-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1056.91', 'ambient_counter': 91, 'ambient_order': 1056, 'ambient_tex': 'C_{264}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 528, 'characteristic': False, 'core_order': 88, 'counter': 27, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '1056.91.12.d1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '12.d1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '12.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 12, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{12}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '88.2', 'subgroup_hash': 2, 'subgroup_order': 88, 'subgroup_tex': 'C_{88}', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '1056.91', 'aut_centralizer_order': 176, 'aut_label': '12.d1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '40.14', 'aut_weyl_index': 352, 'centralizer': '2.b1.a1', 'complements': ['88.d1.a1', '88.d1.b1'], 'conjugacy_class_count': 1, 'contained_in': ['4.d1.a1', '6.b1.a1'], 'contains': ['24.c1.a1', '132.b1.a1'], 'core': '12.d1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [8685, 9133, 3145, 8390, 5609, 9006, 5027, 7709], 'generators': [132, 264, 528, 96], 'label': '1056.91.12.d1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '12.d1.a1', 'normal_contained_in': ['4.d1.a1', '6.b1.a1'], 'normal_contains': ['24.c1.a1', '132.b1.a1'], 'normalizer': '1.a1.a1', 'old_label': '12.d1.a1', 'projective_image': '264.16', 'quotient_action_image': '2.1', 'quotient_action_kernel': '6.2', 'quotient_action_kernel_order': 6, 'quotient_fusion': None, 'short_label': '12.d1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '88.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 10, 'aut_gen_orders': [2, 2, 10], 'aut_gens': [[1], [67], [45], [57]], 'aut_group': '40.14', 'aut_hash': 14, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 40, 'aut_permdeg': 11, 'aut_perms': [3628800, 40320, 753], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [8, 1, 4, 1], [11, 1, 10, 1], [22, 1, 10, 1], [44, 1, 20, 1], [88, 1, 40, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times C_{10}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 10, 'autcent_group': '40.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 40, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\times C_{10}', '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], [8, 1, 4], [11, 1, 10], [22, 1, 10], [44, 1, 20], [88, 1, 40]], 'center_label': '88.2', 'center_order': 88, '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', '11.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['11.1', 1], ['8.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [8, 1, 4, 1], [11, 1, 10, 1], [22, 1, 10, 1], [44, 1, 20, 1], [88, 1, 40, 1]], 'element_repr_type': 'PC', 'elementary': 22, 'eulerian_function': 1, 'exponent': 88, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 11], 'faithful_reps': [[1, 0, 40]], 'familial': True, 'frattini_label': '4.1', 'frattini_quotient': '22.2', 'hash': 2, 'hyperelementary': 22, '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': 40, 'irrQ_dim': 40, 'irrR_degree': 2, 'irrep_stats': [[1, 88]], 'label': '88.2', 'linC_count': 40, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 2, 'linQ_dim': 14, 'linQ_dim_count': 2, 'linR_count': 20, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C88', 'ngens': 4, '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': 88, 'number_divisions': 8, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 88, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 1], [4, 2], [8, 4], [11, 10], [22, 10], [44, 20], [88, 40]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 10, 'outer_gen_orders': [2, 2, 10], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[67], [45], [57]], 'outer_group': '40.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 40, 'outer_permdeg': 11, 'outer_perms': [3628800, 40320, 753], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_{10}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [8, 11], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [4, 1], [10, 2], [20, 1], [40, 1]], 'representations': {'PC': {'code': 44718150009, 'gens': [1], 'pres': [4, -2, -2, -2, -11, 8, 21, 34]}, 'GLFp': {'d': 2, 'p': 23, 'gens': [14100]}, 'Perm': {'d': 19, 'gens': [47039662365696000, 36288000, 19918769962291200, 6423384152908800]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [88], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{88}', 'transitive_degree': 88, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '48.20', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 220, 'aut_gen_orders': [10, 10, 10, 44, 20, 2, 10], 'aut_gens': [[1, 4], [147, 820], [217, 76], [721, 380], [1035, 620], [745, 582], [531, 268], [675, 190]], 'aut_group': None, 'aut_hash': 2218350592113462965, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 14080, 'aut_permdeg': 98, 'aut_perms': [1135018993951749855382771518521903941501433812212955368216646682262484802758343987128202777312995530205574754612582628363763145771883439731041821187576371, 7094594485492621804590315313133544664407496736070487875488271168119561911911686319739872835522628225763861973650326024433143587648550960049063136673229278, 2884652813716858454541180967572521509338526050709853684937914317016100319062686662355615300292352719098835407842297566101940470539459448745167269941365216, 3166983285260480579602293615905649249707567936544525368843350125845653217698546222279945278256462217173013673795452210813652472058854667581328724731501112, 5744798147647918642438880364676633323105685229103890377727807934522024188904627231451228985801081080893185826191469698946074428636453736835710157112879109, 5245078566678281302808382357345904418047216203041391067788085711429385247689116867052995088850855044686822559893543138329305968689498946923410567612911355, 1921559792430562860220083751068381407093737030670569673774443077073127449831022859486006608555825024420268128210247649418183296838238071114203701015117129], 'aut_phi_ratio': 44.