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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1248.1158', 'ambient_counter': 1158, 'ambient_order': 1248, 'ambient_tex': 'C_2\\times C_4\\times F_{13}', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 156, 'counter': 16, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1248.1158.8.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '8.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '8.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times C_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '156.8', 'subgroup_hash': 8, 'subgroup_order': 156, 'subgroup_tex': 'C_{26}:C_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1248.1158', 'aut_centralizer_order': 16, 'aut_label': '8.a1', 'aut_quo_index': 2, 'aut_stab_index': 4, 'aut_weyl_group': '156.7', 'aut_weyl_index': 64, 'centralizer': '156.a1', 'complements': [], 'conjugacy_class_count': 4, 'contained_in': ['4.a1', '4.b1'], 'contains': ['16.a1', '16.c1', '16.d1', '24.a1', '104.a1'], 'core': '8.a1', 'coset_action_label': None, 'count': 4, 'diagramx': [7701, 8656, 7380, 8501], 'generators': [26, 16, 96, 624], 'label': '1248.1158.8.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '8.a1', 'normal_contained_in': ['4.a1', '4.b1'], 'normal_contains': ['16.a1', '16.c1', '16.d1', '24.a1'], 'normalizer': '1.a1', 'old_label': '8.a1', 'projective_image': '624.119', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.a1', 'subgroup_fusion': None, 'weyl_group': '156.7'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 156, 'aut_gen_orders': [12, 26], 'aut_gens': [[1, 6], [79, 42], [55, 6]], 'aut_group': '312.45', 'aut_hash': 45, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 312, 'aut_permdeg': 15, 'aut_perms': [73287406224, 175337097121], 'aut_phi_ratio': 6.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 13, 2, 1], [3, 13, 1, 2], [6, 13, 1, 2], [6, 13, 2, 2], [13, 6, 2, 1], [26, 6, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times F_{13}', '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': 156, 'autcentquo_group': '156.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 156, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{13}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 13, 2], [3, 13, 2], [6, 13, 6], [13, 6, 2], [26, 6, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '78.1', 'commutator_count': 1, 'commutator_label': '13.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '13.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 8, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['78.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 13, 1, 2], [3, 13, 2, 1], [6, 13, 2, 3], [13, 6, 2, 1], [26, 6, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 78, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3, 13], 'factors_of_order': [2, 3, 13], 'faithful_reps': [[6, 1, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '156.8', 'hash': 8, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 78, 'inner_gen_orders': [6, 13], 'inner_gens': [[1, 138], [25, 6]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 78, 'inner_split': False, 'inner_tex': 'C_{13}:C_6', 'inner_used': [1, 2], 'irrC_degree': 6, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 6, 'irrep_stats': [[1, 12], [6, 4]], 'label': '156.8', 'linC_count': 2, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 1, 'linQ_dim': 12, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C26:C6', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 16, 'number_divisions': 10, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 20, 'number_subgroups': 116, 'old_label': None, 'order': 156, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 27], [3, 26], [6, 78], [13, 12], [26, 12]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [5, 0], 'outer_gens': [[1, 90], [79, 150]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [12, 2]], 'representations': {'PC': {'code': 7872593661863232923, 'gens': [1, 3], 'pres': [4, -2, -3, -2, -13, 8, 1658, 330, 34, 1923, 871]}, 'GLFp': {'d': 2, 'p': 13, 'gens': [2211, 2201, 26376]}, 'Perm': {'d': 15, 'gens': [6321455568, 1, 14030031846, 106023793950]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{26}:C_6', 'transitive_degree': 26, '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': '96.161', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 156, 'aut_gen_orders': [2, 4, 4, 12, 26], 'aut_gens': [[1, 4, 48], [3, 4, 1202], [625, 4, 50], [1, 631, 48], [1, 4, 336], [1, 246, 48]], 'aut_group': '9984.a', 'aut_hash': 4147733784568435005, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 9984, 'aut_permdeg': 104, 'aut_perms': [1193346723948723629725742238969269137498030947446840538311237855230665889417874169896115854870600347379614307360129566232407471146493226304064644639635535883836060117, 47074679923585432489927681321654741430197944979563338134596240734026021820749523567647110920729074340037575181381108447561827906003499017190621222713866136239821794, 4889193544814476640477878192304876256312496574270668196710285770390662167844420903228749108032074090385232433597160085354558549665593167909093120255430631701236945128, 6399560433937448009807723402560866662531598972487406953632649211961383997753177427327334822202757092898033859665949183199355139727685198263380220462843456847313714208, 1000799378400333581491010803884701802704606022496734119647050485733929449528157185003947362415734704102624030160017345566829900050572286763725950779795265523302680896], 'aut_phi_ratio': 26.