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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1344.11566', 'ambient_counter': 11566, 'ambient_order': 1344, 'ambient_tex': 'C_{84}.C_2^4', 'central': False, 'central_factor': False, 'centralizer_order': 2, 'characteristic': False, 'core_order': 96, 'counter': 59, 'cyclic': False, 'direct': None, 'hall': 6, 'label': '1344.11566.7.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '7.a1', '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': 7, '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': '192.1527', 'subgroup_hash': 1527, 'subgroup_order': 192, 'subgroup_tex': 'C_{12}.C_2^4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.11566', 'aut_centralizer_order': 6, 'aut_label': '7.a1', 'aut_quo_index': None, 'aut_stab_index': 7, 'aut_weyl_group': None, 'aut_weyl_index': 42, 'centralizer': '672.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1'], 'contains': ['14.a1', '14.b1', '14.c1', '14.d1', '14.e1', '14.f1', '14.g1', '14.h1', '14.i1', '14.j1', '14.k1', '21.a1'], 'core': '14.a1', 'coset_action_label': None, 'count': 7, 'diagramx': [5043, -1, 7727, -1], 'generators': [1, 16, 672, 2, 448, 4, 8], 'label': '1344.11566.7.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '7.a1', 'old_label': '7.a1', 'projective_image': '672.1271', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '7.a1', 'subgroup_fusion': None, 'weyl_group': '96.230'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': None, 'aut_cyclic': None, 'aut_derived_length': None, 'aut_exponent': None, 'aut_gen_orders': None, 'aut_gens': [[1, 2, 4, 8, 16], [97, 2, 4, 8, 80], [1, 98, 4, 8, 80], [1, 2, 100, 8, 80], [1, 2, 4, 104, 80], [1, 2, 4, 8, 176], [1, 2, 52, 56, 80], [1, 147, 53, 57, 80], [107, 2, 4, 8, 82], [1, 2, 4, 8, 80], [1, 2, 68, 8, 16]], 'aut_group': '23040.y', 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 23040, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': None, 'aut_solvable': None, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 5, 1], [2, 6, 10, 1], [3, 2, 1, 1], [4, 2, 10, 1], [4, 3, 2, 1], [4, 6, 5, 1], [6, 2, 1, 1], [6, 4, 5, 1], [12, 4, 10, 1]], 'aut_supersolvable': None, 'aut_tex': 'C_2\\wr S_5\\times S_3', 'autcent_abelian': None, 'autcent_cyclic': None, 'autcent_exponent': None, 'autcent_group': None, 'autcent_hash': None, 'autcent_nilpotent': None, 'autcent_order': None, 'autcent_solvable': None, 'autcent_split': None, 'autcent_supersolvable': None, 'autcent_tex': None, 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': None, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 5], [2, 6, 10], [3, 2, 1], [4, 2, 10], [4, 3, 2], [4, 6, 5], [6, 2, 1], [6, 4, 5], [12, 4, 10]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '96.230', 'commutator_count': 1, 'commutator_label': '6.2', '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'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1527, '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, 6, 1, 10], [3, 2, 1, 1], [4, 2, 1, 10], [4, 3, 2, 1], [4, 6, 1, 5], [6, 2, 1, 1], [6, 4, 1, 5], [12, 4, 1, 10]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3333120, 'exponent': 12, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 1, 1]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '96.230', 'hash': 1527, 'hyperelementary': 2, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': None, 'inner_gens': None, 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': 96, 'inner_split': None, 'inner_tex': 'C_2^3\\times D_6', 'inner_used': None, 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 16, 'irrR_degree': 8, 'irrep_stats': [[1, 32], [2, 16], [4, 2], [8, 1]], 'label': '192.1527', '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': 'C12.C2^4', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 11, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 51, 'number_divisions': 50, 'number_normal_subgroups': 443, 'number_subgroup_autclasses': 72, 'number_subgroup_classes': 810, 'number_subgroups': 1704, 'old_label': None, 'order': 192, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 71], [3, 2], [4, 56], [6, 22], [12, 40]], 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': True, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': '240.189', 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 240, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': 'C_2\\times S_5', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 16], [8, 2]], 'representations': {'PC': {'code': 661849925503755701677210625, 'gens': [1, 2, 3, 4, 5], 'pres': [7, -2, -2, -2, -2, -2, -2, -3, 1373, 360, 1558, 515, 102, 1699, 124, 1588]}, 'Perm': {'d': 19, 'gens': [6514014129696001, 14949610261689720, 21892222364197704, 28609347327856080, 33151571127019440, 41838649437896640, 3]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 