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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '336.227', 'ambient_counter': 227, 'ambient_order': 336, 'ambient_tex': 'C_2^2\\times D_{42}', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 24, 'counter': 9, 'cyclic': False, 'direct': None, 'hall': 6, 'label': '336.227.7.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '7.a1', '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': 7, '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': '48.51', 'subgroup_hash': 51, 'subgroup_order': 48, 'subgroup_tex': 'C_2^2\\times D_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '336.227', 'aut_centralizer_order': 6, 'aut_label': '7.a1', 'aut_quo_index': None, 'aut_stab_index': 7, 'aut_weyl_group': '8064.bp', 'aut_weyl_index': 42, 'centralizer': '42.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1'], 'contains': ['14.a1', '14.b1', '21.a1'], 'core': '14.a1', 'coset_action_label': None, 'count': 7, 'diagramx': [945, -1, 3631, -1], 'generators': [1, 112, 2, 4, 168], 'label': '336.227.7.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '7.a1', 'old_label': '7.a1', 'projective_image': '42.5', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '7.a1', 'subgroup_fusion': None, 'weyl_group': '6.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [14, 4], 'aut_gens': [[41041, 90720, 24, 131784, 3], [2, 131760, 90744, 41040, 4], [29, 41040, 131784, 90720, 4]], 'aut_group': '8064.bp', 'aut_hash': 8982001159746026648, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 8064, 'aut_permdeg': 11, 'aut_perms': [26573066, 11017685], 'aut_phi_ratio': 504.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1], [2, 3, 8, 1], [3, 2, 1, 1], [6, 2, 7, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times C_2^3:\\GL(3,2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '1344.11686', 'autcent_hash': 11686, 'autcent_nilpotent': False, 'autcent_order': 1344, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^3:\\GL(3,2)', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 3, 8], [3, 2, 1], [6, 2, 7]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 51, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 3, 1, 8], [3, 2, 1, 1], [6, 2, 1, 7]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 195, 'exponent': 6, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '48.51', 'hash': 51, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 1, 1, 1, 3], 'inner_gens': [[41041, 90720, 24, 131784, 4], [41041, 90720, 24, 131784, 3], [41041, 90720, 24, 131784, 3], [41041, 90720, 24, 131784, 3], [41045, 90720, 24, 131784, 3]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 8]], 'label': '48.51', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, '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': 'C2^2*D6', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 24, 'number_divisions': 24, 'number_normal_subgroups': 83, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 134, 'number_subgroups': 236, 'old_label': None, 'order': 48, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 31], [3, 2], [6, 14]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 84, 'outer_gen_orders': [4, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[41065, 90720, 41040, 41064, 3], [41041, 90744, 24, 41040, 3]], 'outer_group': '1344.11686', 'outer_hash': 11686, 'outer_nilpotent': False, 'outer_order': 1344, 'outer_permdeg': 8, 'outer_perms': [4498, 25112], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^3:\\GL(3,2)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 8]], 'representations': {'PC': {'code': 1259012353, 'gens': [1, 2, 3, 4], 'pres': [5, -2, -2, -2, -2, -3, 213, 58, 214]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [7115648, 35813302, 7233337, 35931072]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [6594100, 8188666, 16002800, 28816400, 17616532]}, 'Perm': {'d': 9, 'gens': [41041, 90720, 24, 131784, 3]}}, 'schur_multiplier': [2, 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': 'C_2^2\\times D_6', 'transitive_degree': 24, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [84, 42, 12], 'aut_gens': [[1, 2, 4, 8], [169, 35, 1, 181], [172, 23, 1, 153], [1, 46, 172, 137]], 'aut_group': None, 'aut_hash': 3603963603123995132, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 338688, 'aut_permdeg': 168, 'aut_perms': [111450503871314329293998008906305058451765332696049578669878153875861484601995213406943502959855998124941566624426626452846656744851083544148634442944122053795860586751018463208924284037801961137231946799849188310815142972676141114759658965463964571332718477077960948914366856697486461691043505530245143, 209115257440989171934221292086022786540753800468785571329826353713151231623908039051806127914181160965573417418428809966961639228768386926304003230015499344799281290217465583246506252166791866820705716747142203554553962887626458599808420660249367686666557712731279796655429616798370310781930056536296099, 49599384598611236803814004286691154763762974624515621510876212353747726209667939490202714150812786239521294966972962378180716056939064473392207241073648036248577267133450590621210281633813489789072037781988983322468287997020343457615663236119203458464176603026622922839710160751403349862016169577284605], 'aut_phi_ratio': 3528.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1], [2, 21, 8, 1], [3, 2, 1, 1], [6, 2, 7, 1], [7, 2, 3, 1], [14, 2, 21, 1], [21, 2, 6, 1], [42, 2, 42, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times C_2^3.\\PSL(2,7)\\times F_7', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '1344.11686', 'autcent_hash': 11686, 'autcent_nilpotent': False, 'autcent_order': 1344, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^3:\\GL(3,2)', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '252.26', 'autcentquo_hash': 26, 'autcentquo_nilpotent': False, 'autcentquo_order': 252, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 21, 8], [3, 2, 1], [6, 2, 7], [7, 2, 3], [14, 2, 21], [21, 2, 6], [42, 2, 42]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '42.5', 'commutator_count': 1, 'commutator_label': '21.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 227, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['42.5', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 21, 1, 8], [3, 2, 1, 1], [6, 2, 1, 7], [7, 2, 3, 1], [14, 2, 3, 7], [21, 2, 6, 1], [42, 2, 6, 7]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 11115, 'exponent': 42, 'exponents_of_order': [4, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '336.227', 'hash': 227, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [1, 2, 1, 21], 'inner_gens': [[1, 2, 4, 8], [1, 2, 4, 328], [1, 2, 4, 8], [1, 18, 4, 8]], 'inner_hash': 5, 'inner_nilpotent': False, 'inner_order': 42, 'inner_split': True, 'inner_tex': 'D_{21}', 'inner_used': [2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 80]], 'label': '336.227', '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': 'C2^2*D42', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 9, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 96, 'number_divisions': 40, 'number_normal_subgroups': 115, 'number_subgroup_autclasses': 32, 'number_subgroup_classes': 268, 'number_subgroups': 1696, 'old_label': None, 'order': 336, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 175], [3, 2], [6, 14], [7, 6], [14, 42], [21, 12], [42, 84]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 84, 'outer_gen_orders': [42, 42], 'outer_gen_pows': [0, 0], 'outer_gens': [[173, 6, 169, 81], [169, 2, 168, 253]], 'outer_group': '8064.by', 'outer_hash': 2432465494741877190, 'outer_nilpotent': False, 'outer_order': 8064, 'outer_permdeg': 13, 'outer_perms': [1677674911, 1445351071], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_6\\times C_2^3:\\GL(3,2)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 16, '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], [6, 8], [12, 8]], 'representations': {'PC': {'code': 3141937033007820903947, 'gens': [1, 2, 3, 4], 'pres': [6, -2, -2, -2, -2, -3, -7, 3945, 69, 4810, 118, 5195]}, 'GLZN': {'d': 2, 'p': 42, 'gens': [3037819, 963157, 74677, 74971, 2148581, 74341]}, 'Perm': {'d': 16, 'gens': [87660962047, 40327, 16, 23, 840, 1482992582400]}}, 'schur_multiplier': [2, 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': 'C_2^2\\times D_{42}', 'transitive_degree': 168, 'wreath_data': None, 'wreath_product': False}
