-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '368.42', 'ambient_counter': 42, 'ambient_order': 368, 'ambient_tex': 'C_2^3\\times C_{46}', 'central': True, 'central_factor': False, 'centralizer_order': 368, 'characteristic': True, 'core_order': 23, 'counter': 5, 'cyclic': True, 'direct': True, 'hall': 23, 'label': '368.42.16.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, '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': True, 'solvable': True, 'special_labels': ['C4'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '23.1', 'subgroup_hash': 1, 'subgroup_order': 23, 'subgroup_tex': 'C_{23}', 'supersolvable': True, 'sylow': 23}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '368.42', 'aut_centralizer_order': 20160, 'aut_label': '16.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '22.2', 'aut_weyl_index': 20160, 'centralizer': '1.a1', 'complements': ['23.a1'], 'conjugacy_class_count': 1, 'contained_in': ['8.a1'], 'contains': ['368.a1'], 'core': '16.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [1203, 1203, 6653, 6653], 'generators': [16], 'label': '368.42.16.a1', 'mobius_quo': -1, 'mobius_sub': 64, 'normal_closure': '16.a1', 'normal_contained_in': ['8.a1'], 'normal_contains': ['368.a1'], 'normalizer': '1.a1', 'old_label': '16.a1', 'projective_image': '16.14', 'quotient_action_image': '1.1', 'quotient_action_kernel': '16.14', 'quotient_action_kernel_order': 16, 'quotient_fusion': None, 'short_label': '16.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '23.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 22, 'aut_gen_orders': [22], 'aut_gens': [[1], [14]], 'aut_group': '22.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 22, 'aut_permdeg': 13, 'aut_perms': [515289600], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [23, 1, 22, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{22}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 22, 'autcent_group': '22.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 22, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_{22}', '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], [23, 1, 22]], 'center_label': '23.1', 'center_order': 23, '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': ['23.1'], '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], [23, 1, 22, 1]], 'element_repr_type': 'PC', 'elementary': 23, 'eulerian_function': 1, 'exponent': 23, 'exponents_of_order': [1], 'factors_of_aut_order': [2, 11], 'factors_of_order': [23], 'faithful_reps': [[1, 0, 22]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '23.1', 'hash': 1, 'hyperelementary': 23, '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': 22, 'irrQ_dim': 22, 'irrR_degree': 2, 'irrep_stats': [[1, 23]], 'label': '23.1', 'linC_count': 22, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 22, 'linQ_degree_count': 1, 'linQ_dim': 22, 'linQ_dim_count': 1, 'linR_count': 11, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C23', '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': 23, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 23, 'order_factorization_type': 1, 'order_stats': [[1, 1], [23, 22]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 22, 'outer_gen_orders': [22], 'outer_gen_pows': [0], 'outer_gens': [[14]], 'outer_group': '22.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 22, 'outer_permdeg': 13, 'outer_perms': [515289600], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{22}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 23, 'pgroup': 23, 'primary_abelian_invariants': [23], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [22, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -23]}, 'Lie': [{'d': 1, 'q': 23, 'gens': [1177652997443428940313], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 23, 'gens': [12191]}, 'Perm': {'d': 23, 'gens': [24728016011107368960000]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [23], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{23}', 'transitive_degree': 23, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '368.42', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 4620, 'aut_gen_orders': [154, 77], 'aut_gens': [[1, 2, 4, 8], [4, 5, 7, 156], [189, 184, 4, 295]], 'aut_group': None, 'aut_hash': 3432502961775588985, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 443520, 'aut_permdeg': 37, 'aut_perms': [1489165060196231468227160638954428729600000, 3103288569335423574421934378391327861964800], 'aut_phi_ratio': 2520.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 15, 1], [23, 1, 22, 1], [46, 1, 330, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{22}\\times A_8', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4620, 'autcent_group': None, 'autcent_hash': 3432502961775588985, 'autcent_nilpotent': False, 'autcent_order': 443520, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_{22}\\times 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], [23, 1, 22], [46, 1, 330]], 'center_label': '368.42', 'center_order': 368, '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', '23.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 42, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 4], ['23.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15], [23, 1, 22, 1], [46, 1, 22, 15]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 12720, 'exponent': 46, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3, 5, 7, 11], 'factors_of_order': [2, 23], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '368.42', 'hash': 42, '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, 368]], 'label': '368.42', 'linC_count': None, 'linC_degree': 4, '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': 'C2^3*C46', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, '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': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 368, 'number_divisions': 32, 'number_normal_subgroups': 134, 'number_subgroup_autclasses': 10, 'number_subgroup_classes': 134, 'number_subgroups': 134, 'old_label': None, 'order': 368, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 15], [23, 22], [46, 330]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4620, 'outer_gen_orders': [44, 330], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 7, 4, 106], [2, 7, 187, 63]], 'outer_group': None, 'outer_hash': 3432502961775588985, 'outer_nilpotent': False, 'outer_order': 443520, 'outer_permdeg': 37, 'outer_perms': [86243526297158592873091942985970150604800, 2730670332324834709647619819013731442688000], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_{22}\\times A_8', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 31, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 23], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [22, 16]], 'representations': {'PC': {'code': 84478335765, 'gens': [1, 2, 3, 4], 'pres': [5, -2, -2, -2, -2, -23, 58]}, 'GLZN': {'d': 2, 'p': 92, 'gens': [35430349, 782921, 36598383, 36991913, 779057]}, 'Perm': {'d': 31, 'gens': [265252859812191058636308480000000, 304888344611713860501504000000, 403291461126605635584000000, 620448401733239439360000, 24728016011107368960000]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 46], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3\\times C_{46}', 'transitive_degree': 368, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'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}
