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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1088.296', 'ambient_counter': 296, 'ambient_order': 1088, 'ambient_tex': 'C_{68}.D_8', 'central': False, 'central_factor': False, 'centralizer_order': 272, 'characteristic': True, 'core_order': 16, 'counter': 37, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1088.296.68.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '68.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '68.4', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 4, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 68, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_{34}', 'simple': False, 'solvable': True, 'special_labels': ['Phi', 'U1'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '16.2', 'subgroup_hash': 2, 'subgroup_order': 16, 'subgroup_tex': 'C_4^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1088.296', 'aut_centralizer_order': 17408, 'aut_label': '68.a1', 'aut_quo_index': 2, 'aut_stab_index': 1, 'aut_weyl_group': '4.2', 'aut_weyl_index': 17408, 'centralizer': '4.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '34.a1.a1', '34.b1.a1', '34.c1.a1'], 'contains': ['136.a1.a1', '136.b1.a1', '136.c1.a1'], 'core': '68.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5850, 7265, 6510, 5819, 6977, 5905, 7834, 4073], 'generators': [828, 816], 'label': '1088.296.68.a1.a1', 'mobius_quo': 0, 'mobius_sub': -34, 'normal_closure': '68.a1.a1', 'normal_contained_in': ['4.a1.a1', '34.a1.a1'], 'normal_contains': ['136.a1.a1', '136.b1.a1', '136.c1.a1'], 'normalizer': '1.a1.a1', 'old_label': '68.a1.a1', 'projective_image': '272.14', 'quotient_action_image': '4.2', 'quotient_action_kernel': '17.1', 'quotient_action_kernel_order': 17, 'quotient_fusion': None, 'short_label': '68.a1.a1', 'subgroup_fusion': None, 'weyl_group': '4.2'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '16.2', '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, 6, 2, 2], 'aut_gens': [[1, 4], [3, 5], [3, 12], [14, 13], [9, 6], [3, 14]], 'aut_group': '96.195', 'aut_hash': 195, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96, 'aut_permdeg': 8, 'aut_perms': [134, 16, 1447, 11520, 5160], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [4, 1, 12, 1]], 'aut_supersolvable': False, 'aut_tex': '\\GL(2,\\mathbb{Z}/4)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '96.195', 'autcent_hash': 195, 'autcent_nilpotent': False, 'autcent_order': 96, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(2,\\mathbb{Z}/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], [4, 1, 12]], 'center_label': '16.2', 'center_order': 16, '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', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['4.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 6]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 4], [1, 4]], '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, 16]], 'label': '16.2', 'linC_count': 48, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 12, 'linQ_dim': 4, 'linQ_dim_count': 12, 'linR_count': 12, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 16, 'number_divisions': 10, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 15, 'number_subgroups': 15, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 12]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 6, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[3, 5], [3, 12], [14, 13], [9, 6], [3, 14]], 'outer_group': '96.195', 'outer_hash': 195, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 8, 'outer_perms': [134, 16, 1447, 11520, 5160], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,\\mathbb{Z}/4)', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [4, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6]], 'representations': {'PC': {'code': 10245, 'gens': [1, 3], 'pres': [4, 2, 2, 2, 2, 8, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16917782, 35931238]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [251, 377]}, 'Perm': {'d': 8, 'gens': [16560, 22, 5160, 7]}}, 'schur_multiplier': [4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 4], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4^2', 'transitive_degree': 16, '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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 272, 'aut_gen_orders': [16, 68, 8, 16, 16, 8, 16, 8], 'aut_gens': [[1, 2, 16], [713, 982, 112], [157, 694, 16], [1009, 746, 240], [721, 722, 656], [249, 526, 1008], [565, 834, 944], [1021, 738, 112], [93, 898, 240]], 'aut_group': None, 'aut_hash': 5835236701334903880, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 69632, 'aut_permdeg': 288, 'aut_perms': 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'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': 272, 'autcentquo_group': '4352.1118910', 'autcentquo_hash': 1118910, 'autcentquo_nilpotent': False, 'autcentquo_order': 4352, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{17}:(C_2^4\\times C_{16})', 'cc_stats': [[1, 1, 1], [2, 1, 3], [4, 2, 4], [4, 4, 1], [4, 136, 2], [8, 4, 4], [8, 68, 4], [17, 2, 8], [34, 2, 24], [68, 2, 32], [68, 4, 32], [136, 4, 64]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '272.14', 'commutator_count': 1, 'commutator_label': '136.9', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '17.