-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '576.8418', 'ambient_counter': 8418, 'ambient_order': 576, 'ambient_tex': 'D_6^2:C_2^2', 'central': False, 'central_factor': False, 'centralizer_order': 48, 'characteristic': False, 'core_order': 4, 'counter': 125, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '576.8418.24.u1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '24.u1', 'outer_equivalence': True, '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': '24.14', 'subgroup_hash': 14, 'subgroup_order': 24, 'subgroup_tex': 'C_2\\times D_6', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '576.8418', 'aut_centralizer_order': None, 'aut_label': '24.u1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '12.c1', 'complements': None, 'conjugacy_class_count': 4, 'contained_in': ['8.t1', '12.c1', '12.n1', '12.o1'], 'contains': ['48.m1', '48.p1', '48.bb1', '72.m1'], 'core': '144.b1', 'coset_action_label': None, 'count': 8, 'diagramx': None, 'generators': [85806, 6, 126, 7706160], 'label': '576.8418.24.u1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.f1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.a1', 'old_label': '24.u1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '24.u1', 'subgroup_fusion': None, 'weyl_group': None}
-
gps_groups • Show schema
Hide schema
{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 3, 3, 2, 2], 'aut_gens': [[1, 2, 4], [1, 2, 20], [1, 2, 5], [12, 15, 5], [1, 10, 4], [1, 14, 4], [1, 15, 4]], 'aut_group': '144.183', 'aut_hash': 183, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 144, 'aut_permdeg': 7, 'aut_perms': [1, 120, 144, 3, 744, 1680], 'aut_phi_ratio': 18.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 3, 4, 1], [3, 2, 1, 1], [6, 2, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.12', 'autcent_hash': 12, 'autcent_nilpotent': False, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 3, 4], [3, 2, 1], [6, 2, 3]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', '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': 14, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 1, 4], [3, 2, 1, 1], [6, 2, 1, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 28, 'exponent': 6, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '24.14', 'hash': 14, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [1, 2, 3], 'inner_gens': [[1, 2, 4], [1, 2, 20], [1, 10, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 4]], 'label': '24.14', 'linC_count': 12, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 12, 'linQ_dim': 3, 'linQ_dim_count': 12, 'linR_count': 12, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*D6', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, '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': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 12, 'number_divisions': 12, 'number_normal_subgroups': 21, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 32, 'number_subgroups': 54, 'old_label': None, 'order': 24, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 15], [3, 2], [6, 6]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 3, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 2, 5], [13, 14, 17], [1, 15, 4], [1, 14, 4]], 'outer_group': '24.12', 'outer_hash': 12, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 4, 'outer_perms': [2, 4, 16, 7], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4]], 'representations': {'PC': {'code': 5123137, 'gens': [1, 2, 3], 'pres': [4, -2, -2, -2, -3, 126, 34, 135]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [16325, 16295, 3362]}, 'GLFp': {'d': 3, 'p': 3, 'gens': [7426, 8156, 16849, 13286]}, 'Perm': {'d': 7, 'gens': [127, 7, 16, 840]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times D_6', 'transitive_degree': 12, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [6, 4, 4, 12, 4, 4, 12], 'aut_gens': [[85807, 40285008, 1, 6, 806400, 126, 7706160, 84228480], [14923440, 41091534, 1, 120, 94394160, 126, 7706160, 163336320], [83462406, 41091535, 1, 7, 94394286, 126, 84228480, 14883120], [14923566, 207290449, 127, 121, 94394160, 126, 14883120, 163336320], [163699321, 40285008, 127, 6, 174988926, 126, 84228480, 7706160], [14923561, 131725854, 1, 7, 15689646, 126, 14883120, 163336320], [163699321, 41091408, 127, 7, 171319806, 126, 163336320, 7706160], [766087, 131725854, 1, 7, 15689646, 126, 163336320, 14883120]], 'aut_group': None, 'aut_hash': 2223990294858638248, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 18432, 'aut_permdeg': 72, 'aut_perms': [6464273859269510176241041032240668186577286875456618677077327950210980927293480925023882203584301900737, 33467779743933882073771129302479384867960170407424451394412958717389928728791548981028486200733706873950, 43047515369877529313358705274065517866661943618231806339463205215094203430557412869722196350558894993799, 34647826013382181981905618725324596748893974329593203431263325492278122097454558494512342105238254017276, 9808256730783530320500234197083176984502972909229265587393645454806665243990627052782154494406590988214, 9250642075292074633888901348124153466983656347680887594696149533604989997498093621086268047316027251458, 53366014124883569001654661351974522222135210146825727324659826903072965416727612518005446201324492261997], 'aut_phi_ratio': 96.