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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '400.219', 'ambient_counter': 219, 'ambient_order': 400, 'ambient_tex': 'C_{10}^2:C_2^2', 'central': False, 'central_factor': False, 'centralizer_order': 200, 'characteristic': True, 'core_order': 25, 'counter': 15, 'cyclic': False, 'direct': False, 'hall': 5, 'label': '400.219.16.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '16.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '16.14', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 14, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 16, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '25.2', 'subgroup_hash': 2, 'subgroup_order': 25, 'subgroup_tex': 'C_5^2', 'supersolvable': True, 'sylow': 5}
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gps_subgroup_data • Show schema
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{'ambient': '400.219', 'aut_centralizer_order': 6720, 'aut_label': '16.a1', 'aut_quo_index': 15, 'aut_stab_index': 1, 'aut_weyl_group': '16.2', 'aut_weyl_index': 6720, 'centralizer': '2.b1', 'complements': ['25.a1'], 'conjugacy_class_count': 1, 'contained_in': ['8.a1', '8.b1'], 'contains': ['80.a1', '80.b1', '80.c1'], 'core': '16.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [827, 1019, 1954, 2716], 'generators': [8, 80], 'label': '400.219.16.a1', 'mobius_quo': 1, 'mobius_sub': 64, 'normal_closure': '16.a1', 'normal_contained_in': ['8.a1', '8.b1'], 'normal_contains': ['80.a1', '80.b1'], 'normalizer': '1.a1', 'old_label': '16.a1', 'projective_image': '80.51', 'quotient_action_image': '2.1', 'quotient_action_kernel': '8.5', 'quotient_action_kernel_order': 8, 'quotient_fusion': None, 'short_label': '16.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '25.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 120, 'aut_gen_orders': [4, 3], 'aut_gens': [[1, 5], [4, 16], [17, 11]], 'aut_group': '480.218', 'aut_hash': 218, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 480, 'aut_permdeg': 24, 'aut_perms': [429137129243734377940845, 535103329646962732126054], 'aut_phi_ratio': 24.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [5, 1, 24, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(2,5)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 120, 'autcent_group': '480.218', 'autcent_hash': 218, 'autcent_nilpotent': False, 'autcent_order': 480, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(2,5)', '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], [5, 1, 24]], 'center_label': '25.2', 'center_order': 25, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['5.1', '5.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['5.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [5, 1, 4, 6]], 'element_repr_type': 'PC', 'elementary': 5, 'eulerian_function': 1, 'exponent': 5, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '25.2', 'hash': 2, 'hyperelementary': 5, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 5], [1, 5]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 25]], 'label': '25.2', 'linC_count': 240, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 15, 'linQ_dim': 8, 'linQ_dim_count': 15, 'linR_count': 60, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 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': 25, 'number_divisions': 7, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 25, 'order_factorization_type': 2, 'order_stats': [[1, 1], [5, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 120, 'outer_gen_orders': [4, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[4, 16], [17, 11]], 'outer_group': '480.218', 'outer_hash': 218, 'outer_nilpotent': False, 'outer_order': 480, 'outer_permdeg': 24, 'outer_perms': [429137129243734377940845, 535103329646962732126054], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,5)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 10, 'pgroup': 5, 'primary_abelian_invariants': [5, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 6]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -5, 5]}, 'GLFp': {'d': 2, 'p': 11, 'gens': [3997, 5325]}, 'Perm': {'d': 10, 'gens': [1451520, 96]}}, 'schur_multiplier': [5], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [5, 5], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5^2', 'transitive_degree': 25, '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': '80.52', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 420, 'aut_gen_orders': [60, 28, 4], 'aut_gens': [[1, 2, 4, 40], [20, 243, 13, 60], [220, 3, 9, 340], [201, 202, 208, 180]], 'aut_group': None, 'aut_hash': 1303858183256349142, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 107520, 'aut_permdeg': 44, 'aut_perms': [242417467116122700867930635786060378581340506959206431, 66237635112364397354472819189833754577811776344043529, 67378285218142365942882802097063148785424207132086366], 'aut_phi_ratio': 672.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1], [2, 5, 8, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 8, 1], [10, 1, 28, 1], [10, 2, 14, 1], [10, 2, 56, 1], [10, 5, 32, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times C_2^3.