-
gps_subgroup_search • Show schema
Hide schema
{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1568.224', 'ambient_counter': 224, 'ambient_order': 1568, 'ambient_tex': 'C_{14}^2.D_4', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 8, 'counter': 25, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1568.224.14.f1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '14.f1.a1', 'outer_equivalence': False, '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': 14, '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': '112.12', 'subgroup_hash': 12, 'subgroup_order': 112, 'subgroup_tex': 'C_{28}:C_4', 'supersolvable': True, 'sylow': 0}
-
gps_subgroup_data • Show schema
Hide schema
{'ambient': '1568.224', 'aut_centralizer_order': 4, 'aut_label': '14.f1', 'aut_quo_index': None, 'aut_stab_index': 42, 'aut_weyl_group': '672.1093', 'aut_weyl_index': 168, 'centralizer': '392.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.b1.a1'], 'contains': ['28.d1.a1', '28.i1.a1', '28.i1.a2', '98.b1.a1'], 'core': '196.a1.a1', 'coset_action_label': None, 'count': 14, 'diagramx': [2456, -1, 2714, -1, 7530, -1, 2734, -1], 'generators': [787, 840, 784, 868, 688], 'label': '1568.224.14.f1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '14.f1.a1', 'old_label': '14.f1.a1', 'projective_image': '392.21', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '14.f1.a1', 'subgroup_fusion': None, 'weyl_group': '28.3'}
-
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': 84, 'aut_gen_orders': [2, 2, 4, 6, 14], 'aut_gens': [[1, 4], [3, 6], [29, 60], [29, 4], [1, 20], [51, 4]], 'aut_group': '1344.6612', 'aut_hash': 6612, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1344, 'aut_permdeg': 15, 'aut_perms': [55, 18498, 18550, 7312400209, 548496754129], 'aut_phi_ratio': 28.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 2, 1], [4, 14, 4, 1], [7, 2, 3, 1], [14, 2, 3, 3], [28, 2, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\wr C_2\\times F_7', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '84.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 84, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 2], [4, 14, 4], [7, 2, 3], [14, 2, 9], [28, 2, 12]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '28.3', 'commutator_count': 1, 'commutator_label': '14.2', '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': 12, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 2], [4, 14, 2, 2], [7, 2, 3, 1], [14, 2, 3, 3], [28, 2, 6, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, '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': '4.2', 'frattini_quotient': '28.3', 'hash': 12, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 14], 'inner_gens': [[1, 108], [9, 4]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 28, 'inner_split': True, 'inner_tex': 'D_{14}', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 26]], 'label': '112.12', 'linC_count': 48, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 4, 'linQ_dim': 10, 'linQ_dim_count': 4, 'linR_count': 12, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C28:C4', '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': 11, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 34, 'number_divisions': 14, 'number_normal_subgroups': 19, 'number_subgroup_autclasses': 20, 'number_subgroup_classes': 26, 'number_subgroups': 72, 'old_label': None, 'order': 112, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [4, 60], [7, 6], [14, 18], [28, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 12], 'outer_gen_pows': [0, 28, 0], 'outer_gens': [[1, 60], [29, 4], [29, 14]], 'outer_group': '48.45', 'outer_hash': 45, 'outer_nilpotent': True, 'outer_order': 48, 'outer_permdeg': 9, 'outer_perms': [24, 41064, 46083], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6\\times D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [6, 4], [12, 2]], 'representations': {'PC': {'code': 11348901618847063781, 'gens': [1, 3], 'pres': [5, -2, -2, -2, -2, -7, 10, 1622, 42, 2083, 58, 2404]}, 'GLZN': {'d': 2, 'p': 39, 'gens': [1483000, 2294707, 122887, 2204419, 830480]}, 'Perm': {'d': 15, 'gens': [87661324927, 174835584000, 7983360, 274467916800, 973]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{28}:C_4', 'transitive_degree': 112, 'wreath_data': None, 'wreath_product': False}
-