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 6, 8, 1], [2, 9, 2, 2], [2, 12, 2, 1], [2, 18, 2, 1], [3, 4, 1, 2], [4, 12, 2, 1], [4, 36, 4, 1], [6, 4, 1, 2], [6, 4, 2, 2], [6, 4, 4, 1], [6, 8, 2, 1], [6, 12, 8, 1], [6, 24, 2, 1], [12, 24, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^2.C_2^6.C_2^5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.1578', 'autcent_hash': 1578, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '144.186', 'autcentquo_hash': 186, 'autcentquo_nilpotent': False, 'autcentquo_order': 144, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3^2:C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 6, 8], [2, 9, 4], [2, 12, 2], [2, 18, 2], [3, 4, 2], [4, 12, 2], [4, 36, 4], [6, 4, 10], [6, 8, 2], [6, 12, 8], [6, 24, 2], [12, 24, 2]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '144.186', 'commutator_count': 1, 'commutator_label': '36.13', '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', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 8418, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['288.889', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 6, 1, 8], [2, 9, 1, 4], [2, 12, 1, 2], [2, 18, 1, 2], [3, 4, 1, 2], [4, 12, 1, 2], [4, 36, 1, 4], [6, 4, 1, 10], [6, 8, 1, 2], [6, 12, 1, 8], [6, 24, 1, 2], [12, 24, 1, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 1834560, 'exponent': 12, 'exponents_of_order': [6, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '288.1031', 'hash': 8418, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 1, 2, 2, 1, 3, 3], 'inner_gens': [[85807, 41091408, 1, 6, 806400, 126, 14883120, 84228480], [766207, 40285008, 1, 120, 806400, 126, 84228480, 7706160], [85807, 40285008, 1, 6, 806400, 126, 7706160, 84228480], [85807, 40285134, 1, 6, 806400, 126, 7706160, 84228480], [85807, 40285008, 1, 6, 806400, 126, 14883120, 163336320], [85807, 40285008, 1, 6, 806400, 126, 7706160, 84228480], [14923567, 131725728, 1, 6, 15689520, 126, 7706160, 84228480], [85807, 207290448, 1, 6, 163739520, 126, 7706160, 84228480]], 'inner_hash': 186, 'inner_nilpotent': False, 'inner_order': 144, 'inner_split': False, 'inner_tex': 'S_3^2:C_2^2', 'inner_used': [1, 2, 4, 7], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 12], [4, 24], [8, 2]], 'label': '576.8418', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'D6^2:C2^2', 'ngens': 8, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 22, 'number_characteristic_subgroups': 19, 'number_conjugacy_classes': 54, 'number_divisions': 54, 'number_normal_subgroups': 113, 'number_subgroup_autclasses': 286, 'number_subgroup_classes': 1154, 'number_subgroups': 7506, 'old_label': None, 'order': 576, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 151], [3, 8], [4, 168], [6, 200], [12, 48]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 6, 0, 0, 0, 0], 'outer_gens': [[85681, 40285008, 1, 7, 806400, 126, 7706160, 84228480], [85801, 40285009, 1, 6, 806526, 126, 7706160, 84228480], [85801, 40285009, 127, 6, 806526, 126, 7706160, 84228480], [85681, 40285009, 1, 6, 806400, 126, 7706160, 84228480], [85806, 40285008, 1, 6, 806400, 126, 7706160, 84228480], [85680, 40285008, 1, 120, 806400, 126, 7706160, 84228480], [85681, 40285008, 1, 6, 806400, 126, 7706160, 84228480]], 'outer_group': '128.1755', 'outer_hash': 1755, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 10, 'outer_perms': [373681, 126774, 1356175, 848929, 847440, 375894, 848928], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:D_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [4, 24], [8, 2]], 'representations': {'PC': {'code': 6781024398962455775386066899600963726261388890629, 'gens': [1, 2, 4, 5, 7], 'pres': [8, 2, 2, 2, 2, 2, 3, 2, 3, 97, 41, 18884, 4332, 820, 116, 12293, 9997, 789, 14350, 6742, 166, 14351, 6167]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [706116953542, 706461792402, 141602720900, 706589870652]}, 'Perm': {'d': 12, 'gens': [85807, 40285008, 1, 6, 806400, 126, 7706160, 84228480]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'D_6^2:C_2^2', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
