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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1584.122', 'ambient_counter': 122, 'ambient_order': 1584, 'ambient_tex': 'D_{22}.D_{18}', 'central': False, 'central_factor': False, 'centralizer_order': 792, 'characteristic': True, 'core_order': 18, 'counter': 83, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '1584.122.88.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '88.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '88.11', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 11, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 88, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times D_{22}', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '18.2', 'subgroup_hash': 2, 'subgroup_order': 18, 'subgroup_tex': 'C_{18}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1584.122', 'aut_centralizer_order': 7920, 'aut_label': '88.a1', 'aut_quo_index': 24, 'aut_stab_index': 1, 'aut_weyl_group': '6.2', 'aut_weyl_index': 7920, 'centralizer': '2.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['8.a1.a1', '44.a1.a1', '44.b1.a1', '44.c1.a1', '44.d1.a1', '44.e1.a1', '44.f1.a1', '44.g1.a1'], 'contains': ['176.a1.a1', '264.a1.a1'], 'core': '88.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [1417, 3439, 1587, 3075, 2084, 3982, 2142, 4383], 'generators': [4, 1408, 1056], 'label': '1584.122.88.a1.a1', 'mobius_quo': 0, 'mobius_sub': 88, 'normal_closure': '88.a1.a1', 'normal_contained_in': ['8.a1.a1', '44.a1.a1', '44.b1.a1', '44.c1.a1'], 'normal_contains': ['176.a1.a1', '264.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '88.a1.a1', 'projective_image': '792.40', 'quotient_action_image': '2.1', 'quotient_action_kernel': '44.3', 'quotient_action_kernel_order': 44, 'quotient_fusion': None, 'short_label': '88.a1.a1', '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': '18.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 6, 'aut_gen_orders': [6], 'aut_gens': [[1], [11]], 'aut_group': '6.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 6, 'aut_permdeg': 5, 'aut_perms': [27], '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], [9, 1, 6, 1], [18, 1, 6, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_6', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 6, 'autcent_group': '6.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': '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], [3, 1, 2], [6, 1, 2], [9, 1, 6], [18, 1, 6]], 'center_label': '18.2', 'center_order': 18, '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', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [6, 1, 2, 1], [9, 1, 6, 1], [18, 1, 6, 1]], 'element_repr_type': 'PC', 'elementary': 6, 'eulerian_function': 1, 'exponent': 18, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[1, 0, 6]], 'familial': True, 'frattini_label': '3.1', 'frattini_quotient': '6.2', 'hash': 2, 'hyperelementary': 6, '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': 6, 'irrQ_dim': 6, 'irrR_degree': 2, 'irrep_stats': [[1, 18]], 'label': '18.2', 'linC_count': 6, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C18', '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': 18, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 18, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [3, 2], [6, 2], [9, 6], [18, 6]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [6], 'outer_gen_pows': [0], 'outer_gens': [[11]], 'outer_group': '6.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 6, 'outer_permdeg': 5, 'outer_perms': [27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_6', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 9], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 2], [6, 2]], 'representations': {'PC': {'code': 5403, 'gens': [1], 'pres': [3, -2, -3, -3, 6, 22]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [71343819913236817]}, 'GLFp': {'d': 2, 'p': 17, 'gens': [26270]}, 'Perm': {'d': 11, 'gens': [3628800, 357120, 80884]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [18], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{18}', 'transitive_degree': 18, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 990, 'aut_gen_orders': [30, 30, 30, 30, 90, 30], 'aut_gens': [[1, 2, 8], [1441, 6, 924], [581, 534, 760], [1301, 6, 252], [145, 6, 252], [869, 706, 1112], [1297, 534, 1352]], 'aut_group': None, 'aut_hash': 8058055515597410150, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 47520, 'aut_permdeg': 42, 'aut_perms': [869967285960884927124117865756680896309916748164614, 246118559555309919532341778604318213076072297282133, 45892894414582711174678626161428801312118533680005, 45892988921701632351690734665159793058755159242340, 577663689475475672815306654829365258494364235860385, 46201764127634687702165357300842172175832001588947], 'aut_phi_ratio': 99.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 22, 1, 1], [2, 198, 1, 1], [3, 2, 1, 1], [4, 9, 2, 1], [4, 18, 1, 1], [4, 22, 1, 1], [4, 198, 1, 1], [6, 2, 1, 1], [6, 4, 1, 1], [6, 44, 1, 1], [9, 2, 3, 1], [11, 2, 5, 1], [12, 44, 1, 1], [18, 2, 3, 1], [18, 4, 3, 1], [18, 44, 3, 1], [22, 2, 5, 1], [22, 2, 10, 1], [33, 4, 5, 1], [36, 44, 3, 1], [44, 18, 10, 2], [66, 4, 5, 1], [66, 4, 10, 1], [99, 4, 15, 1], [198, 4, 15, 1], [198, 4, 30, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{99}.C_{30}.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 990, 'autcentquo_group': '5940.d', 'autcentquo_hash': 4717361004838527357, 'autcentquo_nilpotent': False, 'autcentquo_order': 5940, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_{99}:C_{30}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 22, 1], [2, 198, 1], [3, 2, 1], [4, 9, 2], [4, 18, 1], [4, 22, 1], [4, 198, 1], [6, 2, 1], [6, 4, 1], [6, 44, 1], [9, 2, 3], [11, 2, 5], [12, 44, 1], [18, 2, 3], [18, 4, 3], [18, 44, 3], [22, 2, 15], [33, 4, 5], [36, 44, 3], [44, 18, 20], [66, 4, 15], [99, 4, 15], [198, 4, 45]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '792.40', 'commutator_count': 1, 'commutator_label': '198.