-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '792.91', 'ambient_counter': 91, 'ambient_order': 792, 'ambient_tex': 'C_{12}\\times D_{33}', 'central': False, 'central_factor': False, 'centralizer_order': 396, 'characteristic': True, 'core_order': 33, 'counter': 34, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '792.91.24.b1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '24.b1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '24.9', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 9, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 24, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times C_{12}', 'simple': False, 'solvable': True, 'special_labels': ['D', 'L1', 'D1', 'C4'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '33.1', 'subgroup_hash': 1, 'subgroup_order': 33, 'subgroup_tex': 'C_{33}', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '792.91', 'aut_centralizer_order': 264, 'aut_label': '24.b1', 'aut_quo_index': 2, 'aut_stab_index': 1, 'aut_weyl_group': '20.5', 'aut_weyl_index': 264, 'centralizer': '2.b1.a1', 'complements': ['33.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['8.a1.a1', '12.b1.a1', '12.c1.a1', '12.c1.b1'], 'contains': ['72.a1.a1', '264.b1.a1'], 'core': '24.b1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [8649, 1226, 8001, 2049, 1595, 2321, 7840, 3515], 'generators': [528, 72], 'label': '792.91.24.b1.a1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '24.b1.a1', 'normal_contained_in': ['8.a1.a1', '12.b1.a1', '12.c1.a1', '12.c1.b1'], 'normal_contains': ['72.a1.a1', '264.b1.a1'], 'normalizer': '1.a1.a1', 'old_label': '24.b1.a1', 'projective_image': '792.91', 'quotient_action_image': '2.1', 'quotient_action_kernel': '12.2', 'quotient_action_kernel_order': 12, 'quotient_fusion': None, 'short_label': '24.b1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '33.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 10, 'aut_gen_orders': [2, 10], 'aut_gens': [[1], [10], [20]], 'aut_group': '20.5', 'aut_hash': 5, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 20, 'aut_permdeg': 9, 'aut_perms': [720, 40353], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [11, 1, 10, 1], [33, 1, 20, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{10}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 10, 'autcent_group': '20.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 20, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_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], [3, 1, 2], [11, 1, 10], [33, 1, 20]], 'center_label': '33.1', 'center_order': 33, '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': ['3.1', '11.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['11.1', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1], [11, 1, 10, 1], [33, 1, 20, 1]], 'element_repr_type': 'PC', 'elementary': 33, 'eulerian_function': 1, 'exponent': 33, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [3, 11], 'faithful_reps': [[1, 0, 20]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '33.1', 'hash': 1, 'hyperelementary': 33, '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': 20, 'irrQ_dim': 20, 'irrR_degree': 2, 'irrep_stats': [[1, 33]], 'label': '33.1', 'linC_count': 20, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 1, 'linQ_dim': 12, 'linQ_dim_count': 1, 'linR_count': 10, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C33', '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': 33, 'number_divisions': 4, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 4, 'old_label': None, 'order': 33, 'order_factorization_type': 11, 'order_stats': [[1, 1], [3, 2], [11, 10], [33, 20]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 10, 'outer_gen_orders': [2, 10], 'outer_gen_pows': [0, 0], 'outer_gens': [[10], [20]], 'outer_group': '20.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 20, 'outer_permdeg': 9, 'outer_perms': [720, 40353], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{10}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [3, 11], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [10, 1], [20, 1]], 'representations': {'PC': {'code': 519, 'gens': [1], 'pres': [2, -3, -11, 6]}, 'GLFp': {'d': 2, 'p': 23, 'gens': [90443]}, 'Perm': {'d': 14, 'gens': [12454041600, 36288000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [33], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{33}', 'transitive_degree': 33, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '24.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 330, 'aut_gen_orders': [2, 30, 2, 10, 2, 22], 'aut_gens': [[1, 6], [113, 258], [65, 150], [617, 654], [149, 294], [197, 786], [301, 534]], 'aut_group': None, 'aut_hash': 6779243509808250168, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5280, 'aut_permdeg': 70, 'aut_perms': [8428614840153996575484958496152439681307945471613590021743948146296123357872376725019886569683975925, 11282106947402510156121667533199464095257078844903278488017358310084998708571240835486324187628938725, 537948926397516136649868999624307090016551909650225386939319434088946796539202421295218376523113795, 6939232002145886265047193715318810802400173687128082342742803895200807181183226037111789262166108126, 8927387706294117808281337348585688736474873829556008883979073309256164605429393624762149497483784327, 6389934303506735945118771037825153671555554353347054585103292683771593446817793797806508459204469132], 'aut_phi_ratio': 22.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 33, 2, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 1, 2, 1], [4, 33, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 33, 4, 1], [11, 2, 5, 1], [12, 1, 4, 1], [12, 2, 2, 1], [12, 2, 4, 1], [12, 33, 4, 1], [22, 2, 5, 1], [33, 2, 10, 2], [33, 2, 20, 1], [44, 2, 10, 1], [66, 2, 10, 2], [66, 2, 20, 1], [132, 2, 20, 2], [132, 2, 40, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{33}.