-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '256.26336', 'ambient_counter': 26336, 'ambient_order': 256, 'ambient_tex': '(D_4\\times C_2^3):C_4', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 64, 'counter': 26, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '256.26336.4.r1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '4.r1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '4.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^2', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '64.33', 'subgroup_hash': 33, 'subgroup_order': 64, 'subgroup_tex': 'C_2^3.D_4', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '256.26336', 'aut_centralizer_order': 8, 'aut_label': '4.r1', 'aut_quo_index': 3, 'aut_stab_index': 4, 'aut_weyl_group': '128.1755', 'aut_weyl_index': 32, 'centralizer': '64.c1', 'complements': ['64.q1', '64.bf1', '64.bg1'], 'conjugacy_class_count': 4, 'contained_in': ['2.b1', '2.f1'], 'contains': ['8.x1', '8.y1', '8.z1'], 'core': '4.r1', 'coset_action_label': None, 'count': 4, 'diagramx': [1892, 3525, 1898, 3509], 'generators': [1, 80], 'label': '256.26336.4.r1', 'mobius_quo': 0, 'mobius_sub': 2, 'normal_closure': '4.r1', 'normal_contained_in': ['2.b1', '2.f1'], 'normal_contains': ['8.x1', '8.y1', '8.z1'], 'normalizer': '1.a1', 'old_label': '4.r1', 'projective_image': '128.1613', 'quotient_action_image': '4.2', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '4.r1', 'subgroup_fusion': None, 'weyl_group': '64.90'}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, '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, 2, 2, 2, 2, 2, 2], 'aut_gens': [[1, 4, 8, 16], [3, 12, 8, 52], [13, 4, 8, 56], [11, 36, 8, 24], [1, 4, 8, 24], [33, 4, 8, 24], [1, 12, 8, 56], [9, 4, 8, 16]], 'aut_group': '128.1755', 'aut_hash': 1755, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 128, 'aut_permdeg': 10, 'aut_perms': [930649, 380521, 26190, 1, 2718174, 1154310, 1315848], 'aut_phi_ratio': 4.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 2], [4, 4, 1, 1], [4, 4, 2, 1], [4, 8, 1, 1], [4, 8, 2, 1], [8, 8, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^4:D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '32.27', 'autcentquo_hash': 27, 'autcentquo_nilpotent': True, 'autcentquo_order': 32, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2\\wr C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 4, 2], [4, 4, 3], [4, 8, 3], [8, 8, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '32.6', 'commutator_count': 1, 'commutator_label': '8.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 33, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 2], [4, 4, 1, 1], [4, 4, 2, 1], [4, 8, 1, 1], [4, 8, 2, 1], [8, 8, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 12, 'exponent': 8, 'exponents_of_order': [6], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[4, 0, 2]], 'familial': False, 'frattini_label': '16.11', 'frattini_quotient': '4.2', 'hash': 33, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [4, 2, 1, 4], 'inner_gens': [[1, 44, 8, 28], [41, 4, 8, 24], [1, 4, 8, 16], [5, 12, 8, 16]], 'inner_hash': 6, 'inner_nilpotent': True, 'inner_order': 32, 'inner_split': True, 'inner_tex': 'C_2^3:C_4', 'inner_used': [1, 2, 4], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 2], [4, 3]], 'label': '64.33', 'linC_count': 2, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 1, 'linQ_dim': 8, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3.D4', 'ngens': 2, 'nilpotency_class': 4, 'nilpotent': True, 'normal_counts': [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': 13, 'number_conjugacy_classes': 13, 'number_divisions': 10, 'number_normal_subgroups': 13, 'number_subgroup_autclasses': 34, 'number_subgroup_classes': 34, 'number_subgroups': 81, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 11], [4, 36], [8, 16]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 4, 8, 24], [39, 36, 8, 48]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 1], [8, 1]], 'representations': {'PC': {'code': 18811608232852756305, 'gens': [1, 3, 4, 5], 'pres': [6, 2, 2, 2, 2, 2, 2, 12, 794, 116, 844, 850, 196, 88, 1445]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [122109387504, 115142802501, 113044404613, 30594431879, 41808022846, 122244534693]}, 'Perm': {'d': 16, 'gens': [14002613452800, 6920136143857, 6227383801, 2789792421136, 5167, 1313941673647]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3.D_4', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '32.