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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1728.47365', 'ambient_counter': 47365, 'ambient_order': 1728, 'ambient_tex': 'Q_8:S_3^3', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 8, 'counter': 252, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1728.47365.18.x1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '18.x1', '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': 18, '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': '96.215', 'subgroup_hash': 215, 'subgroup_order': 96, 'subgroup_tex': 'D_4:D_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1728.47365', 'aut_centralizer_order': None, 'aut_label': '18.x1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '432.g1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['6.w1', '6.z1'], 'contains': ['36.bo1', '36.bv1', '36.bx1', '36.cb1', '36.ck1', '36.cy1', '36.dt1', '54.j1'], 'core': '216.a1', 'coset_action_label': None, 'count': 18, 'diagramx': None, 'generators': [3, 52, 864, 12, 72, 1296], 'label': '1728.47365.18.x1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '2.j1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '18.x1', 'old_label': '18.x1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '18.x1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.14', '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, 6, 2, 2, 6], 'aut_gens': [[1, 2, 4, 8], [1, 50, 4, 8], [49, 50, 77, 8], [29, 2, 1, 40], [49, 2, 4, 8], [49, 2, 52, 56], [1, 66, 52, 8]], 'aut_group': '576.8659', 'aut_hash': 8659, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 576, 'aut_permdeg': 11, 'aut_perms': [7, 766080, 4395007, 3669120, 16, 7984800], 'aut_phi_ratio': 18.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 3, 1], [2, 3, 2, 1], [2, 6, 3, 1], [3, 2, 1, 1], [4, 1, 2, 1], [4, 2, 3, 1], [4, 3, 2, 1], [4, 6, 3, 1], [6, 2, 1, 1], [6, 4, 3, 1], [12, 2, 2, 1], [12, 4, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2:D_6^2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '36.10', 'autcentquo_hash': 10, 'autcentquo_nilpotent': False, 'autcentquo_order': 36, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 3], [2, 3, 2], [2, 6, 3], [3, 2, 1], [4, 1, 2], [4, 2, 3], [4, 3, 2], [4, 6, 3], [6, 2, 1], [6, 4, 3], [12, 2, 2], [12, 4, 3]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '24.14', 'commutator_count': 1, 'commutator_label': '6.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 215, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['16.13', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 3], [2, 3, 1, 2], [2, 6, 1, 3], [3, 2, 1, 1], [4, 1, 2, 1], [4, 2, 1, 3], [4, 3, 2, 1], [4, 6, 1, 3], [6, 2, 1, 1], [6, 4, 1, 3], [12, 2, 2, 1], [12, 4, 1, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 43680, 'exponent': 12, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 0, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '48.51', 'hash': 215, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 2, 3], 'inner_gens': [[1, 2, 52, 8], [1, 2, 4, 40], [49, 2, 4, 8], [1, 66, 4, 8]], 'inner_hash': 14, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': True, 'inner_tex': 'C_2\\times D_6', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 12], [4, 2]], 'label': '96.215', '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': 'D4:D6', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 14, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 30, 'number_divisions': 27, 'number_normal_subgroups': 87, 'number_subgroup_autclasses': 66, 'number_subgroup_classes': 164, 'number_subgroups': 330, 'old_label': None, 'order': 96, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 31], [3, 2], [4, 32], [6, 14], [12, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [0, 1, 0], 'outer_gens': [[1, 50, 4, 8], [1, 2, 29, 8], [4, 2, 29, 56]], 'outer_group': '24.14', 'outer_hash': 14, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 7, 'outer_perms': [7, 127, 856], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 8], [4, 2], [8, 1]], 'representations': {'PC': {'code': 558648945456059905, 'gens': [1, 2, 3, 4], 'pres': [6, -2, -2, -2, -2, -2, -3, 938, 489, 69, 1210, 88, 1163]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101729095570500, 91793140190669372, 125101739157295540, 91678743437649212]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [117168654377, 141124222951, 25586613754, 21872747503, 116436624378, 143688093129]}, 'Perm': {'d': 11, 'gens': [3719545, 8361600, 12506424, 85704, 3719544, 3]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4:D_6', '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': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [6, 12, 4, 2, 12, 6, 2], 'aut_gens': [[1, 2, 12, 24, 144], [54, 97, 444, 800, 588], [1, 970, 1296, 1140, 1452], [870, 105, 444, 220, 1584], [1, 922, 876, 120, 156], [9, 914, 444, 744, 720], [5, 922, 432, 756, 1596], [9, 10, 12, 1176, 720]], 'aut_group': None, 'aut_hash': 942252590584195724, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 82944, 'aut_permdeg': 24, 'aut_perms': [152297894689764724364505, 19191514483363089751212, 466055521751745882620510, 2299786626428730910153, 564183427232389608877075, 278481055957010281196649, 545044274773266031154220], 'aut_phi_ratio': 144.