-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1666.10', 'ambient_counter': 10, 'ambient_order': 1666, 'ambient_tex': 'C_7\\times C_{238}', 'central': True, 'central_factor': False, 'centralizer_order': 1666, 'characteristic': False, 'core_order': 238, 'counter': 10, 'cyclic': True, 'direct': True, 'hall': 0, 'label': '1666.10.7.a1.h1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '7.a1.h1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '7.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 7, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_7', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '238.4', 'subgroup_hash': 4, 'subgroup_order': 238, 'subgroup_tex': 'C_{238}', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '1666.10', 'aut_centralizer_order': 42, 'aut_label': '7.a1', 'aut_quo_index': 1, 'aut_stab_index': 8, 'aut_weyl_group': '96.59', 'aut_weyl_index': 336, 'centralizer': '1.a1.a1', 'complements': ['238.a1.a1', '238.a1.e1', '238.a1.b1', '238.a1.f1', '238.a1.c1', '238.a1.g1', '238.a1.d1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['14.a1.h1', '49.a1.a1', '119.a1.h1'], 'core': '7.a1.h1', 'coset_action_label': None, 'count': 1, 'diagramx': [6888, 6888, 1818, 1818, 9048, 9048, 6582, 6582], 'generators': [833, 1, 98], 'label': '1666.10.7.a1.h1', 'mobius_quo': -1, 'mobius_sub': -1, 'normal_closure': '7.a1.h1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['14.a1.h1', '49.a1.a1', '119.a1.h1'], 'normalizer': '1.a1.a1', 'old_label': '7.a1.h1', 'projective_image': '7.1', 'quotient_action_image': '1.1', 'quotient_action_kernel': '7.1', 'quotient_action_kernel_order': 7, 'quotient_fusion': None, 'short_label': '7.a1.h1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '238.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 48, 'aut_gen_orders': [2, 48], 'aut_gens': [[1], [69], [3]], 'aut_group': '96.59', 'aut_hash': 59, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 96, 'aut_permdeg': 21, 'aut_perms': [2439325392333218664, 19984155209275203], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [7, 1, 6, 1], [14, 1, 6, 1], [17, 1, 16, 1], [34, 1, 16, 1], [119, 1, 96, 1], [238, 1, 96, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{48}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 48, 'autcent_group': '96.59', 'autcent_hash': 59, 'autcent_nilpotent': True, 'autcent_order': 96, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{48}', '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, 1], [7, 1, 6], [14, 1, 6], [17, 1, 16], [34, 1, 16], [119, 1, 96], [238, 1, 96]], 'center_label': '238.4', 'center_order': 238, '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', '7.1', '17.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['17.1', 1], ['2.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [7, 1, 6, 1], [14, 1, 6, 1], [17, 1, 16, 1], [34, 1, 16, 1], [119, 1, 96, 1], [238, 1, 96, 1]], 'element_repr_type': 'PC', 'elementary': 238, 'eulerian_function': 1, 'exponent': 238, 'exponents_of_order': [1, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 7, 17], 'faithful_reps': [[1, 0, 96]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '238.4', 'hash': 4, 'hyperelementary': 238, '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': 96, 'irrQ_dim': 96, 'irrR_degree': 2, 'irrep_stats': [[1, 238]], 'label': '238.4', 'linC_count': 96, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 22, 'linQ_degree_count': 3, 'linQ_dim': 22, 'linQ_dim_count': 3, 'linR_count': 48, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C238', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 238, 'number_divisions': 8, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 238, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [7, 6], [14, 6], [17, 16], [34, 16], [119, 96], [238, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 48, 'outer_gen_orders': [2, 48], 'outer_gen_pows': [0, 0], 'outer_gens': [[69], [3]], 'outer_group': '96.59', 'outer_hash': 59, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 21, 'outer_perms': [2439325392333218664, 19984155209275203], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{48}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 7, 17], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [6, 2], [16, 2], [96, 2]], 'representations': {'PC': {'code': 33435743, 'gens': [1], 'pres': [3, -2, -7, -17, 6, 46]}, 'GLFp': {'d': 2, 'p': 239, 'gens': [1064849869]}, 'Perm': {'d': 26, 'gens': [15511210043330985984000000, 155112100433309859840000, 334764638208000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [238], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{238}', 'transitive_degree': 238, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '1666.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 336, 'aut_gen_orders': [6, 3, 16], 'aut_gens': [[1, 7], [3, 7], [1196, 248], [1, 497]], 'aut_group': None, 'aut_hash': 1521251412974810190, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 32256, 'aut_permdeg': 64, 'aut_perms': [7966005636825307936002447828462175774461781175925357430557817454883684103795058877922080, 119103126033049518096338041325483192948349504370776473376633332576236887415512676317960744, 1461704175441132686037446674575607440598673040761364382547249436210], 'aut_phi_ratio': 48.