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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1152.32531', 'ambient_counter': 32531, 'ambient_order': 1152, 'ambient_tex': 'C_{12}^2:D_4', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 96, 'counter': 31, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1152.32531.6.j1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '6.j1', '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': 6, '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': '192.886', 'subgroup_hash': 886, 'subgroup_order': 192, 'subgroup_tex': 'C_3\\times D_4:D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1152.32531', 'aut_centralizer_order': 4, 'aut_label': '6.j1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '512.7530050', 'aut_weyl_index': 12, 'centralizer': '192.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.d1', '3.b1'], 'contains': ['12.c1', '12.z1', '12.bb1', '12.bc1', '12.be1', '18.d1'], 'core': '12.c1', 'coset_action_label': None, 'count': 3, 'diagramx': [8823, -1, 8422, -1], 'generators': [1, 576, 600, 288, 48, 12, 8], 'label': '1152.32531.6.j1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.d1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.b1', 'old_label': '6.j1', 'projective_image': '192.803', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.j1', 'subgroup_fusion': None, 'weyl_group': '64.138'}
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gps_groups • Show schema
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{'Agroup': False, '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': 8, 'aut_gen_orders': [2, 2, 2, 2, 2, 2, 2, 2, 2], 'aut_gens': [[1, 2, 12, 48], [1, 3, 132, 72], [37, 34, 36, 84], [1, 10, 12, 180], [1, 34, 12, 48], [1, 2, 12, 72], [97, 110, 36, 144], [1, 2, 36, 144], [25, 26, 12, 48], [1, 26, 12, 48]], 'aut_group': '512.7530050', 'aut_hash': 6209376761179717675, 'aut_nilpotency_class': 4, 'aut_nilpotent': True, 'aut_order': 512, 'aut_permdeg': 12, 'aut_perms': [207381000, 22186936, 126727, 94484903, 94484887, 262459440, 137520, 83589120, 94484880], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 2, 2], [2, 8, 1, 1], [3, 1, 2, 1], [4, 2, 2, 1], [4, 4, 2, 2], [6, 1, 2, 1], [6, 2, 2, 1], [6, 4, 4, 2], [6, 8, 2, 1], [8, 8, 2, 1], [12, 2, 4, 1], [12, 4, 4, 2], [24, 8, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4^2:C_2^3', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '32.27', 'autcentquo_hash': 27, 'autcentquo_nilpotent': True, 'autcentquo_order': 32, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\wr C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 4, 4], [2, 8, 1], [3, 1, 2], [4, 2, 2], [4, 4, 4], [6, 1, 2], [6, 2, 2], [6, 4, 8], [6, 8, 2], [8, 8, 2], [12, 2, 4], [12, 4, 8], [24, 8, 4]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '32.27', 'commutator_count': 1, 'commutator_label': '8.2', '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'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 886, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['64.134', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 4], [2, 8, 1, 1], [3, 1, 2, 1], [4, 2, 1, 2], [4, 4, 1, 4], [6, 1, 2, 1], [6, 2, 2, 1], [6, 4, 2, 4], [6, 8, 2, 1], [8, 8, 1, 2], [12, 2, 2, 2], [12, 4, 2, 4], [24, 8, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 4368, 'exponent': 24, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 0, 4]], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '24.15', 'hash': 886, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 2, 2, 4], 'inner_gens': [[1, 2, 36, 144], [1, 2, 12, 156], [25, 2, 12, 48], [97, 134, 12, 48]], 'inner_hash': 27, 'inner_nilpotent': True, 'inner_order': 32, 'inner_split': False, 'inner_tex': 'C_2^2\\wr C_2', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 24], [2, 18], [4, 6]], 'label': '192.886', 'linC_count': 4, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 16, 'linQ_dim': 6, 'linQ_dim_count': 16, 'linR_count': 16, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3*D4:D4', 'ngens': 7, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [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': 22, 'number_conjugacy_classes': 48, 'number_divisions': 32, 'number_normal_subgroups': 54, 'number_subgroup_autclasses': 96, 'number_subgroup_classes': 168, 'number_subgroups': 370, 'old_label': None, 'order': 192, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 27], [3, 2], [4, 20], [6, 54], [8, 16], [12, 40], [24, 32]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 36, 0, 0], 'outer_gens': [[1, 3, 132, 48], [13, 2, 12, 48], [1, 2, 12, 72], [1, 10, 12, 48]], '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': 4, '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, 14], [4, 8], [8, 2]], 'representations': {'PC': {'code': 3482972028997189922624784, 'gens': [1, 2, 4, 6], 'pres': [7, 2, 2, 3, 2, 2, 2, 2, 36, 1011, 80, 6053, 3288, 124, 2953]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101729095570500, 125101739157295540, 108470323050979754]}, 'Perm': {'d': 11, 'gens': [42000, 16148160, 7985040, 4, 11613864, 3669120, 3669864]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times D_4:D_4', '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': '24.