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': 84, 'aut_gen_orders': [42, 42, 12, 28, 42, 6, 6, 42], 'aut_gens': [[1, 2, 112], [873, 498, 112], [809, 638, 112], [21, 626, 1456], [829, 1314, 112], [45, 694, 112], [89, 890, 560], [785, 870, 336], [821, 1222, 112]], 'aut_group': None, 'aut_hash': 5504141904905321301, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 112896, 'aut_permdeg': 112, 'aut_perms': [146635635622377477874774916800114994233933997486502451657800938310280343621925439348942653703590309300566158037161240350864209174187384035759422297374276745826585216060986631220948901, 12216578774976126564932652975957883253071778389124973531487246156209605967538733926297175211564954647905454011010544475675304255074123190148828524264331110173270195897204851132922109, 142224679711313345245953403081049679219368357644632811807930429860162071024689383756938670597710967895896910120314295495460866834863404458432380461303498601914943099773090626604372822, 109636581761326630741251512549965583147761841973973866789855810681212440457532584589350417290065888805722896185246972766784760173642049188511075243224848266753466759355885728086862584, 122768816167058319366179331592501921363603892766334959131986909155879406615231475347121198684906036078563308468113123939792778211674165849706220808118120064830090196305051927370208425, 88185866814466712110077129569880562100601492917753487214633881437333826271063394276303648724035839202196428461149024237996264719461076506237127957968309324900569910037561634585888175, 32005652792141763975990714859411208340432774760128874803150213811568378242543118161013382914495466152864563391128200969348586176670524868346973458109811532090324182092218417689250076, 56772873256489053373707128814697871475531097990081490571501893314447628603783621787955680228887421671307054223766224605975400208267809793775684579555739133869576751449297435556295844], 'aut_phi_ratio': 168.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 2], [4, 28, 2, 1], [4, 196, 2, 1], [7, 2, 3, 2], [7, 4, 9, 1], [8, 14, 4, 1], [14, 2, 3, 6], [14, 4, 9, 3], [28, 2, 6, 2], [28, 4, 3, 2], [28, 4, 18, 2], [28, 28, 12, 1], [56, 14, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_7^2.C_6^2.C_2^6', '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': 42, 'autcentquo_group': None, 'autcentquo_hash': 8777691769077315760, 'autcentquo_nilpotent': False, 'autcentquo_order': 7056, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_7^2.C_6^2.C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 2], [4, 28, 2], [4, 196, 2], [7, 2, 6], [7, 4, 9], [8, 14, 4], [14, 2, 18], [14, 4, 27], [28, 2, 12], [28, 4, 42], [28, 28, 12], [56, 14, 24]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '392.21', 'commutator_count': 1, 'commutator_label': '196.7', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '7.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 224, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 2, 1, 2], [4, 28, 1, 2], [4, 196, 2, 1], [7, 2, 3, 2], [7, 4, 3, 3], [8, 14, 4, 1], [14, 2, 3, 6], [14, 4, 3, 9], [28, 2, 6, 2], [28, 4, 3, 2], [28, 4, 6, 6], [28, 28, 6, 2], [56, 14, 24, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 56, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '196.9', 'hash': 224, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 28, 'inner_gen_orders': [2, 28, 7], 'inner_gens': [[1, 894, 112], [789, 2, 1456], [1, 226, 112]], 'inner_hash': 21, 'inner_nilpotent': False, 'inner_order': 392, 'inner_split': True, 'inner_tex': 'C_7:D_{28}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 78], [4, 78]], 'label': '1568.224', 'linC_count': 288, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 20, 'linQ_dim': 18, 'linQ_dim_count': 30, 'linR_count': 54, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C14^2.D4', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 29, 'number_characteristic_subgroups': 38, 'number_conjugacy_classes': 164, 'number_divisions': 43, 'number_normal_subgroups': 42, 'number_subgroup_autclasses': 87, 'number_subgroup_classes': 123, 'number_subgroups': 1110, 'old_label': None, 'order': 1568, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 3], [4, 452], [7, 48], [8, 56], [14, 144], [28, 528], [56, 336]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 6, 6], 'outer_gen_pows': [0, 28, 0, 0, 0], 'outer_gens': [[57, 26, 1456], [869, 82, 1456], [785, 82, 112], [1, 78, 1232], [57, 50, 1456]], 'outer_group': '288.1045', 'outer_hash': 1045, 'outer_nilpotent': True, 'outer_order': 288, 'outer_permdeg': 16, 'outer_perms': [5040, 1307674373040, 6227020800, 367923, 39921984], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_6^2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 2], [6, 6], [12, 19], [24, 8]], 'representations': {'PC': {'code': 16890166761833378269068293192079991421186958137, 'gens': [1, 2, 6], 'pres': [7, -2, -2, -2, -2, -7, -2, -7, 392, 12517, 36, 2270, 58, 2915, 80, 3364, 30588, 124, 32941]}, 'Perm': {'d': 26, 'gens': [17427733587721250906480040, 31050624841308874611532809, 47854489245407278866432016, 47854489245407278866432000, 64634537850123552030720000, 397707843456000, 4364760]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{14}^2.D_4', 'transitive_degree': 224, 'wreath_data': None, 'wreath_product': False}
