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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '11664.bg', 'ambient_counter': 33, 'ambient_order': 11664, 'ambient_tex': 'C_3^5:(C_2\\times S_4)', 'central': False, 'central_factor': False, 'centralizer_order': 9, 'characteristic': False, 'core_order': 27, 'counter': 407, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '11664.bg.144.v1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '144.v1', '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': 144, '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': '81.7', 'subgroup_hash': 7, 'subgroup_order': 81, 'subgroup_tex': 'C_3\\wr C_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '11664.bg', 'aut_centralizer_order': None, 'aut_label': '144.v1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1296.c1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['36.bk1', '48.c1', '72.x1'], 'contains': ['432.a1', '432.cg1', '432.ci1'], 'core': '432.a1', 'coset_action_label': None, 'count': 24, 'diagramx': [7869, -1, 1068, -1], 'generators': [5669613555880346, 43545604, 93448857748, 711374936070963], 'label': '11664.bg.144.v1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '24.e1', 'old_label': '144.v1', 'projective_image': '11664.bg', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '144.v1', 'subgroup_fusion': None, 'weyl_group': '54.5'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '9.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 6, 'aut_gen_orders': [6, 6, 3, 3], 'aut_gens': [[1, 3, 9], [55, 6, 80], [32, 33, 70], [1, 3, 42], [1, 3, 36]], 'aut_group': '324.117', 'aut_hash': 117, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 324, 'aut_permdeg': 12, 'aut_perms': [3852987, 171866280, 45600, 79914480], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 3, 2, 1], [3, 3, 6, 1], [3, 9, 2, 1], [9, 9, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3^3:D_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 3, 'autcent_group': '9.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 9, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_3^2', '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], [3, 1, 2], [3, 3, 8], [3, 9, 2], [9, 9, 4]], 'center_label': '3.1', 'center_order': 3, 'central_product': False, 'central_quotient': '27.3', 'commutator_count': 1, 'commutator_label': '9.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '3.1', '3.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 7, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [3, 3, 2, 4], [3, 9, 2, 1], [9, 9, 2, 2]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 12, 'exponent': 9, 'exponents_of_order': [4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [3], 'faithful_reps': [[3, 0, 6]], 'familial': True, 'frattini_label': '9.2', 'frattini_quotient': '9.2', 'hash': 7, 'hyperelementary': 3, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 3, 'inner_gen_orders': [3, 3, 3], 'inner_gens': [[1, 3, 66], [1, 3, 36], [61, 57, 9]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 27, 'inner_split': True, 'inner_tex': '\\He_3', 'inner_used': [1, 2, 3], 'irrC_degree': 3, 'irrQ_degree': 6, 'irrQ_dim': 6, 'irrR_degree': 6, 'irrep_stats': [[1, 9], [3, 8]], 'label': '81.7', 'linC_count': 6, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 3, 'linQ_dim': 6, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C3wrC3', 'ngens': 2, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 17, 'number_divisions': 9, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 14, 'number_subgroup_classes': 20, 'number_subgroups': 50, 'old_label': None, 'order': 81, 'order_factorization_type': 3, 'order_stats': [[1, 1], [3, 44], [9, 36]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [2, 0], 'outer_gens': [[1, 6, 77], [56, 6, 37]], 'outer_group': '12.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [720, 28], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 3, 'primary_abelian_invariants': [3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 4], [6, 4]], 'representations': {'PC': {'code': 3774548, 'gens': [1, 2, 3], 'pres': [4, 3, 3, 3, 3, 794, 150, 46]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [41624334336939899, 69483172573970095]}, 'Lie': [{'d': 1, 'q': 27, 'gens': [954503806467903513674092321, 1542188426621362927188600585, 611014341166651000646291555, 1351483486413078361230720], 'family': 'ASigmaL'}], 'GLFp': {'d': 3, 'p': 7, 'gens': [36712269, 9534373, 17540971, 23068812]}, 'Perm': {'d': 9, 'gens': [276480, 80883, 243, 80884]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 3], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\wr C_3', 'transitive_degree': 9, 'wreath_data': ['C_3', 'C_3', '3T1'], 'wreath_product': True}