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 296, '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, 4], [4, 4, 1, 1], [4, 136, 1, 2], [8, 4, 4, 1], [8, 68, 4, 1], [17, 2, 8, 1], [34, 2, 8, 3], [68, 2, 16, 2], [68, 4, 8, 2], [68, 4, 16, 1], [136, 4, 64, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 136, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 17], 'factors_of_order': [2, 17], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.2', 'frattini_quotient': '68.4', 'hash': 296, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 68, 'inner_gen_orders': [2, 4, 34], 'inner_gens': [[1, 830, 1072], [277, 2, 1072], [33, 34, 16]], 'inner_hash': 14, 'inner_nilpotent': False, 'inner_order': 272, 'inner_split': True, 'inner_tex': 'D_{34}:C_4', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 138], [4, 33]], 'label': '1088.296', 'linC_count': 256, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 24, 'linQ_degree_count': 16, 'linQ_dim': 24, 'linQ_dim_count': 4, 'linR_count': 96, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C68.D8', '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': 22, 'number_characteristic_subgroups': 31, 'number_conjugacy_classes': 179, 'number_divisions': 23, 'number_normal_subgroups': 33, 'number_subgroup_autclasses': 58, 'number_subgroup_classes': 64, 'number_subgroups': 674, 'old_label': None, 'order': 1088, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 3], [4, 284], [8, 288], [17, 16], [34, 48], [68, 192], [136, 256]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 16, 'outer_gen_orders': [2, 2, 2, 2, 16], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[553, 2, 528], [1, 554, 528], [817, 2, 1072], [1, 818, 1072], [1, 2, 656]], 'outer_group': '256.55608', 'outer_hash': 55608, 'outer_nilpotent': True, 'outer_order': 256, 'outer_permdeg': 24, 'outer_perms': [357001369769647, 25852016738884976640000, 51090942171709440000, 121645100408832000, 2803205272625], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4\\times C_{16}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 33, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 5], [16, 2], [32, 5], [64, 3]], 'representations': {'PC': {'code': 1488381084151465884233196904084921646426839734116417551, 'gens': [1, 2, 5], 'pres': [7, -2, -2, -2, -2, -2, -2, -17, 3808, 11621, 36, 254, 58, 37524, 18771, 102, 44357, 22188, 124, 50182, 25101]}, 'Perm': {'d': 33, 'gens': [280664987410134249599667452274857647, 33431007390643814075175283861673647, 568171931010670179676498322571264000, 822301875607863273727030198272000000, 1969256573772393185280000, 1102673843946265328692125788897280000, 23550673235293]}}, '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_{68}.D_8', 'transitive_degree': 1088, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 272, 'aut_gen_orders': [16, 34], 'aut_gens': [[1, 2], [35, 6], [31, 2]], 'aut_group': '544.242', 'aut_hash': 242, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 544, 'aut_permdeg': 19, 'aut_perms': [3007368056298528, 12904176190351327], 'aut_phi_ratio': 17.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 17, 2, 1], [17, 2, 8, 1], [34, 2, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times F_{17}', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 272, 'autcentquo_group': '272.50', 'autcentquo_hash': 50, 'autcentquo_nilpotent': False, 'autcentquo_order': 272, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{17}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 17, 2], [17, 2, 8], [34, 2, 8]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '34.1', 'commutator_count': 1, 'commutator_label': '17.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '17.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [['2.1', 1], ['34.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 17, 1, 2], [17, 2, 8, 1], [34, 2, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 34, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 17], 'factors_of_order': [2, 17], 'faithful_reps': [[2, 1, 8]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '68.4', 'hash': 4, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 34, 'inner_gen_orders': [2, 17], 'inner_gens': [[1, 66], [5, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 34, 'inner_split': False, 'inner_tex': 'D_{17}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 16]], 'label': '68.4', 'linC_count': 8, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 1, 'linQ_dim': 16, 'linQ_dim_count': 1, 'linR_count': 8, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D34', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 20, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 10, 'number_subgroups': 58, 'old_label': None, 'order': 68, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 35], [17, 16], [34, 16]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 8, 'outer_gen_orders': [2, 8], 'outer_gen_pows': [0, 1], 'outer_gens': [[35, 66], [1, 14]], 'outer_group': '16.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 10, 'outer_perms': [362880, 5913], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_8', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [16, 2]], 'representations': {'PC': {'code': 9233948687, 'gens': [1, 2], 'pres': [3, -2, -2, -17, 397, 16, 578]}, 'GLFp': {'d': 2, 'p': 17, 'gens': [4931, 78609, 78624]}, 'Perm': {'d': 19, 'gens': [357001369769646, 1, 7116376445267286]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{34}', 'transitive_degree': 34, 'wreath_data': None, 'wreath_product': False}