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_{12}.C_2^4', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': None, 'aut_cyclic': None, 'aut_derived_length': None, 'aut_exponent': None, 'aut_gen_orders': None, 'aut_gens': [[1, 2, 4, 8, 32], [1, 2, 4, 8, 34], [17, 2, 4, 8, 32], [1, 2, 28, 8, 32], [1, 2, 4, 24, 1312], [1, 2, 4, 8, 1312], [897, 2, 900, 8, 928], [1, 2, 4, 8, 1184], [1, 2, 20, 8, 32], [683, 696, 686, 8, 34], [897, 2, 900, 8, 32], [1, 2, 388, 8, 32], [17, 18, 20, 8, 32], [1, 2, 4, 8, 48]], 'aut_group': None, 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 96768, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': None, 'aut_solvable': None, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 3, 1], [2, 6, 1, 1], [2, 6, 3, 1], [2, 14, 2, 1], [2, 42, 6, 1], [3, 2, 1, 1], [4, 2, 1, 1], [4, 2, 3, 1], [4, 3, 2, 1], [4, 6, 3, 1], [4, 14, 6, 1], [4, 42, 2, 1], [6, 2, 1, 1], [6, 4, 3, 1], [6, 28, 2, 1], [7, 2, 3, 1], [12, 4, 1, 1], [12, 4, 3, 1], [12, 28, 6, 1], [14, 2, 3, 1], [14, 4, 9, 1], [14, 6, 6, 1], [14, 12, 9, 1], [21, 4, 3, 1], [28, 2, 6, 1], [28, 4, 9, 1], [28, 6, 6, 1], [28, 12, 9, 1], [42, 4, 3, 1], [42, 8, 9, 1], [84, 4, 6, 1], [84, 8, 9, 1]], 'aut_supersolvable': None, 'aut_tex': None, 'autcent_abelian': None, 'autcent_cyclic': None, 'autcent_exponent': None, 'autcent_group': None, 'autcent_hash': None, 'autcent_nilpotent': None, 'autcent_order': None, 'autcent_solvable': None, 'autcent_split': None, 'autcent_supersolvable': None, 'autcent_tex': None, 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': None, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 3], [2, 6, 4], [2, 14, 2], [2, 42, 6], [3, 2, 1], [4, 2, 4], [4, 3, 2], [4, 6, 3], [4, 14, 6], [4, 42, 2], [6, 2, 1], [6, 4, 3], [6, 28, 2], [7, 2, 3], [12, 4, 4], [12, 28, 6], [14, 2, 3], [14, 4, 9], [14, 6, 6], [14, 12, 9], [21, 4, 3], [28, 2, 6], [28, 4, 9], [28, 6, 6], [28, 12, 9], [42, 4, 3], [42, 8, 9], [84, 4, 6], [84, 8, 9]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '672.1271', 'commutator_count': 1, 'commutator_label': '42.6', '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', '7.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 11566, '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, 3], [2, 6, 1, 4], [2, 14, 1, 2], [2, 42, 1, 6], [3, 2, 1, 1], [4, 2, 1, 4], [4, 3, 2, 1], [4, 6, 1, 3], [4, 14, 1, 6], [4, 42, 1, 2], [6, 2, 1, 1], [6, 4, 1, 3], [6, 28, 1, 2], [7, 2, 3, 1], [12, 4, 1, 4], [12, 28, 1, 6], [14, 2, 3, 1], [14, 4, 3, 3], [14, 6, 6, 1], [14, 12, 3, 3], [21, 4, 3, 1], [28, 2, 6, 1], [28, 4, 3, 3], [28, 6, 6, 1], [28, 12, 3, 3], [42, 4, 3, 1], [42, 8, 3, 3], [84, 4, 6, 1], [84, 8, 3, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 13332480000, 'exponent': 84, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[8, 1, 6]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '672.1271', 'hash': 11566, 'hyperelementary': 2, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': None, 'inner_gens': None, 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': 672, 'inner_split': None, 'inner_tex': 'C_{21}:C_2^5', 'inner_used': None, 'irrC_degree': 8, 'irrQ_degree': 48, 'irrQ_dim': 96, 'irrR_degree': 8, 'irrep_stats': [[1, 32], [2, 64], [4, 38], [8, 7]], 'label': '1344.11566', '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': 'C84.C2^4', '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, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 34, 'number_characteristic_subgroups': 40, 'number_conjugacy_classes': 141, 'number_divisions': 76, 'number_normal_subgroups': 528, 'number_subgroup_autclasses': 404, 'number_subgroup_classes': 1620, 'number_subgroups': 11652, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 311], [3, 2], [4, 200], [6, 70], [7, 6], [12, 184], [14, 186], [21, 12], [28, 192], [42, 84], [84, 96]], 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': True, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': '144.195', 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 144, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': 'C_6^2:C_2^2', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 26, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 16], [6, 16], [8, 2], [12, 8], [48, 2]], 'representations': {'PC': {'code': 5559847065097708954379003499261975844813588965899, 'gens': [1, 2, 3, 4, 6], 'pres': [8, -2, -2, -2, -2, -2, 2, -3, -7, 137, 211, 91, 45317, 1165, 15765, 141, 28678, 17942, 222, 18455]}, 'Perm': {'d': 26, 'gens': [621574835364366654960196, 358828940499169, 4579447041886, 5987254304875, 6767426431180, 8698611191114, 6758061133824000, 16754357281155028254720000]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 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_{84}.C_2^4', 'transitive_degree': 336, 'wreath_data': None, 'wreath_product': False}