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 8, 'aut_gen_orders': [2, 2, 4, 2, 2, 2, 2, 2, 2], 'aut_gens': [[1, 4, 8, 16, 64], [1, 4, 8, 16, 96], [63, 164, 136, 144, 64], [41, 52, 136, 48, 232], [49, 132, 40, 48, 224], [19, 132, 168, 16, 64], [129, 52, 40, 48, 192], [15, 36, 136, 144, 64], [171, 36, 136, 16, 64], [17, 164, 40, 48, 224]], 'aut_group': '4096.brb', 'aut_hash': 3950354226804192420, 'aut_nilpotency_class': 4, 'aut_nilpotent': True, 'aut_order': 4096, 'aut_permdeg': 72, 'aut_perms': [702466085250537595749320915094215707867512477365932516494181352190878463513, 52238239686147942489636743933578865024124246581203626797213690859879962491889387094446226389762169148118, 46441026328871235344859229398390262842099743492288938950663905908150457610394425698523769991643564902294, 55773409737792977339768048662388518438278779395213176355113580144778904737082267224950395072499731441226, 9748663126008800151946056462404152536625914375447144487812370317595215104714015033794421388420823006691, 30925652839709627472186517022484340209235241910714946897250878941472171487601726596232969760224303759230, 41507327971767281436299443439664095885462444072758822385125686298296644058352911659783983109112715636440, 22260701330655778895999178737593260185261726313267075357645365566691030914724507873277401833639757732413, 3034003033165853770725421500206094214642242017612248239783078550770004245627246073517842833080049108996], 'aut_phi_ratio': 32.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 2, 4, 1], [2, 4, 1, 3], [2, 4, 2, 3], [2, 8, 2, 1], [4, 2, 2, 1], [4, 4, 1, 3], [4, 4, 2, 1], [4, 8, 1, 3], [4, 8, 2, 1], [4, 8, 8, 1], [8, 8, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^4.(C_2^2\\times D_4^2)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '256.29080', 'autcentquo_hash': 29080, 'autcentquo_nilpotent': True, 'autcentquo_order': 256, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2:D_4^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 5], [2, 4, 9], [2, 8, 2], [4, 2, 2], [4, 4, 5], [4, 8, 13], [8, 8, 8]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '128.1613', 'commutator_count': 1, 'commutator_label': '8.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 26336, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 5], [2, 4, 1, 9], [2, 8, 1, 2], [4, 2, 1, 2], [4, 4, 1, 5], [4, 8, 1, 5], [4, 8, 2, 4], [8, 8, 2, 4]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 322560, 'exponent': 8, 'exponents_of_order': [8], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[8, 1, 2]], 'familial': False, 'frattini_label': '16.11', 'frattini_quotient': '16.14', 'hash': 26336, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [4, 2, 2, 2, 4], 'inner_gens': [[1, 4, 168, 16, 200], [1, 4, 8, 48, 64], [161, 4, 8, 16, 96], [1, 36, 8, 16, 64], [169, 4, 40, 16, 64]], 'inner_hash': 1613, 'inner_nilpotent': True, 'inner_order': 128, 'inner_split': None, 'inner_tex': 'C_2^5:C_4', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 32], [2, 8], [4, 4], [8, 2]], 'label': '256.26336', 'linC_count': None, 'linC_degree': None, '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': '(D4*C2^3):C4', 'ngens': 4, 'nilpotency_class': 4, 'nilpotent': True, 'normal_counts': [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': 51, 'number_conjugacy_classes': 46, 'number_divisions': 38, 'number_normal_subgroups': 181, 'number_subgroup_autclasses': 335, 'number_subgroup_classes': 705, 'number_subgroups': 2223, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 63], [4, 128], [8, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 4, 8, 16, 96], [19, 132, 168, 16, 96], [45, 36, 40, 144, 96], [181, 52, 136, 144, 104]], 'outer_group': '32.46', 'outer_hash': 46, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 16, 'outer_perms': [1334672640450, 10888245730006, 10894473113747, 19328471502467], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times D_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 4], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 16], [4, 4], [8, 2]], 'representations': {'PC': {'code': 651574923237232375847145550688849, 'gens': [1, 3, 4, 5, 7], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 16, 5379, 651, 500, 116, 11206, 5390, 702, 166, 10247]}, 'Perm': {'d': 16, 'gens': [14002653732480, 1313941697263, 5612462133817, 2789792421136, 5606234750016, 2789832700823, 5167, 1313941673647]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': '(D_4\\times C_2^3):C_4', 'transitive_degree': 16, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', '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]], 'center_label': '4.2', 'center_order': 4, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', '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, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, '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': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
