, 864, 2, 1], [30, 1296, 2, 1], [30, 2592, 2, 1], [45, 1728, 2, 1], [60, 1296, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\wr S_3\\times S_5', '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': '155520.m', 'autcentquo_hash': 3820719417594226523, 'autcentquo_nilpotent': False, 'autcentquo_order': 155520, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\wr S_3\\times S_5', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 15, 1], [2, 18, 1], [2, 27, 2], [2, 54, 1], [2, 135, 1], [2, 270, 1], [2, 405, 2], [2, 810, 1], [3, 6, 1], [3, 8, 1], [3, 12, 1], [3, 20, 1], [3, 72, 1], [3, 120, 1], [3, 160, 1], [3, 240, 1], [3, 1440, 1], [4, 54, 1], [4, 162, 1], [4, 810, 1], [4, 2430, 1], [5, 12, 2], [6, 36, 4], [6, 54, 1], [6, 72, 1], [6, 90, 1], [6, 108, 1], [6, 120, 1], [6, 180, 2], [6, 216, 1], [6, 360, 1], [6, 540, 6], [6, 720, 4], [6, 810, 1], [6, 1080, 4], [6, 1440, 1], [6, 1620, 1], [6, 2160, 1], [6, 3240, 1], [6, 4320, 1], [9, 144, 1], [9, 2880, 1], [10, 108, 2], [10, 216, 2], [10, 324, 4], [10, 648, 2], [12, 108, 1], [12, 1080, 1], [12, 1620, 1], [12, 2160, 1], [12, 3240, 1], [15, 72, 2], [15, 96, 2], [15, 144, 2], [15, 864, 2], [18, 2160, 1], [20, 648, 2], [20, 1944, 2], [30, 432, 8], [30, 648, 2], [30, 864, 2], [30, 1296, 2], [30, 2592, 2], [45, 1728, 2], [60, 1296, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '77760.p', 'commutator_count': 1, 'commutator_label': '19440.i', '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', '60.5'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 16, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [['1296.3490', 1], ['60.5', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 15, 1, 1], [2, 18, 1, 1], [2, 27, 1, 2], [2, 54, 1, 1], [2, 135, 1, 1], [2, 270, 1, 1], [2, 405, 1, 2], [2, 810, 1, 1], [3, 6, 1, 1], [3, 8, 1, 1], [3, 12, 1, 1], [3, 20, 1, 1], [3, 72, 1, 1], [3, 120, 1, 1], [3, 160, 1, 1], [3, 240, 1, 1], [3, 1440, 1, 1], [4, 54, 1, 1], [4, 162, 1, 1], [4, 810, 1, 1], [4, 2430, 1, 1], [5, 12, 2, 1], [6, 36, 1, 4], [6, 54, 1, 1], [6, 72, 1, 1], [6, 90, 1, 1], [6, 108, 1, 1], [6, 120, 1, 1], [6, 180, 1, 2], [6, 216, 1, 1], [6, 360, 1, 1], [6, 540, 1, 6], [6, 720, 1, 4], [6, 810, 1, 1], [6, 1080, 1, 4], [6, 1440, 1, 1], [6, 1620, 1, 1], [6, 2160, 1, 1], [6, 3240, 1, 1], [6, 4320, 1, 1], [9, 144, 1, 1], [9, 2880, 1, 1], [10, 108, 2, 1], [10, 216, 2, 1], [10, 324, 2, 2], [10, 648, 2, 1], [12, 108, 1, 1], [12, 1080, 1, 1], [12, 1620, 1, 1], [12, 2160, 1, 1], [12, 3240, 1, 1], [15, 72, 2, 1], [15, 96, 2, 1], [15, 144, 2, 1], [15, 864, 2, 1], [18, 2160, 1, 1], [20, 648, 2, 1], [20, 1944, 2, 1], [30, 432, 2, 4], [30, 648, 2, 1], [30, 864, 2, 1], [30, 1296, 2, 1], [30, 2592, 2, 1], [45, 1728, 2, 1], [60, 1296, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 4446, 'exponent': 180, 'exponents_of_order': [6, 5, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[18, 1, 8], [24, 1, 8], [30, 1, 4], [32, 1, 2], [36, 1, 10], [40, 1, 2], [48, 1, 7], [60, 1, 5], [64, 1, 1], [80, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '77760.p', 'hash': 4970818228607773946, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 180, 'inner_gen_orders': [6, 20], 'inner_gens': [[7276964411, 1605466851], [21607332561, 13067354201]], 'inner_hash': 4970818228607773946, 'inner_nilpotent': False, 'inner_order': 77760, 'inner_split': True, 'inner_tex': 'A_5\\times S_3\\wr S_3', 'inner_used': [1, 2], 'irrC_degree': 18, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 18, 'irrep_stats': [[1, 4], [2, 2], [3, 12], [4, 4], [5, 4], [6, 8], [8, 4], [9, 8], [10, 2], [12, 9], [15, 4], [16, 1], [18, 8], [24, 8], [30, 4], [32, 2], [36, 10], [40, 2], [48, 7], [60, 5], [64, 1], [80, 1]], 'label': '77760.p', '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': False, 'name': 'A5*S3wrS3', '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, 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': 88, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 110, 'number_divisions': 88, 'number_normal_subgroups': 20, 'number_subgroup_autclasses': 3463, 'number_subgroup_classes': 3585, 'number_subgroups': 1091772, 'old_label': None, 'order': 77760, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 2175], [3, 2078], [4, 3456], [5, 24], [6, 25554], [9, 3024], [10, 3240], [12, 8208], [15, 2352], [18, 2160], [20, 5184], [30, 14256], [45, 3456], [60, 2592]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [60], 'outer_gens': [[8357610963, 12988246405]], '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': None, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2], [3, 4], [4, 4], [5, 4], [6, 8], [8, 4], [10, 2], [12, 11], [15, 4], [16, 1], [18, 4], [24, 4], [30, 4], [32, 2], [36, 4], [40, 2], [48, 7], [60, 5], [64, 1], [72, 5], [80, 1], [96, 1]], 'representations': {'Perm': {'d': 14, 'gens': [7276964411, 13067354201]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'A_5\\times S_3\\wr S_3', 'transitive_degree': 45, 'wreath_data': None, 'wreath_product': False}