\\PSL(2,7)\\times F_5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 84, 'autcent_group': '5376.bu', 'autcent_hash': 5218989228857211849, 'autcent_nilpotent': False, 'autcent_order': 5376, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_4\\times C_2^3:\\GL(3,2)', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 5, 8], [5, 1, 4], [5, 2, 10], [10, 1, 28], [10, 2, 70], [10, 5, 32]], 'center_label': '40.14', 'center_order': 40, 'central_product': True, 'central_quotient': '10.1', 'commutator_count': 1, 'commutator_label': '5.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '5.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 219, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['10.1', 1], ['2.1', 3], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 5, 1, 8], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 4, 2], [10, 1, 4, 7], [10, 2, 2, 7], [10, 2, 4, 14], [10, 5, 4, 8]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 72540, 'exponent': 10, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '400.219', 'hash': 219, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [1, 2, 1, 5], 'inner_gens': [[1, 2, 4, 40], [1, 2, 4, 360], [1, 2, 4, 40], [1, 82, 4, 40]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 10, 'inner_split': False, 'inner_tex': 'D_5', 'inner_used': [2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 80], [2, 80]], 'label': '400.219', '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': 'C10^2:C2^2', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 160, 'number_divisions': 56, 'number_normal_subgroups': 166, 'number_subgroup_autclasses': 36, 'number_subgroup_classes': 300, 'number_subgroups': 740, 'old_label': None, 'order': 400, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 47], [5, 24], [10, 328]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 84, 'outer_gen_orders': [4, 4, 4, 4, 4, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 2, 0], 'outer_gens': [[1, 2, 28, 280], [220, 23, 25, 40], [1, 223, 12, 40], [1, 3, 12, 280], [200, 3, 213, 81], [1, 2, 36, 40], [1, 2, 4, 280], [1, 22, 28, 280]], 'outer_group': '10752.bw', 'outer_hash': 5516294691169544409, 'outer_nilpotent': False, 'outer_order': 10752, 'outer_permdeg': 14, 'outer_perms': [150, 19813253040, 45190580761, 39885270120, 43709990040, 288, 1, 6321455430], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_4\\times C_2^4:\\GL(3,2)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 5], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [4, 24], [8, 16]], 'representations': {'PC': {'code': 827384260888444979, 'gens': [1, 2, 3, 5], 'pres': [6, -2, -2, -2, -5, -2, -5, 50, 5410, 88, 5771]}, 'GLZN': {'d': 2, 'p': 33, 'gens': [898450, 35962, 826574, 359380, 794398, 35960]}, 'Perm': {'d': 16, 'gens': [1313901388807, 1313941305600, 2789705318400, 39916800, 408960, 37]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 10], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{10}^2:C_2^2', 'transitive_degree': 80, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 420, 'aut_gen_orders': [6, 3], 'aut_gens': [[1, 2, 4, 8], [12, 4, 5, 3], [10, 14, 7, 5]], 'aut_group': '20160.a', 'aut_hash': 3764836782182912467, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20160, 'aut_permdeg': 8, 'aut_perms': [5193, 5760], 'aut_phi_ratio': 2520.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 15, 1]], 'aut_supersolvable': False, 'aut_tex': 'A_8', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 420, 'autcent_group': '20160.a', 'autcent_hash': 3764836782182912467, 'autcent_nilpotent': False, 'autcent_order': 20160, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'A_8', '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], [2, 1, 15]], 'center_label': '16.14', 'center_order': 16, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 14, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 4]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [4], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '16.14', 'hash': 14, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1, 1], 'inner_gens': [[1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 8]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16]], 'label': '16.14', 'linC_count': 840, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 840, 'linQ_dim': 4, 'linQ_dim_count': 840, 'linR_count': 840, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^4', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 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': 16, 'number_divisions': 16, 'number_normal_subgroups': 67, 'number_subgroup_autclasses': 5, 'number_subgroup_classes': 67, 'number_subgroups': 67, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 15]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 420, 'outer_gen_orders': [6, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[12, 4, 5, 3], [10, 14, 7, 5]], 'outer_group': '20160.a', 'outer_hash': 3764836782182912467, 'outer_nilpotent': False, 'outer_order': 20160, 'outer_permdeg': 8, 'outer_perms': [5193, 5760], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'A_8', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16]], 'representations': {'PC': {'code': 0, 'gens': [1, 2, 3, 4], 'pres': [4, -2, 2, 2, 2]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [7233746, 7115648, 35812976, 7115160]}, 'GLFp': {'d': 4, 'p': 2, 'gens': [33837, 18465, 27183, 18467]}, 'Perm': {'d': 8, 'gens': [5040, 120, 6, 1]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^4', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