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 24, 'aut_gen_orders': [2, 8, 4, 24, 4, 12, 2], 'aut_gens': [[1, 2, 24, 96], [265, 842, 72, 1056], [85, 902, 648, 96], [397, 442, 600, 1128], [853, 518, 648, 144], [1033, 698, 72, 1104], [265, 502, 72, 168], [37, 670, 648, 480]], 'aut_group': None, 'aut_hash': 522063148052194636, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6144, 'aut_permdeg': 42, 'aut_perms': [16686551434011235802165840157397285677897975180960, 10870456221099264701822156103799752051236804503766, 1374879517194988527765802503131420495771745437447217, 10870459268170933371163992942071795088175993420980, 31799750281927590679920732029443741390856430200942, 1381575382729025674706086148430340798262326496683595, 1387466077521722320094420240802517787848907161221779], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 2, 1], [2, 4, 4, 1], [2, 8, 1, 1], [2, 24, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 4, 1, 1], [4, 4, 2, 1], [4, 8, 2, 1], [4, 24, 1, 1], [4, 48, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 3], [6, 2, 4, 1], [6, 4, 4, 1], [6, 4, 8, 2], [6, 4, 16, 1], [6, 8, 2, 3], [6, 8, 4, 2], [6, 24, 2, 1], [8, 48, 1, 1], [12, 4, 2, 1], [12, 4, 4, 2], [12, 4, 8, 1], [12, 8, 1, 1], [12, 8, 2, 1], [12, 8, 4, 2], [12, 8, 8, 1], [12, 24, 2, 1], [12, 48, 4, 1], [24, 48, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3:(C_2^4.C_2^5.C_2^2)', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '384.12882', 'autcentquo_hash': 12882, 'autcentquo_nilpotent': False, 'autcentquo_order': 384, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\wr C_2^2\\times S_3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 4, 6], [2, 8, 1], [2, 24, 1], [3, 1, 2], [3, 2, 3], [4, 4, 3], [4, 8, 2], [4, 24, 1], [4, 48, 2], [6, 1, 2], [6, 2, 11], [6, 4, 36], [6, 8, 14], [6, 24, 2], [8, 48, 1], [12, 4, 18], [12, 8, 19], [12, 24, 2], [12, 48, 4], [24, 48, 2]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '192.803', 'commutator_count': 1, 'commutator_label': '48.45', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 32531, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['384.4485', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 6], [2, 8, 1, 1], [2, 24, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 4, 1, 3], [4, 8, 1, 2], [4, 24, 1, 1], [4, 48, 1, 2], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 5], [6, 4, 2, 18], [6, 8, 1, 2], [6, 8, 2, 6], [6, 24, 2, 1], [8, 48, 1, 1], [12, 4, 2, 9], [12, 8, 1, 1], [12, 8, 2, 9], [12, 24, 2, 1], [12, 48, 2, 2], [24, 48, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 69888, 'exponent': 24, 'exponents_of_order': [7, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 0, 16]], 'familial': False, 'frattini_label': '16.11', 'frattini_quotient': '72.48', 'hash': 32531, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 4, 2, 12], 'inner_gens': [[1, 14, 24, 1128], [13, 2, 600, 1128], [1, 578, 24, 96], [217, 218, 24, 96]], 'inner_hash': 803, 'inner_nilpotent': False, 'inner_order': 192, 'inner_split': True, 'inner_tex': '(C_2\\times C_{12}):D_4', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 24], [2, 66], [4, 42], [8, 3]], 'label': '1152.32531', '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': True, 'name': 'C12^2:D4', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 44, 'number_characteristic_subgroups': 46, 'number_conjugacy_classes': 135, 'number_divisions': 80, 'number_normal_subgroups': 90, 'number_subgroup_autclasses': 443, 'number_subgroup_classes': 922, 'number_subgroups': 3830, 'old_label': None, 'order': 1152, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 59], [3, 8], [4, 148], [6, 328], [8, 48], [12, 464], [24, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [0, 12, 300, 0], 'outer_gens': [[49, 10, 24, 96], [13, 10, 600, 744], [61, 370, 600, 744], [61, 322, 600, 696]], 'outer_group': '32.46', 'outer_hash': 46, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 8, 'outer_perms': [16, 23, 11520, 16567], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 18], [4, 34], [8, 19], [16, 1]], 'representations': {'PC': {'code': '261887069814523580338998718089704782967316850798388029140001', 'gens': [1, 2, 5, 7], 'pres': [9, 2, 2, 2, 3, 2, 2, 2, 2, 3, 253, 46, 74, 13513, 832, 130, 71070, 35547, 10608, 186, 72583, 36304, 214, 62216, 31121]}, 'Perm': {'d': 14, 'gens': [6354396856, 1480636080, 14374770480, 20277815040, 12055680, 12944332800, 14382023040, 325, 435]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 10, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}^2:D_4', 'transitive_degree': 24, 'wreath_data': ['C_3\times D_4', 'C_2', '2T1'], 'wreath_product': True}