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [3, 1, 2, 1], [4, 1, 2, 2], [4, 22, 4, 1], [6, 1, 2, 1], [6, 1, 4, 1], [8, 2, 4, 1], [8, 22, 4, 1], [11, 2, 5, 1], [12, 1, 4, 2], [12, 22, 8, 1], [22, 2, 5, 1], [22, 2, 10, 1], [24, 2, 8, 1], [24, 22, 8, 1], [33, 2, 10, 1], [44, 2, 10, 2], [66, 2, 10, 1], [66, 2, 20, 1], [88, 2, 40, 1], [132, 2, 20, 2], [264, 2, 80, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_2^3\\times C_{11}:C_5).C_2^5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.2163', 'autcent_hash': 2163, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4:D_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 110, 'autcentquo_group': '110.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 110, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [3, 1, 2], [4, 1, 4], [4, 22, 4], [6, 1, 6], [8, 2, 4], [8, 22, 4], [11, 2, 5], [12, 1, 8], [12, 22, 8], [22, 2, 15], [24, 2, 8], [24, 22, 8], [33, 2, 10], [44, 2, 20], [66, 2, 30], [88, 2, 40], [132, 2, 40], [264, 2, 80]], 'center_label': '24.9', 'center_order': 24, 'central_product': True, 'central_quotient': '44.3', 'commutator_count': 1, 'commutator_label': '22.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 91, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['352.21', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [4, 1, 2, 2], [4, 22, 2, 2], [6, 1, 2, 3], [8, 2, 2, 2], [8, 22, 2, 2], [11, 2, 5, 1], [12, 1, 4, 2], [12, 22, 4, 2], [22, 2, 5, 3], [24, 2, 4, 2], [24, 22, 4, 2], [33, 2, 10, 1], [44, 2, 10, 2], [66, 2, 10, 3], [88, 2, 20, 2], [132, 2, 20, 2], [264, 2, 40, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 264, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 5, 11], 'factors_of_order': [2, 3, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '132.7', 'hash': 91, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 22, 'inner_gen_orders': [2, 22], 'inner_gens': [[1, 436], [625, 4]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 44, 'inner_split': False, 'inner_tex': 'D_{22}', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 48], [2, 252]], 'label': '1056.91', 'linC_count': 2560, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 16, 'linQ_dim': 18, 'linQ_dim_count': 80, 'linR_count': 640, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C264:C4', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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': 28, 'number_characteristic_subgroups': 34, 'number_conjugacy_classes': 300, 'number_divisions': 40, 'number_normal_subgroups': 58, 'number_subgroup_autclasses': 60, 'number_subgroup_classes': 80, 'number_subgroups': 308, 'old_label': None, 'order': 1056, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 92], [6, 6], [8, 96], [11, 10], [12, 184], [22, 30], [24, 192], [33, 20], [44, 40], [66, 60], [88, 80], [132, 80], [264, 160]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 20, 'outer_gen_orders': [2, 2, 2, 2, 20], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[3, 964], [1, 620], [531, 260], [531, 358], [267, 894]], 'outer_group': '320.1629', 'outer_hash': 1629, 'outer_nilpotent': True, 'outer_order': 320, 'outer_permdeg': 15, 'outer_perms': [274471586640, 274467957840, 186810624000, 720, 91504], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{20}:C_2^4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [4, 4, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [4, 8], [8, 2], [10, 4], [20, 6], [40, 4], [80, 2]], 'representations': {'PC': {'code': 19731723705600264671119429744019414370000019, 'gens': [1, 3], 'pres': [7, -2, -2, -2, -2, -2, -3, -11, 14, 9158, 58, 24419, 80, 24084, 102, 13445, 166, 47046]}, 'Perm': {'d': 26, 'gens': [17476014528073522594217649, 33559443123727648038912016, 376610217984000, 46639286320049153482752000, 64036574450394981507072016, 64036574450394981507072000, 99712902360]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 12], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{264}:C_4', 'transitive_degree': 1056, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '12.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2, 2], 'aut_gens': [[1], [5], [7]], 'aut_group': '4.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [1, 6], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [6, 1, 2, 1], [12, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', '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], [3, 1, 2], [4, 1, 2], [6, 1, 2], [12, 1, 4]], 'center_label': '12.2', 'center_order': 12, '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', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [6, 1, 2, 1], [12, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 6, 'eulerian_function': 1, 'exponent': 12, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '6.2', 'hash': 2, 'hyperelementary': 6, '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': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 12]], 'label': '12.2', 'linC_count': 4, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 3, 'linQ_dim': 4, 'linQ_dim_count': 3, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C12', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 12, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 2], [6, 2], [12, 4]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[5], [7]], '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': 1, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [4, 3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 1]], 'representations': {'PC': {'code': 3865, 'gens': [1], 'pres': [3, -2, -2, -3, 6, 16]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [20970031]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [619]}, 'Perm': {'d': 7, 'gens': [2400, 4, 744]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [12], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}', 'transitive_degree': 12, 'wreath_data': None, 'wreath_product': False}
