-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '608.44', 'ambient_counter': 44, 'ambient_order': 608, 'ambient_tex': 'C_{38}.C_4^2', 'central': True, 'central_factor': False, 'centralizer_order': 608, 'characteristic': True, 'core_order': 1, 'counter': 76, 'cyclic': True, 'direct': True, 'hall': 1, 'label': '608.44.608.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '608.a1.a1', 'outer_equivalence': False, 'perfect': True, 'proper': False, 'quotient': '608.44', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 44, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 608, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{38}.C_4^2', 'simple': False, 'solvable': True, 'special_labels': ['PC', 'U2', 'L2', 'D2', 'C6'], 'split': True, 'standard_generators': False, 'stem': True, 'subgroup': '1.1', 'subgroup_hash': 1, 'subgroup_order': 1, 'subgroup_tex': 'C_1', 'supersolvable': True, 'sylow': 1}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '608.44', 'aut_centralizer_order': None, 'aut_label': '608.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1.a1.a1', 'complements': ['1.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['32.a1.a1', '304.a1.a1', '304.b1.a1', '304.b1.b1', '304.b1.c1', '304.c1.a1', '304.c1.b1', '304.c1.c1'], 'contains': [], 'core': '608.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4147, 2259, 3933, 4297, 4264, 5655, 4151, 5632], 'generators': [], 'label': '608.44.608.a1.a1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '608.a1.a1', 'normal_contained_in': ['32.a1.a1', '304.c1.c1', '304.c1.a1', '304.c1.b1', '304.b1.c1', '304.b1.a1', '304.b1.b1', '304.a1.a1'], 'normal_contains': [], 'normalizer': '1.a1.a1', 'old_label': '608.a1.a1', 'projective_image': '608.44', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '608.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '1.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': [], 'composition_length': 0, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1, 'exponent': 1, 'exponents_of_order': [], 'factors_of_aut_order': [], 'factors_of_order': [], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '1.1', 'hash': 1, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [], 'inner_gens': [], '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, 1]], 'label': '1.1', 'linC_count': 1, 'linC_degree': 0, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 0, 'linQ_degree_count': 1, 'linQ_dim': 0, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 0, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C1', 'ngens': 0, 'nilpotency_class': 0, 'nilpotent': True, 'normal_counts': [1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 1, 'number_characteristic_subgroups': 1, 'number_conjugacy_classes': 1, 'number_divisions': 1, 'number_normal_subgroups': 1, 'number_subgroup_autclasses': 1, 'number_subgroup_classes': 1, 'number_subgroups': 1, 'old_label': None, 'order': 1, 'order_factorization_type': 0, 'order_stats': [[1, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 0, 'perfect': True, 'permutation_degree': 1, 'pgroup': 1, 'primary_abelian_invariants': [], 'quasisimple': False, 'rank': 0, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 1]], 'representations': {'PC': {'code': 0, 'gens': [], 'pres': []}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_1', 'transitive_degree': 1, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '304.19', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 36, 'aut_gen_orders': [36, 9, 36, 36], 'aut_gens': [[1, 4, 16], [309, 308, 400], [319, 305, 560], [305, 5, 464], [6, 1, 368]], 'aut_group': None, 'aut_hash': 1181304392295744613, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6912, 'aut_permdeg': 42, 'aut_perms': [1058519276773108879954847482789345302467466962856810, 882949774613999092463319079033358646877389892095763, 785125181233714591185391732125370250822079121158435, 1400027960175805543202270834854254180620702969186296], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 3, 2], [4, 2, 12, 1], [19, 1, 18, 1], [38, 1, 18, 1], [38, 1, 54, 2], [76, 2, 216, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{18}\\times C_2^4:C_3.D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 18, 'autcent_group': '1152.2328', 'autcent_hash': 6921629745565123870, 'autcent_nilpotent': True, 'autcent_order': 1152, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6\times C_{18}', '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], [4, 2, 12], [19, 1, 18], [38, 1, 126], [76, 2, 216]], 'center_label': '152.12', 'center_order': 152, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '19.