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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'ambient': '1272.44', 'ambient_counter': 44, 'ambient_order': 1272, 'ambient_tex': 'C_2^2\\times C_{318}', 'central': True, 'central_factor': False, 'centralizer_order': 1272, 'characteristic': False, 'core_order': 4, 'counter': 49, 'cyclic': False, 'direct': True, 'hall': 0, 'label': '1272.44.318.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '318.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '318.4', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 4, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 318, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{318}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '4.2', 'subgroup_hash': 2, 'subgroup_order': 4, 'subgroup_tex': 'C_2^2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1272.44', 'aut_centralizer_order': 416, 'aut_label': '318.a1', 'aut_quo_index': 1, 'aut_stab_index': 7, 'aut_weyl_group': '6.1', 'aut_weyl_index': 2912, 'centralizer': '1.a1.a1', 'complements': ['4.a1.d1', '4.a1.f1', '4.a1.e1', '4.a1.g1'], 'conjugacy_class_count': 1, 'contained_in': ['6.a1.a1', '106.a1.a1', '159.a1.a1'], 'contains': ['636.a1.a1', '636.a1.b1', '636.a1.c1'], 'core': '318.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5841, 5841, 1311, 1311, 5857, 5857, 6600, 6600], 'generators': [2, 636], 'label': '1272.44.318.a1.a1', 'mobius_quo': 2, 'mobius_sub': -1, 'normal_closure': '318.a1.a1', 'normal_contained_in': ['6.a1.a1', '106.a1.a1', '159.a1.a1'], 'normal_contains': ['636.a1.a1', '636.a1.b1', '636.a1.c1'], 'normalizer': '1.a1.a1', 'old_label': '318.a1.a1', 'projective_image': '318.4', 'quotient_action_image': '1.1', 'quotient_action_kernel': '318.4', 'quotient_action_kernel_order': 318, 'quotient_fusion': None, 'short_label': '318.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', '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]], 'center_label': '4.2', 'center_order': 4, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', '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, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, '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': '1272.44', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 1092, 'aut_gen_orders': [52, 156, 156], 'aut_gens': [[1, 2, 4], [2, 639, 33], [639, 1, 868], [638, 2, 1133]], 'aut_group': None, 'aut_hash': 3608738247080918090, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 17472, 'aut_permdeg': 61, 'aut_perms': [8330510939055884734014587055928673360624068904059230817875450325301042761262820930, 11956875951866986840807359489530594030581494055562767747590134578544640000000000, 14185856751185061972137317692185647773406973955756196078937712139173888000000000], 'aut_phi_ratio': 42.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 7, 1], [3, 1, 2, 1], [6, 1, 14, 1], [53, 1, 52, 1], [106, 1, 364, 1], [159, 1, 104, 1], [318, 1, 728, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times C_{52}\\times \\PSL(2,7)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 1092, 'autcent_group': None, 'autcent_hash': 3608738247080918090, 'autcent_nilpotent': False, 'autcent_order': 17472, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2\\times C_{52}\\times \\PSL(2,7)', '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, 7], [3, 1, 2], [6, 1, 14], [53, 1, 52], [106, 1, 364], [159, 1, 104], [318, 1, 728]], 'center_label': '1272.44', 'center_order': 1272, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '53.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 44, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['3.1', 1], ['53.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 1, 2, 1], [6, 1, 2, 7], [53, 1, 52, 1], [106, 1, 52, 7], [159, 1, 104, 1], [318, 1, 104, 7]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 37219, 'exponent': 318, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 7, 13], 'factors_of_order': [2, 3, 53], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1272.44', 'hash': 44, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1], 'inner_gens': [[1, 2, 4], [1, 2, 4], [1, 2, 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': None, 'irrep_stats': [[1, 1272]], 'label': '1272.44', 'linC_count': None, 'linC_degree': 3, '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': 'C2^2*C318', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 1272, 'number_divisions': 32, 'number_normal_subgroups': 64, 'number_subgroup_autclasses': 16, 'number_subgroup_classes': 64, 'number_subgroups': 64, 'old_label': None, 'order': 1272, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 7], [3, 2], [6, 14], [53, 52], [106, 364], [159, 104], [318, 728]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 1092, 'outer_gen_orders': [78, 52, 156], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[638, 3, 