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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': 36, 'aut_gen_orders': [6, 12, 2], 'aut_gens': [[2623281964445044, 3778528263794394], [2623845573658805, 6381450516891144], [4912757848540229, 2268358803763677], [6401072733615092, 4113856079751721]], 'aut_group': '23328.p', 'aut_hash': 1081498020297448794, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 23328, 'aut_permdeg': 135, 'aut_perms': [120875311180325239071252980060678499368587111686751580244209555391872282096796785035879044143360816955504428068179629560633322721726902431050764411845375801852094048364848009519742479516421072313565365916200521184551041504393296853, 237919874876919006620254095907794278700964293934836889684293267331717873388474216663632556471832564033692124193382632045267347181533926704634257785812418225042940382436339088413354915583366187055444591005547038994895767649917760510, 1486542415749864104574041965193144972603460653693283077549965857567557222728528366973223147271137180516184757951936570864431008766549270230605353164461405970272532595882644899558705829673322167885708817674180323963315761164699885], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 27, 1, 1], [2, 54, 1, 1], [2, 81, 1, 1], [2, 162, 1, 1], [2, 243, 1, 1], [3, 2, 1, 1], [3, 6, 1, 2], [3, 8, 1, 1], [3, 8, 2, 1], [3, 12, 1, 3], [3, 24, 1, 3], [3, 48, 1, 2], [3, 216, 1, 1], [3, 432, 1, 1], [4, 162, 1, 1], [4, 486, 1, 1], [6, 54, 1, 3], [6, 108, 1, 6], [6, 108, 2, 1], [6, 216, 1, 3], [6, 216, 2, 1], [6, 324, 1, 4], [6, 648, 1, 1], [6, 1944, 1, 1], [9, 432, 1, 1], [9, 432, 2, 1], [12, 324, 1, 2], [12, 324, 2, 1], [12, 972, 1, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_3^2\\times S_3^3):D_6', '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': 36, 'autcentquo_group': '23328.p', 'autcentquo_hash': 1081498020297448794, 'autcentquo_nilpotent': False, 'autcentquo_order': 23328, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '(C_3^2\\times S_3^3):D_6', 'cc_stats': [[1, 1, 1], [2, 27, 1], [2, 54, 1], [2, 81, 1], [2, 162, 1], [2, 243, 1], [3, 2, 1], [3, 6, 2], [3, 8, 3], [3, 12, 3], [3, 24, 3], [3, 48, 2], [3, 216, 1], [3, 432, 1], [4, 162, 1], [4, 486, 1], [6, 54, 3], [6, 108, 8], [6, 216, 5], [6, 324, 4], [6, 648, 1], [6, 1944, 1], [9, 432, 3], [12, 324, 4], [12, 972, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '11664.bg', 'commutator_count': 1, 'commutator_label': '2916.ei', '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', '3.1', '3.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 33, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 27, 1, 1], [2, 54, 1, 1], [2, 81, 1, 1], [2, 162, 1, 1], [2, 243, 1, 1], [3, 2, 1, 1], [3, 6, 1, 2], [3, 8, 1, 3], [3, 12, 1, 3], [3, 24, 1, 3], [3, 48, 1, 2], [3, 216, 1, 1], [3, 432, 1, 1], [4, 162, 1, 1], [4, 486, 1, 1], [6, 54, 1, 3], [6, 108, 1, 8], [6, 216, 1, 5], [6, 324, 1, 4], [6, 648, 1, 1], [6, 1944, 1, 1], [9, 432, 1, 1], [9, 432, 2, 1], [12, 324, 1, 2], [12, 324, 2, 1], [12, 972, 1, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 702, 'exponent': 36, 'exponents_of_order': [6, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 1, 4], [24, 1, 8], [48, 1, 1]], 'familial': False, 'frattini_label': '3.1', 'frattini_quotient': '3888.by', 'hash': 338129259051800079, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 36, 'inner_gen_orders': [6, 4], 'inner_gens': [[2623281964445044, 1510927676248347], [1155890524435652, 3778528263794394]], 'inner_hash': 338129259051800079, 'inner_nilpotent': False, 'inner_order': 11664, 'inner_split': True, 'inner_tex': 'C_3^5:(C_2\\times S_4)', 'inner_used': [1, 2], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [2, 4], [3, 4], [4, 1], [6, 10], [8, 2], [12, 14], [16, 4], [24, 10], [48, 1]], 'label': '11664.bg', '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': 'C3^5:(C2*S4)', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 49, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 54, 'number_divisions': 52, 'number_normal_subgroups': 18, 'number_subgroup_autclasses': 1273, 'number_subgroup_classes': 1482, 'number_subgroups': 152004, 'old_label': None, 'order': 11664, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 567], [3, 890], [4, 648], [6, 5994], [9, 1296], [12, 2268]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[2288517326282404, 3401824683963453]], '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': 7, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [3, 4], [4, 1], [6, 10], [8, 2], [12, 12], [16, 2], [24, 11], [32, 1], [48, 1]], 'representations': {'PC': {'code': '480541041753821878247196777555564857057596248298414470622764259752942445743087438527506893399317345415582644858676751490802676017137233071', 'gens': [1, 2, 4, 6, 8, 9, 10], 'pres': [10, 2, 2, 3, 2, 3, 2, 3, 3, 3, 3, 23760, 126441, 51, 299402, 322083, 181213, 71783, 113, 196804, 195614, 97524, 643685, 57255, 107485, 40715, 175, 292326, 166336, 158786, 24396, 103687, 103697, 80667, 1017, 77768, 362898, 9748, 19478, 3298, 388819]}, 'Perm': {'d': 18, 'gens': [2623281964445044, 3778528263794394]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^5:(C_2\\times S_4)', 'transitive_degree': 18, 'wreath_data': None, 'wreath_product': False}
