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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '336.190', 'ambient_counter': 190, 'ambient_order': 336, 'ambient_tex': 'C_{28}.D_6', 'central': False, 'central_factor': False, 'centralizer_order': 56, 'characteristic': False, 'core_order': 14, 'counter': 34, 'cyclic': True, 'direct': None, 'hall': 0, 'label': '336.190.12.c1.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '12.c1.c1', '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': 12, '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': '28.2', 'subgroup_hash': 2, 'subgroup_order': 28, 'subgroup_tex': 'C_{28}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '336.190', 'aut_centralizer_order': 16, 'aut_label': '12.c1', 'aut_quo_index': None, 'aut_stab_index': 9, 'aut_weyl_group': '12.5', 'aut_weyl_index': 144, 'centralizer': '6.b1.c1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.c1.c1', '6.b1.c1', '6.c1.a1', '6.c1.b1'], 'contains': ['24.a1.a1', '84.c1.c1'], 'core': '24.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [4206, -1, 4627, -1, 3461, -1, 6274, -1], 'generators': [31, 168, 48], 'label': '336.190.12.c1.c1', 'mobius_quo': None, 'mobius_sub': -2, 'normal_closure': '4.c1.c1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '12.c1.c1', 'projective_image': '24.14', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '12.c1.c1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '28.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1], [13], [11]], 'aut_group': '12.5', 'aut_hash': 5, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 12, 'aut_permdeg': 7, 'aut_perms': [24, 723], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [7, 1, 6, 1], [14, 1, 6, 1], [28, 1, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '12.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 12, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_6', '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], [7, 1, 6], [14, 1, 6], [28, 1, 12]], 'center_label': '28.2', 'center_order': 28, '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', '7.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['4.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1], [7, 1, 6, 1], [14, 1, 6, 1], [28, 1, 12, 1]], 'element_repr_type': 'PC', 'elementary': 14, 'eulerian_function': 1, 'exponent': 28, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 7], 'faithful_reps': [[1, 0, 12]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '14.2', 'hash': 2, 'hyperelementary': 14, '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': 12, 'irrQ_dim': 12, 'irrR_degree': 2, 'irrep_stats': [[1, 28]], 'label': '28.2', 'linC_count': 12, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 2, 'linQ_dim': 8, 'linQ_dim_count': 2, 'linR_count': 6, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C28', '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': 28, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 28, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [4, 2], [7, 6], [14, 6], [28, 12]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[13], [11]], 'outer_group': '12.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 12, 'outer_permdeg': 7, 'outer_perms': [24, 723], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_6', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [4, 7], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1], [6, 2], [12, 1]], 'representations': {'PC': {'code': 242573, 'gens': [1], 'pres': [3, -2, -2, -7, 6, 16]}, 'GLFp': {'d': 2, 'p': 13, 'gens': [14163]}, 'Perm': {'d': 11, 'gens': [11612160, 4320, 3669120]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [28], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{28}', 'transitive_degree': 28, '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': '56.13', '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, 2, 6, 6, 6], 'aut_gens': [[1, 2, 4], [1, 170, 4], [169, 2, 173], [1, 2, 52], [252, 2, 117], [1, 2, 316], [169, 226, 4]], 'aut_group': '1728.47893', 'aut_hash': 47893, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1728, 'aut_permdeg': 18, 'aut_perms': [3591061671134880, 210997893662640, 1480550423, 3266751728958600, 285487949093, 4513085228620080], 'aut_phi_ratio': 18.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [3, 2, 1, 1], [4, 2, 3, 1], [4, 6, 3, 1], [6, 2, 1, 1], [7, 1, 6, 1], [12, 4, 3, 1], [14, 1, 6, 1], [14, 3, 12, 1], [21, 2, 6, 1], [28, 2, 18, 1], [28, 6, 18, 1], [42, 2, 6, 1], [84, 4, 18, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6\\times D_6\\times S_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '48.52', 'autcent_hash': 52, 'autcent_nilpotent': True, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\times C_6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '36.10', 'autcentquo_hash': 10, 'autcentquo_nilpotent': False, 'autcentquo_order': 36, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [3, 2, 1], [4, 2, 3], [4, 6, 3], [6, 2, 1], [7, 1, 6], [12, 4, 3], [14, 1, 6], [14, 3, 12], [21, 2, 6], [28, 2, 18], [28, 6, 18], [42, 2, 6], [84, 4, 18]], 'center_label': '14.2', 'center_order': 14, 'central_product': True, 'central_quotient': '24.14', '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', '3.1', '7.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 190, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['6.1', 1], ['7.1', 1], ['8.4', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 2, 1, 1], [4, 2, 1, 3], [4, 6, 1, 3], [6, 2, 1, 1], [7, 1, 6, 1], [12, 4, 1, 3], [14, 1, 6, 1], [14, 3, 6, 2], [21, 2, 6, 1], [28, 2, 6, 3], [28, 6, 6, 3], [42, 2, 6, 1], [84, 4, 6, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6384, 'exponent': 84, 'exponents_of_order': [4, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[4, 0, 6]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '168.55', 'hash': 190, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 6], 'inner_gens': [[1, 2, 172], [1, 2, 116], [169, 226, 4]], 'inner_hash': 14, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': True, 'inner_tex': 'C_2\\times D_6', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 24, 'irrQ_dim': 48, 'irrR_degree': 8, 'irrep_stats': [[1, 56], [2, 42], [4, 7]], 'label': '336.190', 'linC_count': 390, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 72, 'linQ_dim': 12, 'linQ_dim_count': 64, 'linR_count': 24, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C28.D6', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 16, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 105, 'number_divisions': 30, 'number_normal_subgroups': 50, 'number_subgroup_autclasses': 40, 'number_subgroup_classes': 76, 'number_subgroups': 128, 'old_label': None, 'order': 336, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 24], [6, 2], [7, 6], [12, 12], [14, 42], [21, 12], [28, 144], [42, 12], [84, 72]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6, 6], 'outer_gen_pows': [0, 1, 0], 'outer_gens': [[1, 170, 4], [1, 2, 149], [84, 2, 221]], 'outer_group': '72.48', 'outer_hash': 48, 'outer_nilpotent': False, 'outer_order': 72, 'outer_permdeg': 10, 'outer_perms': [367920, 1174465, 806403], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6\\times D_6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 7], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 1], [6, 8], [12, 6], [24, 1]], 'representations': {'PC': {'code': 214144119478392332617688102219, 'gens': [1, 2, 3], 'pres': [6, -2, -2, -2, -2, -3, -7, 1008, 3098, 1052, 50, 2793, 69, 1930, 118]}, 'GLZN': {'d': 2, 'p': 42, 'gens': [136445, 2173837, 3088189, 2148581, 74341, 1667863]}, 'Perm': {'d': 18, 'gens': [362880, 7039, 12593, 378011776665600, 18619, 3991680]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 14], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{28}.D_6', 'transitive_degree': 168, 'wreath_data': None, 'wreath_product': False}