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 44, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['19.1', 1], ['32.2', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [4, 2, 2, 6], [19, 1, 18, 1], [38, 1, 18, 7], [76, 2, 36, 6]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 20, 'exponent': 76, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 19], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '76.4', 'hash': 44, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 1], 'inner_gens': [[1, 308, 16], [305, 4, 16], [1, 4, 16]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 304], [2, 76]], 'label': '608.44', 'linC_count': 1316736, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 24, 'linQ_degree_count': 304, 'linQ_dim': 24, 'linQ_dim_count': 196, 'linR_count': 13176, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C38.C4^2', 'ngens': 6, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 380, 'number_divisions': 28, 'number_normal_subgroups': 52, 'number_subgroup_autclasses': 26, 'number_subgroup_classes': 76, 'number_subgroups': 100, 'old_label': None, 'order': 608, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 7], [4, 24], [19, 18], [38, 126], [76, 432]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 36, 'outer_gen_orders': [2, 4, 6, 2, 18], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[9, 14, 592], [15, 6, 16], [310, 309, 592], [307, 318, 16], [11, 12, 272]], 'outer_group': '1728.30448', 'outer_hash': 30448, 'outer_nilpotent': False, 'outer_order': 1728, 'outer_permdeg': 19, 'outer_perms': [367920, 408383, 806657, 94, 6780385522310844], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_{18}\\times \\GL(2,\\mathbb{Z}/4)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 31, 'pgroup': 0, 'primary_abelian_invariants': [4, 4, 19], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [18, 4], [36, 10]], 'representations': {'PC': {'code': 224455797762020607115601, 'gens': [1, 3, 5], 'pres': [6, -2, -2, -2, -2, -2, -19, 12, 5546, 50, 88]}, 'Perm': {'d': 31, 'gens': [274399510151719999023311093760000, 419424246005299646791680000, 115242726703104000, 2350183339898634240000, 548189243614211704521602826240000, 837605342339873243136000000]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 76], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{38}.C_4^2', 'transitive_degree': 608, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '304.19', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 36, 'aut_gen_orders': [36, 9, 36, 36], 'aut_gens': [[1, 4, 16], [309, 308, 400], [319, 305, 560], [305, 5, 464], [6, 1, 368]], 'aut_group': None, 'aut_hash': 1181304392295744613, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6912, 'aut_permdeg': 42, 'aut_perms': [1058519276773108879954847482789345302467466962856810, 882949774613999092463319079033358646877389892095763, 785125181233714591185391732125370250822079121158435, 1400027960175805543202270834854254180620702969186296], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 3, 2], [4, 2, 12, 1], [19, 1, 18, 1], [38, 1, 18, 1], [38, 1, 54, 2], [76, 2, 216, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{18}\\times C_2^4:C_3.D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 18, 'autcent_group': '1152.2328', 'autcent_hash': 6921629745565123870, 'autcent_nilpotent': True, 'autcent_order': 1152, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6\times C_{18}', '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], [4, 2, 12], [19, 1, 18], [38, 1, 126], [76, 2, 216]], 'center_label': '152.12', 'center_order': 152, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '19.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 44, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['19.1', 1], ['32.2', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [4, 2, 2, 6], [19, 1, 18, 1], [38, 1, 18, 7], [76, 2, 36, 6]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 20, 'exponent': 76, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 19], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '76.4', 'hash': 44, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2, 1], 'inner_gens': [[1, 308, 16], [305, 4, 16], [1, 4, 16]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 304], [2, 76]], 'label': '608.44', 'linC_count': 1316736, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 24, 'linQ_degree_count': 304, 'linQ_dim': 24, 'linQ_dim_count': 196, 'linR_count': 13176, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C38.C4^2', 'ngens': 6, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 380, 'number_divisions': 28, 'number_normal_subgroups': 52, 'number_subgroup_autclasses': 26, 'number_subgroup_classes': 76, 'number_subgroups': 100, 'old_label': None, 'order': 608, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 7], [4, 24], [19, 18], [38, 126], [76, 432]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 36, 'outer_gen_orders': [2, 4, 6, 2, 18], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[9, 14, 592], [15, 6, 16], [310, 309, 592], [307, 318, 16], [11, 12, 272]], 'outer_group': '1728.30448', 'outer_hash': 30448, 'outer_nilpotent': False, 'outer_order': 1728, 'outer_permdeg': 19, 'outer_perms': [367920, 408383, 806657, 94, 6780385522310844], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_{18}\\times \\GL(2,\\mathbb{Z}/4)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 31, 'pgroup': 0, 'primary_abelian_invariants': [4, 4, 19], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [18, 4], [36, 10]], 'representations': {'PC': {'code': 224455797762020607115601, 'gens': [1, 3, 5], 'pres': [6, -2, -2, -2, -2, -2, -19, 12, 5546, 50, 88]}, 'Perm': {'d': 31, 'gens': [274399510151719999023311093760000, 419424246005299646791680000, 115242726703104000, 2350183339898634240000, 548189243614211704521602826240000, 837605342339873243136000000]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 76], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{38}.C_4^2', 'transitive_degree': 608, 'wreath_data': None, 'wreath_product': False}
