-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '448.652', 'ambient_counter': 652, 'ambient_order': 448, 'ambient_tex': '(C_2\\times C_8).D_{14}', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': False, 'core_order': 112, 'counter': 10, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '448.652.4.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '4.b1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '4.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '112.35', 'subgroup_hash': 35, 'subgroup_order': 112, 'subgroup_tex': 'C_2^2.D_{14}', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '448.652', 'aut_centralizer_order': 16, 'aut_label': '4.b1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '672.1266', 'aut_weyl_index': 32, 'centralizer': '28.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1'], 'contains': ['8.a1.a1', '8.d1.a1', '8.e1.b1', '8.h1.a1', '8.h1.b1', '28.c1.a1'], 'core': '4.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [8171, 9223, 1361, 416, 1690, 1732, 1480, 1728], 'generators': [338, 224, 232, 20, 64], 'label': '448.652.4.b1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '4.b1.a1', 'normal_contained_in': ['2.a1.a1'], 'normal_contains': ['8.a1.a1', '8.d1.a1', '8.e1.b1'], 'normalizer': '1.a1.a1', 'old_label': '4.b1.a1', 'projective_image': '112.28', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.b1.a1', 'subgroup_fusion': None, 'weyl_group': '28.3'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 84, 'aut_gen_orders': [2, 6, 2, 2, 3, 2, 12, 42, 2, 2], 'aut_gens': [[1, 2, 8], [1, 3, 12], [56, 38, 105], [5, 63, 12], [1, 7, 13], [60, 3, 9], [1, 6, 12], [60, 98, 17], [60, 27, 13], [1, 6, 8], [5, 2, 8]], 'aut_group': '8064.cl', 'aut_hash': 3465494477509594995, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 8064, 'aut_permdeg': 15, 'aut_perms': [562998159360, 88706784436, 462878841600, 648656023680, 561473377920, 87178331520, 2087329903, 374100294867, 87660961920, 3669120], 'aut_phi_ratio': 168.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [4, 7, 8, 1], [7, 2, 3, 1], [14, 2, 3, 1], [14, 2, 18, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^3:S_4\\times F_7', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '192.1493', 'autcent_hash': 1493, 'autcent_nilpotent': False, 'autcent_order': 192, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^3:S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '42.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 42, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_7', 'cc_stats': [[1, 1, 1], [2, 1, 7], [4, 7, 8], [7, 2, 3], [14, 2, 21]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '14.1', 'commutator_count': 1, 'commutator_label': '7.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '7.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 35, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['28.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [4, 7, 2, 4], [7, 2, 3, 1], [14, 2, 3, 7]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 56, 'exponent': 28, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '56.12', 'hash': 35, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [1, 2, 7], 'inner_gens': [[1, 2, 8], [1, 2, 104], [1, 18, 8]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 14, 'inner_split': True, 'inner_tex': 'D_7', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 24]], 'label': '112.35', 'linC_count': 1008, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 48, 'linQ_dim': 9, 'linQ_dim_count': 48, 'linR_count': 144, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2.D14', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, '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': 7, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 40, 'number_divisions': 20, 'number_normal_subgroups': 43, 'number_subgroup_autclasses': 18, 'number_subgroup_classes': 54, 'number_subgroups': 120, 'old_label': None, 'order': 112, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 7], [4, 56], [7, 6], [14, 42]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 3, 3, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 2, 0, 0, 0, 0, 0, 0], 'outer_gens': [[57, 6, 8], [1, 2, 24], [56, 2, 9], [5, 63, 12], [1, 7, 12], [1, 3, 8], [1, 59, 8], [1, 6, 8]], 'outer_group': '576.8479', 'outer_hash': 8479, 'outer_nilpotent': False, 'outer_order': 576, 'outer_permdeg': 11, 'outer_perms': [7983360, 3, 19596240, 19777680, 24174240, 27803064, 19818024, 3669864], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_3\\times C_2^3:S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [6, 8]], 'representations': {'PC': {'code': 904557920481221, 'gens': [1, 2, 4], 'pres': [5, -2, -2, -2, -2, -7, 26, 1048, 58, 1209]}, 'GLZN': {'d': 2, 'p': 28, 'gens': [285389, 22345, 329295, 22065, 22049]}, 'Perm': {'d': 15, 'gens': [4032127, 87657292800, 186818607360, 7983360, 973]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2.D_{14}', 'transitive_degree': 112, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 84, 'aut_gen_orders': [12, 12, 2, 6, 4, 6, 12], 'aut_gens': [[1, 4, 16], [113, 212, 80], [1, 20, 144], [1, 212, 432], [11, 44, 304], [225, 