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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '27216.a', 'ambient_counter': 1, 'ambient_order': 27216, 'ambient_tex': 'C_3\\times S_3\\times {}^2G(2,3)', 'central': False, 'central_factor': False, 'centralizer_order': 324, 'characteristic': False, 'core_order': 1, 'counter': 537, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '27216.a.9072.f1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '9072.f1.b1', 'outer_equivalence': False, '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': 9072, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': True, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '3.1', 'subgroup_hash': 1, 'subgroup_order': 3, 'subgroup_tex': 'C_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '27216.a', 'aut_centralizer_order': None, 'aut_label': '9072.f1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '84.b1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1296.d1.b1', '2268.j1.b1', '3024.c1.a1', '3024.h1.b1', '3024.k1.b1', '3024.l1.b1', '3024.t1.a1', '3024.v1.b1', '3024.w1.b1', '4536.e1.b1', '4536.i1.b1', '4536.m1.b1'], 'contains': ['27216.a1.a1'], 'core': '27216.a1.a1', 'coset_action_label': None, 'count': 84, 'diagramx': [5541, -1, 6704, -1, 4562, -1, 4284, -1], 'generators': [93405368531], 'label': '27216.a.9072.f1.b1', 'mobius_quo': None, 'mobius_sub': 18, 'normal_closure': '18.c1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '84.b1.a1', 'old_label': '9072.f1.b1', 'projective_image': '27216.a', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '9072.f1.b1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '3.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [2]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', '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], [3, 1, 2]], 'center_label': '3.1', 'center_order': 3, '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': ['3.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [3], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '3.1', 'hash': 1, 'hyperelementary': 3, '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 3]], 'label': '3.1', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3', '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': 3, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 3, 'order_factorization_type': 1, 'order_stats': [[1, 1], [3, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[2]], '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': 1, 'perfect': False, 'permutation_degree': 3, 'pgroup': 3, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -3]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [7]}, 'Lie': [{'d': 1, 'q': 3, 'gens': [3], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [7]}, 'Perm': {'d': 3, 'gens': [4]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [3], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3', 'transitive_degree': 3, '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': '18.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 126, 'aut_gen_orders': [3, 2, 3, 6, 2, 6], 'aut_gens': [[93405312000, 79833600, 39916800, 3908762, 93485288672], [93405312000, 79833600, 83462400, 40196762, 93485288672], [93405312000, 43545600, 39916800, 83742362, 93449000672], [93405312000, 79833600, 39916800, 3908762, 79976672], [93405312000, 43545600, 3628800, 40172338, 43748138], [174356582400, 79833600, 39916800, 3908762, 93485288672], [174356582400, 43545600, 39916800, 83681974, 43814421]], 'aut_group': '54432.a', 'aut_hash': 6326599111135439777, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 54432, 'aut_permdeg': 255, 'aut_perms': 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2667746072338120752823735846846209800367258591096884756287034885211365058911174577247111719014625615124305569594506009867544857260414742088421345654423315867495400231725289283140187898533036560691371193110324801047170088047036629739345113355898726018453275213569157427370136865958638371711163138960759129926362835094310474917910431665903366128370582158559977630830874365800633351685724337276471369550437302040076819915651837383906277497396050802412844197628646629489705804888285450306858969231888774088852], 'aut_phi_ratio': 7.