-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '192.82', 'ambient_counter': 82, 'ambient_order': 192, 'ambient_tex': 'C_{12}.C_4^2', 'central': False, 'central_factor': False, 'centralizer_order': 96, 'characteristic': False, 'core_order': 2, 'counter': 75, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '192.82.24.h1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '24.h1.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': 24, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '8.2', 'subgroup_hash': 2, 'subgroup_order': 8, 'subgroup_tex': 'C_2\\times C_4', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '192.82', 'aut_centralizer_order': 48, 'aut_label': '24.h1', 'aut_quo_index': None, 'aut_stab_index': 8, 'aut_weyl_group': '4.2', 'aut_weyl_index': 384, 'centralizer': '2.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['8.h1.b1', '12.c1.a1'], 'contains': ['48.h1.a1', '48.j1.b1', '48.j1.d1'], 'core': '96.a1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [9650, -1, 2413, -1, 9296, -1, 8865, -1], 'generators': [2, 108], 'label': '192.82.24.h1.b1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '6.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1.a1', 'old_label': '24.h1.b1', 'projective_image': '96.12', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '24.h1.b1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 3], [5, 3]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [1, 22], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.3', 'autcent_hash': 3, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_4', '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, 3], [4, 1, 4]], 'center_label': '8.2', 'center_order': 8, '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'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 8]], 'label': '8.2', 'linC_count': 12, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 4, 'linQ_dim': 3, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C4', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 8, 'number_divisions': 6, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 4]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 3], [5, 3]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [1, 22], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2]], 'representations': {'PC': {'code': 34, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16426]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [251, 504]}, 'Perm': {'d': 6, 'gens': [22, 120, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_4', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [4, 2, 6, 4, 12, 2, 4, 2], 'aut_gens': [[1, 4, 48], [145, 78, 48], [131, 68, 144], [153, 124, 144], [139, 140, 144], [89, 54, 144], [129, 46, 144], [171, 116, 48], [1, 124, 144]], 'aut_group': '1536.408494914', 'aut_hash': 3343536447219507245, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1536, 'aut_permdeg': 64, 'aut_perms': [42331350666134608947599568497226559597137743211203857873808726284642700696453923445146127, 44256578590021456621278073496526094234894640031652161469249051059050043989209556134836316, 106805993121502167791253920365072474510961341813139398915167730932666842719953931102939104, 108715473674812972610723207556882432410600417820460605609723530075479411963130378710012707, 82472969862460375139707739999742880139531870316796153891758796275448099076421786297230741, 38214834657563368007631084901884320872897339883356780139095164459209016887678796926961935, 116813469414326328514621493076955690422954608567401331715273181472518601710307290757030904, 37661103799555028711682462513347231112315314532819833693517537703237800976242978], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 1, 2], [3, 2, 1, 1], [4, 1, 2, 2], [4, 2, 1, 2], [4, 2, 8, 1], [4, 12, 4, 1], [6, 2, 1, 1], [6, 2, 2, 3], [8, 12, 4, 1], [12, 2, 2, 4], [12, 2, 16, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_4^2:D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '64.202', 'autcent_hash': 202, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3:D_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '24.14', 'autcentquo_hash': 14, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_6', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [3, 2, 1], [4, 1, 4], [4, 2, 10], [4, 12, 4], [6, 2, 7], [8, 12, 4], [12, 2, 24]], 'center_label': '8.2', 'center_order': 8, 'central_product': False, 'central_quotient': '24.6', 'commutator_count': 1, 'commutator_label': '12.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 82, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [3, 2, 1, 1], [4, 1, 2, 2], [4, 2, 1, 2], [4, 2, 2, 4], [4, 12, 2, 2], [6, 2, 1, 3], [6, 2, 2, 2], [8, 12, 2, 2], [12, 2, 2, 4], [12, 2, 4, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 24, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.10', 'frattini_quotient': '12.4', 'hash': 82, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 12, 1], 'inner_gens': [[1, 92, 48], [153, 4, 48], [1, 4, 48]], 'inner_hash': 6, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': True, 'inner_tex': 'D_{12}', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 44]], 'label': '192.82', 'linC_count': 128, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 8, 'linQ_dim': 8, 'linQ_dim_count': 56, 'linR_count': 40, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C12.C4^2', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 22, 'number_characteristic_subgroups': 39, 'number_conjugacy_classes': 60, 'number_divisions': 32, 'number_normal_subgroups': 51, 'number_subgroup_autclasses': 80, 'number_subgroup_classes': 110, 'number_subgroups': 232, 'old_label': None, 'order': 192, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 72], [6, 14], [8, 48], [12, 48]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 24, 0], 'outer_gens': [[1, 148, 144], [3, 4, 48], [3, 28, 144], [51, 28, 144], [51, 30, 144]], 'outer_group': '64.261', 'outer_hash': 261, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 32, 'outer_perms': [43328107513216344878530623767946840, 17774154876782283574170730336362004, 204650948727546386496987892949991340, 204651253615891151503949092984995101, 254872281173311478503587874963428575], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_2^3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [4, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 14], [4, 10], [8, 4]], 'representations': {'PC': {'code': 3483461946002348282336900, 'gens': [1, 3, 6], 'pres': [7, -2, -2, -2, -2, -3, -2, -2, 14, 1934, 58, 3811, 80, 1124, 124]}, 'GLFp': {'d': 3, 'p': 13, 'gens': [9865148832, 3149638242, 6256627175, 9789111396, 9761058284, 759059844, 2651075743]}, 'Perm': {'d': 15, 'gens': [95005612929, 193165418880, 199395302416, 25437928320, 282219759360, 199395302400, 840]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}.C_4^2', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
