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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '800.1199', 'ambient_counter': 1199, 'ambient_order': 800, 'ambient_tex': 'D_{10}^2.C_2', 'central': False, 'central_factor': True, 'centralizer_order': 4, 'characteristic': False, 'core_order': 400, 'counter': 2, 'cyclic': False, 'direct': True, 'hall': 0, 'label': '800.1199.2.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '2.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '2.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 2, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '400.210', 'subgroup_hash': 210, 'subgroup_order': 400, 'subgroup_tex': 'D_{10}:F_5', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '800.1199', 'aut_centralizer_order': 2, 'aut_label': '2.a1', 'aut_quo_index': 1, 'aut_stab_index': 12, 'aut_weyl_group': '3200.ih', 'aut_weyl_index': 24, 'centralizer': '200.a1', 'complements': ['400.a1', '400.b1', '400.c1'], 'conjugacy_class_count': 12, 'contained_in': ['1.a1'], 'contains': ['4.a1', '4.c1', '4.d1', '4.g1', '10.a1'], 'core': '2.a1', 'coset_action_label': None, 'count': 12, 'diagramx': [7506, 7513, 2491, 7998], 'generators': [2, 160, 400, 4, 40, 656], 'label': '800.1199.2.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.a1', 'normal_contained_in': ['1.a1'], 'normal_contains': ['4.a1', '4.c1', '4.d1', '4.g1'], 'normalizer': '1.a1', 'old_label': '2.a1', 'projective_image': '400.210', 'quotient_action_image': '1.1', 'quotient_action_kernel': '2.1', 'quotient_action_kernel_order': 2, 'quotient_fusion': None, 'short_label': '2.a1', 'subgroup_fusion': None, 'weyl_group': '200.42'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 40, 'aut_gen_orders': [4, 4, 4, 8], 'aut_gens': [[1, 4, 40], [25, 12, 40], [105, 68, 280], [201, 228, 360], [337, 190, 232]], 'aut_group': '3200.ih', 'aut_hash': 478294559689971653, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3200, 'aut_permdeg': 70, 'aut_perms': [1706630822478158861477103351910596501739177439155479225397641467951912305508749334304568987317416109, 9771731289606709789190088979285914845664706075185220954741554415776573404450180993971923689485833592, 10122729324546702701564025981309912609399616374549718461996852903095882068954081629106978903345027320, 1500927953539068915923772367550427624020818621691523586786017891808915841863296088496565967697203996], 'aut_phi_ratio': 20.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 4, 1], [2, 25, 1, 2], [4, 25, 2, 2], [4, 25, 4, 1], [5, 4, 2, 1], [5, 8, 2, 1], [10, 4, 2, 1], [10, 8, 2, 1], [10, 20, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_5^2:C_2^3', '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': 40, 'autcentquo_group': '800.1191', 'autcentquo_hash': 1191, 'autcentquo_nilpotent': False, 'autcentquo_order': 800, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\wr C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 5, 4], [2, 25, 2], [4, 25, 8], [5, 4, 2], [5, 8, 2], [10, 4, 2], [10, 8, 2], [10, 20, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '200.42', 'commutator_count': 1, 'commutator_label': '25.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '5.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 210, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['200.42', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 5, 1, 4], [2, 25, 1, 2], [4, 25, 2, 4], [5, 4, 1, 2], [5, 8, 1, 2], [10, 4, 1, 2], [10, 8, 1, 2], [10, 20, 1, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6048, 'exponent': 20, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[8, 1, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '400.210', 'hash': 210, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [4, 10, 5], 'inner_gens': [[1, 28, 280], [17, 4, 360], [161, 84, 40]], 'inner_hash': 42, 'inner_nilpotent': False, 'inner_order': 200, 'inner_split': False, 'inner_tex': 'D_5:F_5', 'inner_used': [1, 2, 3], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [4, 8], [8, 4]], 'label': '400.210', 'linC_count': 14, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 14, 'linQ_dim': 8, 'linQ_dim_count': 14, 'linR_count': 14, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D10:F5', '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': 13, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 28, 'number_divisions': 24, 'number_normal_subgroups': 39, 'number_subgroup_autclasses': 57, 'number_subgroup_classes': 124, 'number_subgroups': 892, 'old_label': None, 'order': 400, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 71], [4, 200], [5, 24], [10, 104]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [2, 2, 20, 22], 'outer_gens': [[1, 342, 232], [1, 228, 280], [201, 236, 280], [1, 12, 40]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 8], [8, 4]], 'representations': {'PC': {'code': 216894713722374911594438654418908922859571, 'gens': [1, 3, 5], 'pres': [6, -2, -2, -2, -5, -2, -5, 12, 506, 332, 50, 387, 393, 8404, 5410, 2716, 88, 5765, 5771, 2897]}, 'GLFp': {'d': 3, 'p': 5, 'gens': [371499, 1816751, 25977, 867531, 1646908, 1565004]}, 'Perm': {'d': 12, 'gens': [4032009, 736, 720, 7983376, 46, 51528960]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{10}:F_5', 'transitive_degree': 20, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '32.