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 63, 1, 1], [2, 189, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 56, 1, 1], [3, 56, 2, 1], [3, 84, 3, 2], [3, 112, 1, 1], [3, 112, 2, 1], [3, 168, 3, 2], [6, 3, 2, 1], [6, 63, 2, 1], [6, 126, 1, 1], [6, 126, 2, 1], [6, 168, 1, 1], [6, 168, 2, 1], [6, 189, 2, 1], [6, 252, 3, 4], [6, 504, 3, 2], [6, 756, 3, 2], [7, 216, 1, 1], [9, 168, 1, 1], [9, 168, 2, 1], [9, 168, 3, 2], [9, 336, 1, 1], [9, 336, 2, 1], [9, 336, 3, 2], [14, 648, 1, 1], [18, 504, 1, 1], [18, 504, 2, 1], [18, 504, 3, 2], [21, 216, 2, 1], [21, 432, 1, 1], [21, 432, 2, 1], [42, 648, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3^2\\times {}^2G(2,3)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 126, 'autcentquo_group': '9072.b', 'autcentquo_hash': 6831087576112572302, 'autcentquo_nilpotent': False, 'autcentquo_order': 9072, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\times {}^2G(2,3)', 'cc_stats': [[1, 1, 1], [2, 3, 1], [2, 63, 1], [2, 189, 1], [3, 1, 2], [3, 2, 3], [3, 56, 3], [3, 84, 6], [3, 112, 3], [3, 168, 6], [6, 3, 2], [6, 63, 2], [6, 126, 3], [6, 168, 3], [6, 189, 2], [6, 252, 12], [6, 504, 6], [6, 756, 6], [7, 216, 1], [9, 168, 9], [9, 336, 9], [14, 648, 1], [18, 504, 9], [21, 216, 2], [21, 432, 3], [42, 648, 2]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '9072.b', 'commutator_count': 1, 'commutator_label': '1512.780', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '3.1', '3.1', '3.1', '504.156'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['1512.779', 1], ['3.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [2, 63, 1, 1], [2, 189, 1, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [3, 56, 1, 1], [3, 56, 2, 1], [3, 84, 2, 3], [3, 112, 1, 1], [3, 112, 2, 1], [3, 168, 2, 3], [6, 3, 2, 1], [6, 63, 2, 1], [6, 126, 1, 1], [6, 126, 2, 1], [6, 168, 1, 1], [6, 168, 2, 1], [6, 189, 2, 1], [6, 252, 2, 6], [6, 504, 2, 3], [6, 756, 2, 3], [7, 216, 1, 1], [9, 168, 1, 1], [9, 168, 2, 4], [9, 336, 1, 1], [9, 336, 2, 4], [14, 648, 1, 1], [18, 504, 1, 1], [18, 504, 2, 4], [21, 216, 2, 1], [21, 432, 1, 1], [21, 432, 2, 1], [42, 648, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 3408, 'exponent': 126, 'exponents_of_order': [5, 4, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[14, 0, 6], [16, 0, 6], [42, 0, 2], [54, 0, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '27216.a', 'hash': 2848810383865508517, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 126, 'inner_gen_orders': [1, 3, 2, 6, 3], 'inner_gens': [[93405312000, 79833600, 39916800, 3908762, 93485288672], [93405312000, 79833600, 83462400, 40196762, 93485288672], [93405312000, 43545600, 39916800, 83742362, 93449000672], [93405312000, 43545600, 83462400, 3908762, 93449184065], [93405312000, 79833600, 83462400, 40011692, 93485288672]], 'inner_hash': 6831087576112572302, 'inner_nilpotent': False, 'inner_order': 9072, 'inner_split': True, 'inner_tex': 'S_3\\times {}^2G(2,3)', 'inner_used': [2, 3, 4, 5], 'irrC_degree': 14, 'irrQ_degree': 28, 'irrQ_dim': 28, 'irrR_degree': 28, 'irrep_stats': [[1, 18], [2, 9], [7, 18], [8, 18], [14, 9], [16, 9], [21, 6], [27, 6], [42, 3], [54, 3]], 'label': '27216.a', 'linC_count': 144, 'linC_degree': 9, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 11, 'linQ_degree_count': 18, 'linQ_dim': 11, 'linQ_dim_count': 18, 'linR_count': 18, 'linR_degree': 11, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'C3*S3*2G(2,3)', 'ngens': 5, '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, 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': 48, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 99, 'number_divisions': 57, 'number_normal_subgroups': 24, 'number_subgroup_autclasses': 340, 'number_subgroup_classes': 547, 'number_subgroups': 57796, 'old_label': None, 'order': 27216, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 255], [3, 2024], [6, 11976], [7, 216], [9, 4536], [14, 648], [18, 4536], [21, 1728], [42, 1296]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[174356582400, 79833600, 39916800, 3908762, 93485288672], [93405312000, 79833600, 39916800, 3908762, 174436559072]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': None, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 9], [4, 4], [7, 2], [8, 2], [14, 9], [16, 9], [21, 2], [27, 2], [28, 4], [32, 4], [42, 3], [54, 3], [84, 1], [108, 1]], 'representations': {'Perm': {'d': 15, 'gens': [93405312000, 79833600, 39916800, 3908762, 93485288672]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3, 6], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3\\times S_3\\times {}^2G(2,3)', 'transitive_degree': 54, 'wreath_data': None, 'wreath_product': False}