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [7, 1, 48, 1], [14, 1, 48, 1], [17, 1, 16, 1], [34, 1, 16, 1], [119, 1, 768, 1], [238, 1, 768, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_{16}\\times C_6.\\SO(3,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 336, 'autcent_group': None, 'autcent_hash': 1521251412974810190, 'autcent_nilpotent': False, 'autcent_order': 32256, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_{16}\\times C_6.\\SO(3,7)', '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, 1], [7, 1, 48], [14, 1, 48], [17, 1, 16], [34, 1, 16], [119, 1, 768], [238, 1, 768]], 'center_label': '1666.10', 'center_order': 1666, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '7.1', '7.1', '17.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 10, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['17.1', 1], ['2.1', 1], ['7.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [7, 1, 6, 8], [14, 1, 6, 8], [17, 1, 16, 1], [34, 1, 16, 1], [119, 1, 96, 8], [238, 1, 96, 8]], 'element_repr_type': 'PC', 'elementary': 7, 'eulerian_function': 54, 'exponent': 238, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7, 17], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1666.10', 'hash': 10, 'hyperelementary': 7, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 7], [1, 7]], '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': None, 'irrep_stats': [[1, 1666]], 'label': '1666.10', 'linC_count': None, 'linC_degree': 2, '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': True, 'monomial': True, 'name': 'C7*C238', 'ngens': 4, 'nilpotency_class': 1, '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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 1666, 'number_divisions': 36, 'number_normal_subgroups': 40, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 40, 'number_subgroups': 40, 'old_label': None, 'order': 1666, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 1], [7, 48], [14, 48], [17, 16], [34, 16], [119, 768], [238, 768]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 336, 'outer_gen_orders': [6, 3, 16], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[3, 7], [1196, 248], [1, 497]], 'outer_group': None, 'outer_hash': 1521251412974810190, 'outer_nilpotent': False, 'outer_order': 32256, 'outer_permdeg': 64, 'outer_perms': [7966005636825307936002447828462175774461781175925357430557817454883684103795058877922080, 119103126033049518096338041325483192948349504370776473376633332576236887415512676317960744, 1461704175441132686037446674575607440598673040761364382547249436210], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_{16}\\times C_6.\\SO(3,7)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 33, 'pgroup': 0, 'primary_abelian_invariants': [2, 7, 7, 17], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 2], [6, 16], [16, 2], [96, 16]], 'representations': {'PC': {'code': 56472522847, 'gens': [1, 2], 'pres': [4, -7, -2, -7, -17, 21, 94]}, 'GLFp': {'d': 2, 'p': 239, 'gens': [1187717014, 518773117]}, 'Perm': {'d': 33, 'gens': [263130836933693530167218012160000000, 1591517158873146351817850880000000, 155112100433309859840000, 334764638208000]}}, 'schur_multiplier': [7], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [7, 238], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_7\\times C_{238}', 'transitive_degree': 1666, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '7.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 6, 'aut_gen_orders': [6], 'aut_gens': [[1], [3]], 'aut_group': '6.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 6, 'aut_permdeg': 5, 'aut_perms': [27], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [7, 1, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_6', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 6, 'autcent_group': '6.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_6', '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], [7, 1, 6]], 'center_label': '7.1', 'center_order': 7, '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': ['7.1'], '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], [7, 1, 6, 1]], 'element_repr_type': 'PC', 'elementary': 7, 'eulerian_function': 1, 'exponent': 7, 'exponents_of_order': [1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [7], 'faithful_reps': [[1, 0, 6]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '7.1', 'hash': 1, 'hyperelementary': 7, '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': 6, 'irrQ_dim': 6, 'irrR_degree': 2, 'irrep_stats': [[1, 7]], 'label': '7.1', 'linC_count': 6, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C7', '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': 7, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 7, 'order_factorization_type': 1, 'order_stats': [[1, 1], [7, 6]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[3]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 7, 'pgroup': 7, 'primary_abelian_invariants': [7], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [6, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -7]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [74473229056828135]}, 'Lie': [{'d': 1, 'q': 7, 'gens': [873], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 7, 'gens': [351]}, 'Perm': {'d': 7, 'gens': [4320]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [7], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_7', 'transitive_degree': 7, 'wreath_data': None, 'wreath_product': False}