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 120, 'aut_gen_orders': [4, 30, 8, 8], 'aut_gens': [[1, 2, 8, 80], [1, 2, 780, 464], [401, 530, 620, 49], [1, 482, 652, 433], [1, 723, 317, 448]], 'aut_group': None, 'aut_hash': 7210635723483188091, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 76800, 'aut_permdeg': 140, 'aut_perms': [7942222498410828028657015020916095893097783738963353230430870269598255791851922022203664272307868953142365766695116630912600960334888016003761075543653642485130949451184517781602270862447820398336489256653181441656083053811323609802168690208, 1975024803878963235691728009503483032794795180870465175304659851180117161096478684098157483576056266647517081785093830659315457288346843177961266763512646114414058223288232023822215892286730705804624322389484476389083582608893964393833416173, 12978107219321058902080422601188465159113160852556914629683358872320455062291354913530056082853157915000796590041011267793429000014222780200380450717681338026174203628186106486420118329565799792578996305550631130943973718043683076497988714886, 9781294936098949183404847853618679638390216157331414489837517890435964433465683130459966963048814196233286165321317321178086798542543526214298519514268781109482291409902100140868325003493675691700905823641009953664471802952300558894200397791], 'aut_phi_ratio': 240.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 5, 8, 1], [2, 25, 1, 1], [2, 25, 3, 1], [4, 25, 4, 2], [4, 25, 8, 1], [5, 4, 2, 1], [5, 8, 2, 1], [10, 4, 6, 1], [10, 8, 6, 1], [10, 20, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{10}^2.A_4.C_4\\wr C_2.C_2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '96.227', 'autcent_hash': 227, 'autcent_nilpotent': False, 'autcent_order': 96, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^2:S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 40, 'autcentquo_group': '800.1191', 'autcentquo_hash': 1191, 'autcentquo_nilpotent': False, 'autcentquo_order': 800, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\wr C_2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 5, 8], [2, 25, 4], [4, 25, 16], [5, 4, 2], [5, 8, 2], [10, 4, 6], [10, 8, 6], [10, 20, 8]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '200.42', 'commutator_count': 1, 'commutator_label': '25.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1199, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['200.42', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 5, 1, 8], [2, 25, 1, 4], [4, 25, 2, 8], [5, 4, 1, 2], [5, 8, 1, 2], [10, 4, 1, 6], [10, 8, 1, 6], [10, 20, 1, 8]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1614480, 'exponent': 20, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '800.1199', 'hash': 1199, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [1, 4, 10, 5], 'inner_gens': [[1, 2, 8, 80], [1, 2, 376, 560], [1, 514, 8, 720], [1, 322, 168, 80]], 'inner_hash': 42, 'inner_nilpotent': False, 'inner_order': 200, 'inner_split': False, 'inner_tex': 'D_5:F_5', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [4, 16], [8, 8]], 'label': '800.1199', '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': 'D10^2.C2', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 13, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 56, 'number_divisions': 48, 'number_normal_subgroups': 155, 'number_subgroup_autclasses': 89, 'number_subgroup_classes': 526, 'number_subgroups': 4020, 'old_label': None, 'order': 800, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 143], [4, 400], [5, 24], [10, 232]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 3, 2, 2, 2, 2], 'outer_gen_pows': [124, 6, 124, 120, 0, 0, 0, 4], 'outer_gens': [[400, 403, 617, 321], [1, 2, 652, 432], [1, 2, 24, 80], [401, 402, 568, 321], [1, 2, 632, 720], [1, 2, 73, 80], [1, 3, 233, 720], [1, 402, 296, 240]], 'outer_group': '384.20163', 'outer_hash': 20163, 'outer_nilpotent': False, 'outer_order': 384, 'outer_permdeg': 12, 'outer_perms': [265929916, 213071683, 136105698, 92355916, 40723254, 766369, 1881858, 136105446], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^2\\wr S_3', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 14, '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, 16], [8, 8]], 'representations': {'PC': {'code': 454103470539396089848733524438044673248275890227, 'gens': [1, 2, 4, 6], 'pres': [7, -2, -2, -2, -2, -5, -2, -5, 36, 5274, 1641, 80, 571, 578, 11772, 7579, 3806, 124, 7853, 7860, 3947]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [106564013487, 122109387504, 36964419377, 149448494378, 89223856252, 12021289379, 98019097504]}, 'Perm': {'d': 14, 'gens': [522588249, 1037877840, 90720, 41040, 1037836816, 7743859200, 46]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{10}^2.C_2', 'transitive_degree': 40, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, 'wreath_data': None, 'wreath_product': False}