20, 216], [225, 140, 48], [345, 84, 88]], 'aut_group': None, 'aut_hash': 4705075075878970669, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 21504, 'aut_permdeg': 72, 'aut_perms': [27351219204123924510432601177022596590918084998732432753431912837559894654497094773012324836285351585640, 507241509500623957870098877873938680265482894088264625979140482516320412944925696317659934349496126453, 133491071402531105700500608340529221851302095353752939964029695553948364381302735489674954993820002277, 11776159414774950190384847240363038699393493057065887596749031701407313683600922841219846576930815136288, 31603753229253516112548775106028838225428053901959129214869552973771290304700095227385259161252539480809, 31814400502947194383476763426707916057775886447652502298003676591259851480644772209072951659226245673011, 46463820622914009173276199549802659612160021595780752809781730825491545609040706955516869411884189674746], 'aut_phi_ratio': 112.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [4, 1, 2, 2], [4, 2, 2, 1], [4, 14, 4, 2], [7, 2, 3, 1], [8, 4, 4, 1], [8, 28, 4, 1], [14, 2, 3, 1], [14, 2, 6, 1], [14, 4, 6, 1], [28, 2, 6, 2], [28, 4, 6, 1], [56, 4, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_7.(C_2^4\\times C_6).C_2^5', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '128.2328', 'autcent_hash': 2328, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^7', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '168.47', 'autcentquo_hash': 47, 'autcentquo_nilpotent': False, 'autcentquo_order': 168, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [4, 1, 4], [4, 2, 2], [4, 14, 8], [7, 2, 3], [8, 4, 4], [8, 28, 4], [14, 2, 9], [14, 4, 6], [28, 2, 12], [28, 4, 6], [56, 4, 24]], 'center_label': '8.2', 'center_order': 8, 'central_product': False, 'central_quotient': '56.12', 'commutator_count': 1, 'commutator_label': '28.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 652, '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], [4, 1, 2, 2], [4, 2, 1, 2], [4, 14, 1, 4], [4, 14, 2, 2], [7, 2, 3, 1], [8, 4, 2, 2], [8, 28, 2, 2], [14, 2, 3, 3], [14, 4, 3, 2], [28, 2, 6, 2], [28, 4, 3, 2], [56, 4, 12, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1344, 'exponent': 56, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '56.12', 'hash': 652, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 2, 14], 'inner_gens': [[1, 236, 240], [233, 4, 208], [225, 260, 16]], 'inner_hash': 12, 'inner_nilpotent': False, 'inner_order': 56, 'inner_split': False, 'inner_tex': 'C_2\\times D_{14}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 60], [4, 12]], 'label': '448.652', 'linC_count': 192, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 64, 'linQ_dim': 12, 'linQ_dim_count': 32, 'linR_count': 168, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': '(C2*C8).D14', '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': 19, 'number_characteristic_subgroups': 25, 'number_conjugacy_classes': 88, 'number_divisions': 32, 'number_normal_subgroups': 67, 'number_subgroup_autclasses': 72, 'number_subgroup_classes': 126, 'number_subgroups': 420, 'old_label': None, 'order': 448, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 7], [4, 120], [7, 6], [8, 128], [14, 42], [28, 48], [56, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 4, 12], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 228, 240], [3, 228, 24], [3, 4, 16], [1, 116, 16], [121, 4, 280]], 'outer_group': '384.19922', 'outer_hash': 19922, 'outer_nilpotent': True, 'outer_order': 384, 'outer_permdeg': 13, 'outer_perms': [39917544, 1680, 41064, 961632744, 2404], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6\\times D_4^2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [4, 4], [6, 4], [12, 6], [24, 2]], 'representations': {'PC': {'code': 136931638248764895885538320935565176453, 'gens': [1, 3, 5], 'pres': [7, -2, -2, -2, -2, -2, -2, -7, 14, 1576, 4958, 58, 8404, 1838, 102, 4387, 124, 4724]}, 'GLZN': {'d': 2, 'p': 105, 'gens': [74088064, 105045060, 58731730, 9261008, 1157689, 1159201, 49777918]}, 'Perm': {'d': 19, 'gens': [357001373400319, 87091847, 40292146, 6786, 87109819, 18619, 7116370137676800]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(C_2\\times C_8).D_{14}', 'transitive_degree': 224, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [3]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', '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], [4, 1, 2]], 'center_label': '4.1', 'center_order': 4, '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': ['2.1', '2.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [2], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4]], 'label': '4.1', 'linC_count': 2, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4', 'ngens': 1, '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': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 1], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -2, -2, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [56, 15], 'family': 'CSOPlus'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [21]}, 'Perm': {'d': 4, 'gens': [22, 7]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