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 13, 2, 2], [3, 13, 1, 2], [4, 1, 4, 1], [4, 13, 4, 1], [4, 13, 8, 2], [6, 13, 1, 2], [6, 13, 2, 6], [12, 13, 4, 4], [12, 13, 8, 4], [13, 12, 1, 1], [26, 12, 1, 1], [26, 12, 2, 1], [52, 12, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\wr C_2^2\\times F_{13}', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '64.138', 'autcent_hash': 138, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\wr C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 156, 'autcentquo_group': '156.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 156, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{13}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 13, 4], [3, 13, 2], [4, 1, 4], [4, 13, 20], [6, 13, 14], [12, 13, 48], [13, 12, 1], [26, 12, 3], [52, 12, 4]], 'center_label': '8.2', 'center_order': 8, 'central_product': True, 'central_quotient': '156.7', 'commutator_count': 1, 'commutator_label': '13.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '13.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1158, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['156.7', 1], ['2.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 13, 1, 4], [3, 13, 2, 1], [4, 1, 2, 2], [4, 13, 2, 10], [6, 13, 2, 7], [12, 13, 4, 12], [13, 12, 1, 1], [26, 12, 1, 3], [52, 12, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 61152, 'exponent': 156, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 3, 13], 'factors_of_order': [2, 3, 13], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '624.239', 'hash': 1158, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 156, 'inner_gen_orders': [1, 12, 13], 'inner_gens': [[1, 4, 48], [1, 4, 720], [1, 580, 48]], 'inner_hash': 7, 'inner_nilpotent': False, 'inner_order': 156, 'inner_split': True, 'inner_tex': 'F_{13}', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 96], [12, 8]], 'label': '1248.1158', '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': 'C2*C4*F13', '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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 31, 'number_characteristic_subgroups': 24, 'number_conjugacy_classes': 104, 'number_divisions': 46, 'number_normal_subgroups': 116, 'number_subgroup_autclasses': 92, 'number_subgroup_classes': 216, 'number_subgroups': 1416, 'old_label': None, 'order': 1248, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 55], [3, 26], [4, 264], [6, 182], [12, 624], [13, 12], [26, 36], [52, 48]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0], 'outer_gens': [[625, 4, 48], [1, 6, 50], [3, 5, 1200], [3, 4, 1200], [1, 628, 1200], [1, 6, 1200]], 'outer_group': '64.138', 'outer_hash': 138, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 8, 'outer_perms': [5183, 5046, 23616, 7, 11536, 5167], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\wr C_2^2', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 4, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 20], [4, 12], [12, 4], [24, 2]], 'representations': {'PC': {'code': 70666608603136569130229859322274228876329115, 'gens': [1, 3, 6], 'pres': [7, -2, -2, -2, -2, -3, -2, -13, 14, 58, 80, 7579, 4310, 411, 124, 2372, 2379, 916]}, 'GLFp': {'d': 3, 'p': 13, 'gens': [10376931726, 10564211863, 9615033913, 4606310026, 9789111396, 1171582646, 3893903330]}, 'Perm': {'d': 19, 'gens': [401737695937368, 1, 246, 802067782844160, 1179562530666654, 414, 7876147837824000]}}, 'schur_multiplier': [2, 2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4, 12], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_4\\times F_{13}', 'transitive_degree': 104, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 3], [5, 3]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [1, 22], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.3', 'autcent_hash': 3, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_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, 4]], 'center_label': '8.2', 'center_order': 8, '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'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '2.1', '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, 2], [1, 2]], '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, 8]], 'label': '8.2', 'linC_count': 12, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 4, 'linQ_dim': 3, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C4', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 8, 'number_divisions': 6, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 4]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 3], [5, 3]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [1, 22], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2]], 'representations': {'PC': {'code': 34, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16426]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [251, 504]}, 'Perm': {'d': 6, 'gens': [22, 120, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_4', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}