68], [1, 637, 570], [2, 639, 1052]], 'outer_group': None, 'outer_hash': 3608738247080918090, 'outer_nilpotent': False, 'outer_order': 17472, 'outer_permdeg': 61, 'outer_perms': [8335213509108443883278347184270559296040006115289910084110657405444777204205312000, 41948921012327764754023682774071346820823195960210107635612337674649600000000, 8330512399373152073746916328670880705337733446380174682609302575903801737216000000], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2\\times C_{52}\\times \\PSL(2,7)', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 62, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3, 53], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 8], [52, 8], [104, 8]], 'representations': {'PC': {'code': 8253777852330087, 'gens': [1, 2, 3], 'pres': [5, -2, -2, -2, -3, -53, 42, 78]}, 'Perm': {'d': 62, 'gens': [507580213877224798800856812176625227226004528988036003099405939480985600000000000000, 138683118545689835737939019720389406345902876772687432540821294940160000000000000, 40526919504877216755680601905432322134980384796226602145184481280000000000000, 25392806707316551851930201695133033919160642102898873524551680000000000000, 4194225108889081685726353116532995882715054282925930446848000000000000]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 318], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times C_{318}', 'transitive_degree': 1272, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '318.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 52, 'aut_gen_orders': [2, 52], 'aut_gens': [[1], [107], [79]], 'aut_group': '104.9', 'aut_hash': 9, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 104, 'aut_permdeg': 54, 'aut_perms': [414809920819603055174400, 102389954100451748626002193949037214374350251018278997074033567141802595], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1], [53, 1, 52, 1], [106, 1, 52, 1], [159, 1, 104, 1], [318, 1, 104, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times C_{52}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 52, 'autcent_group': '104.9', 'autcent_hash': 9, 'autcent_nilpotent': True, 'autcent_order': 104, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2\\times C_{52}', '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], [3, 1, 2], [6, 1, 2], [53, 1, 52], [106, 1, 52], [159, 1, 104], [318, 1, 104]], 'center_label': '318.4', 'center_order': 318, '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', '3.1', '53.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 4, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['3.1', 1], ['53.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1], [53, 1, 52, 1], [106, 1, 52, 1], [159, 1, 104, 1], [318, 1, 104, 1]], 'element_repr_type': 'PC', 'elementary': 318, 'eulerian_function': 1, 'exponent': 318, 'exponents_of_order': [1, 1, 1], 'factors_of_aut_order': [2, 13], 'factors_of_order': [2, 3, 53], 'faithful_reps': [[1, 0, 104]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '318.4', 'hash': 4, 'hyperelementary': 318, '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': 104, 'irrQ_dim': 104, 'irrR_degree': 2, 'irrep_stats': [[1, 318]], 'label': '318.4', 'linC_count': 104, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 54, 'linQ_degree_count': 3, 'linQ_dim': 54, 'linQ_dim_count': 3, 'linR_count': 52, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C318', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 8, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 318, 'number_divisions': 8, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 318, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 1], [3, 2], [6, 2], [53, 52], [106, 52], [159, 104], [318, 104]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 52, 'outer_gen_orders': [2, 52], 'outer_gen_pows': [0, 0], 'outer_gens': [[107], [79]], 'outer_group': '104.9', 'outer_hash': 9, 'outer_nilpotent': True, 'outer_order': 104, 'outer_permdeg': 54, 'outer_perms': [414809920819603055174400, 102389954100451748626002193949037214374350251018278997074033567141802595], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{52}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 58, 'pgroup': 0, 'primary_abelian_invariants': [2, 3, 53], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2], [52, 2], [104, 2]], 'representations': {'PC': {'code': 1669720103, 'gens': [1], 'pres': [3, -2, -3, -53, 6, 22]}, 'GLFp': {'d': 2, 'p': 107, 'gens': [101456840]}, 'Perm': {'d': 58, 'gens': [40526919504877216755680601905432322134980384796226602145184481280000000000000, 25392806707316551851930201695133033919160642102898873524551680000000000000, 4194225108889081685726353116532995882715054282925930446848000000000000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [318], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{318}', 'transitive_degree': 318, 'wreath_data': None, 'wreath_product': False}