(C_2^4\\times C_{10})', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 330, 'autcentquo_group': '660.15', 'autcentquo_hash': 15, 'autcentquo_nilpotent': False, 'autcentquo_order': 660, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 33, 2], [3, 1, 2], [3, 2, 3], [4, 1, 2], [4, 33, 2], [6, 1, 2], [6, 2, 3], [6, 33, 4], [11, 2, 5], [12, 1, 4], [12, 2, 6], [12, 33, 4], [22, 2, 5], [33, 2, 40], [44, 2, 10], [66, 2, 40], [132, 2, 80]], 'center_label': '12.2', 'center_order': 12, 'central_product': True, 'central_quotient': '66.3', 'commutator_count': 1, 'commutator_label': '33.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '11.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 91, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['4.1', 1], ['66.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 33, 1, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 1, 2, 1], [4, 33, 2, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 33, 2, 2], [11, 2, 5, 1], [12, 1, 4, 1], [12, 2, 2, 1], [12, 2, 4, 1], [12, 33, 4, 1], [22, 2, 5, 1], [33, 2, 10, 2], [33, 2, 20, 1], [44, 2, 10, 1], [66, 2, 10, 2], [66, 2, 20, 1], [132, 2, 20, 2], [132, 2, 40, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 132, 'exponents_of_order': [3, 2, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 11], 'faithful_reps': [[2, 0, 40]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '396.27', 'hash': 91, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 66, 'inner_gen_orders': [2, 33], 'inner_gens': [[1, 390], [409, 6]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 66, 'inner_split': True, 'inner_tex': 'D_{33}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 80, 'irrQ_dim': 80, 'irrR_degree': 4, 'irrep_stats': [[1, 24], [2, 192]], 'label': '792.91', 'linC_count': 40, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 56, 'linQ_dim': 16, 'linQ_dim_count': 56, 'linR_count': 100, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C12*D33', 'ngens': 6, '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': 30, 'number_conjugacy_classes': 216, 'number_divisions': 30, 'number_normal_subgroups': 34, 'number_subgroup_autclasses': 62, 'number_subgroup_classes': 70, 'number_subgroups': 516, 'old_label': None, 'order': 792, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 67], [3, 8], [4, 68], [6, 140], [11, 10], [12, 148], [22, 10], [33, 80], [44, 20], [66, 80], [132, 160]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 10, 'outer_gen_orders': [2, 2, 2, 10], 'outer_gen_pows': [726, 0, 0, 66], 'outer_gens': [[529, 654], [397, 654], [1, 258], [533, 438]], 'outer_group': '80.52', 'outer_hash': 52, 'outer_nilpotent': True, 'outer_order': 80, 'outer_permdeg': 13, 'outer_perms': [720, 479001600, 3628800, 40353], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_{10}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 5], [8, 1], [10, 2], [20, 5], [40, 4], [80, 1]], 'representations': {'PC': {'code': 1850115134966174437789926365010019, 'gens': [1, 3], 'pres': [6, -2, -3, -2, -2, -3, -11, 12, 7022, 50, 18723, 69, 23044, 118, 25925]}, 'GLFp': {'d': 2, 'p': 397, 'gens': [9323045302, 8572196043, 158006]}, 'Perm': {'d': 21, 'gens': [122002101778273920, 864, 3, 1680, 403200, 2676906211693190400]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 12], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}\\times D_{33}', 'transitive_degree': 264, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '24.9', '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], 'aut_gens': [[1, 2], [13, 2], [1, 23], [1, 10], [1, 14]], 'aut_group': '16.11', 'aut_hash': 11, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 16, 'aut_permdeg': 6, 'aut_perms': [288, 6, 127, 126], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [3, 1, 2, 1], [4, 1, 4, 1], [6, 1, 2, 1], [6, 1, 4, 1], [12, 1, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '16.11', 'autcent_hash': 11, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times 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], [3, 1, 2], [4, 1, 4], [6, 1, 6], [12, 1, 8]], 'center_label': '24.9', 'center_order': 24, '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', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 9, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [4, 1, 2, 2], [6, 1, 2, 3], [12, 1, 4, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 12, 'exponent': 12, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '12.5', 'hash': 9, '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, 24]], 'label': '24.9', 'linC_count': 96, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 4, 'linQ_dim': 4, 'linQ_dim_count': 4, 'linR_count': 8, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C12', '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': 24, 'number_divisions': 12, 'number_normal_subgroups': 16, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 16, 'number_subgroups': 16, 'old_label': None, 'order': 24, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 4], [6, 6], [12, 8]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[13, 2], [1, 23], [1, 10], [1, 14]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [288, 6, 127, 126], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [4, 2]], 'representations': {'PC': {'code': 221281, 'gens': [1, 2], 'pres': [4, -2, -2, -2, -3, 21, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [35931072, 26129103]}, 'GLFp': {'d': 2, 'p': 13, 'gens': [10993, 4396]}, 'Perm': {'d': 9, 'gens': [2400, 40320, 4, 744]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 12], '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_{12}', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