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 4, 1], [2, 6, 3, 1], [2, 9, 2, 1], [2, 18, 6, 1], [2, 54, 3, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 4, 1, 1], [3, 4, 2, 1], [3, 8, 1, 1], [4, 2, 3, 1], [4, 3, 2, 1], [4, 6, 6, 1], [4, 9, 4, 1], [4, 18, 3, 1], [4, 27, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 4, 1, 1], [6, 4, 2, 1], [6, 6, 4, 2], [6, 8, 1, 1], [6, 12, 4, 1], [6, 12, 6, 1], [6, 18, 2, 1], [6, 24, 3, 1], [6, 36, 6, 1], [12, 4, 3, 1], [12, 4, 6, 1], [12, 6, 4, 1], [12, 8, 3, 1], [12, 8, 6, 1], [12, 12, 2, 1], [12, 12, 6, 2], [12, 16, 3, 1], [12, 18, 4, 1], [12, 24, 6, 1], [12, 36, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6^2.C_6^2.C_2^6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '32.51', 'autcent_hash': 51, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^5', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '2592.fs', 'autcentquo_hash': 3641402233686599739, 'autcentquo_nilpotent': False, 'autcentquo_order': 2592, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3^4:C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 4], [2, 6, 3], [2, 9, 2], [2, 18, 6], [2, 54, 3], [3, 2, 3], [3, 4, 3], [3, 8, 1], [4, 2, 3], [4, 3, 2], [4, 6, 6], [4, 9, 4], [4, 18, 3], [4, 27, 2], [6, 2, 3], [6, 4, 3], [6, 6, 8], [6, 8, 1], [6, 12, 10], [6, 18, 2], [6, 24, 3], [6, 36, 6], [12, 4, 9], [12, 6, 4], [12, 8, 9], [12, 12, 14], [12, 16, 3], [12, 18, 4], [12, 24, 6], [12, 36, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '864.4704', 'commutator_count': 1, 'commutator_label': '54.15', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 47365, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['48.41', 1], ['6.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 4], [2, 6, 1, 3], [2, 9, 1, 2], [2, 18, 1, 6], [2, 54, 1, 3], [3, 2, 1, 3], [3, 4, 1, 3], [3, 8, 1, 1], [4, 2, 1, 3], [4, 3, 2, 1], [4, 6, 1, 6], [4, 9, 2, 2], [4, 18, 1, 3], [4, 27, 2, 1], [6, 2, 1, 3], [6, 4, 1, 3], [6, 6, 1, 8], [6, 8, 1, 1], [6, 12, 1, 10], [6, 18, 1, 2], [6, 24, 1, 3], [6, 36, 1, 6], [12, 4, 1, 9], [12, 6, 2, 2], [12, 8, 1, 9], [12, 12, 1, 12], [12, 12, 2, 1], [12, 16, 1, 3], [12, 18, 2, 2], [12, 24, 1, 6], [12, 36, 1, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 53329920000, 'exponent': 12, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[16, 1, 1]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '864.4704', 'hash': 47365, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 6, 2, 6, 6], 'inner_gens': [[1, 10, 12, 24, 144], [5, 2, 12, 120, 144], [1, 2, 12, 24, 1008], [1, 50, 12, 24, 1584], [1, 2, 876, 312, 144]], 'inner_hash': 4704, 'inner_nilpotent': False, 'inner_order': 864, 'inner_split': True, 'inner_tex': 'S_3\\times D_6^2', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': 16, 'irrQ_degree': 16, 'irrQ_dim': 32, 'irrR_degree': 16, 'irrep_stats': [[1, 32], [2, 56], [4, 36], [8, 10], [16, 1]], 'label': '1728.47365', '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': 'Q8:S3^3', 'ngens': 9, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 42, 'number_characteristic_subgroups': 32, 'number_conjugacy_classes': 135, 'number_divisions': 126, 'number_normal_subgroups': 664, 'number_subgroup_autclasses': 864, 'number_subgroup_classes': 4092, 'number_subgroups': 30756, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 319], [3, 26], [4, 192], [6, 518], [12, 672]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 2, 2, 3], 'outer_gen_pows': [12, 0, 7, 0, 0, 7], 'outer_gens': [[1, 2, 12, 120, 1596], [1, 2, 876, 120, 720], [6, 97, 876, 940, 720], [865, 2, 876, 120, 720], [865, 866, 12, 24, 720], [1, 10, 432, 1428, 1020]], 'outer_group': '96.209', 'outer_hash': 209, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 9, 'outer_perms': [5040, 143, 21, 136, 23, 80640], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times D_6', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 48], [4, 32], [8, 12], [16, 2]], 'representations': {'PC': {'code': 6391719674496748598971563130793809448715173835076777033, 'gens': [1, 2, 4, 5, 7], 'pres': [9, -2, -2, -3, -2, -2, -3, -2, -2, -3, 181, 46, 218, 2622, 2713, 130, 2606, 5325, 4200, 186, 4363, 214, 3932]}, 'Perm': {'d': 17, 'gens': [21109604140936, 21109604140943, 45787363781040, 68355447955200, 86325272812800, 21109604140800, 325, 435, 45360]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'Q_8:S_3^3', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