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 122, '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, 1], [2, 22, 1, 1], [2, 198, 1, 1], [3, 2, 1, 1], [4, 9, 2, 1], [4, 18, 1, 1], [4, 22, 1, 1], [4, 198, 1, 1], [6, 2, 1, 1], [6, 4, 1, 1], [6, 44, 1, 1], [9, 2, 3, 1], [11, 2, 5, 1], [12, 44, 1, 1], [18, 2, 3, 1], [18, 4, 3, 1], [18, 44, 3, 1], [22, 2, 5, 1], [22, 2, 10, 1], [33, 4, 5, 1], [36, 44, 3, 1], [44, 18, 10, 2], [66, 4, 5, 1], [66, 4, 10, 1], [99, 4, 15, 1], [198, 4, 15, 1], [198, 4, 30, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24192, 'exponent': 396, 'exponents_of_order': [4, 2, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 11], 'faithful_reps': [[4, 0, 30]], 'familial': False, 'frattini_label': '6.2', 'frattini_quotient': '264.34', 'hash': 122, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 198, 'inner_gen_orders': [2, 2, 198], 'inner_gens': [[1, 2, 876], [1, 2, 712], [725, 882, 8]], 'inner_hash': 40, 'inner_nilpotent': False, 'inner_order': 792, 'inner_split': True, 'inner_tex': 'D_9\\times D_{22}', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 120, 'irrQ_dim': 240, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 58], [4, 84]], 'label': '1584.122', 'linC_count': 270, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 20, 'linQ_degree_count': 32, 'linQ_dim': 20, 'linQ_dim_count': 16, 'linR_count': 120, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D22.D18', '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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 30, 'number_characteristic_subgroups': 41, 'number_conjugacy_classes': 150, 'number_divisions': 30, 'number_normal_subgroups': 41, 'number_subgroup_autclasses': 120, 'number_subgroup_classes': 120, 'number_subgroups': 1848, 'old_label': None, 'order': 1584, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 223], [3, 2], [4, 256], [6, 50], [9, 6], [11, 10], [12, 44], [18, 150], [22, 30], [33, 20], [36, 132], [44, 360], [66, 60], [99, 60], [198, 180]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 30, 'outer_gen_orders': [2, 30], 'outer_gen_pows': [1408, 0], 'outer_gens': [[1, 1410, 872], [1, 6, 808]], 'outer_group': '60.13', 'outer_hash': 13, 'outer_nilpotent': True, 'outer_order': 60, 'outer_permdeg': 12, 'outer_perms': [362880, 39922593], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{30}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 28, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 2], [6, 4], [10, 4], [12, 1], [20, 2], [40, 2], [60, 2], [120, 1]], 'representations': {'PC': {'code': 2407735622346666050835907763435875011745860000119, 'gens': [1, 2, 4], 'pres': [7, -2, -2, -2, 2, -3, -3, -11, 36, 24531, 9978, 80, 5604, 24931, 137, 20165, 23196, 166, 70566]}, 'Perm': {'d': 28, 'gens': [6446835113609647, 868708690492852967966208000, 1445242190515200, 20713823523993600, 12145280938112229701713920000, 4364893, 23454917710790789398855680000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{22}.D_{18}', 'transitive_degree': 792, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 660, 'aut_gen_orders': [20, 30, 2, 22], 'aut_gens': [[1, 2, 4], [45, 15, 76], [44, 11, 69], [1, 46, 4], [1, 67, 4]], 'aut_group': '2640.bv', 'aut_hash': 5844094576270439126, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2640, 'aut_permdeg': 15, 'aut_perms': [667250454382, 125786712251, 7, 838021661423], 'aut_phi_ratio': 66.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 11, 4, 1], [11, 2, 5, 1], [22, 2, 15, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_4\\times F_{11}', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '24.12', 'autcent_hash': 12, 'autcent_nilpotent': False, 'autcent_order': 24, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 110, 'autcentquo_group': '110.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 110, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 11, 4], [11, 2, 5], [22, 2, 15]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '22.1', 'commutator_count': 1, 'commutator_label': '11.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '11.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 11, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['22.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 11, 1, 4], [11, 2, 5, 1], [22, 2, 5, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 84, 'exponent': 22, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 11], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '88.11', 'hash': 11, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 22, 'inner_gen_orders': [1, 2, 11], 'inner_gens': [[1, 2, 4], [1, 2, 84], [1, 10, 4]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 22, 'inner_split': True, 'inner_tex': 'D_{11}', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 20]], 'label': '88.11', 'linC_count': 60, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 11, 'linQ_degree_count': 12, 'linQ_dim': 11, 'linQ_dim_count': 12, 'linR_count': 60, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*D22', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, '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': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 28, 'number_divisions': 12, 'number_normal_subgroups': 21, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 32, 'number_subgroups': 142, 'old_label': None, 'order': 88, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 47], [11, 10], [22, 30]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 60, 'outer_gen_orders': [10, 2, 5, 2, 15], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 47, 20], [1, 3, 4], [1, 2, 12], [44, 47, 41], [44, 47, 61]], 'outer_group': '120.37', 'outer_hash': 37, 'outer_nilpotent': False, 'outer_order': 120, 'outer_permdeg': 9, 'outer_perms': [90753, 131760, 64, 10800, 5850], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_5\\times S_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [10, 4]], 'representations': {'PC': {'code': 361356200009, 'gens': [1, 2, 3], 'pres': [4, -2, -2, -2, -11, 510, 34, 647]}, 'GLFp': {'d': 3, 'p': 11, 'gens': [922111610, 2296050935, 214518602, 2143735230]}, 'Perm': {'d': 15, 'gens': [6267305647, 7, 16, 99712902360]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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_2\\times D_{22}', 'transitive_degree': 44, 'wreath_data': None, 'wreath_product': False}
