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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '2880.hl', 'ambient_counter': 194, 'ambient_order': 2880, 'ambient_tex': 'S_3\\times F_5\\times S_4', 'central': False, 'central_factor': False, 'centralizer_order': 240, 'characteristic': False, 'core_order': 4, 'counter': 862, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '2880.hl.360.bm1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '360.bm1.a1', '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': 360, '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': '8.3', 'subgroup_hash': 3, 'subgroup_order': 8, 'subgroup_tex': 'D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '2880.hl', 'aut_centralizer_order': 240, 'aut_label': '360.bm1', 'aut_quo_index': None, 'aut_stab_index': 3, 'aut_weyl_group': '4.2', 'aut_weyl_index': 720, 'centralizer': '12.h1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['72.bm1.a1', '120.b1.a1', '120.bq1.a1', '180.bh1.a1', '180.bj1.a1', '180.bp1.a1'], 'contains': ['720.a1.a1', '720.d1.a1', '720.l1.a1'], 'core': '720.a1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [1140, -1, 1130, -1, 1035, -1, 2664, -1], 'generators': [23, 5, 7], 'label': '2880.hl.360.bm1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '120.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.b1.a1', 'old_label': '360.bm1.a1', 'projective_image': '2880.hl', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '360.bm1.a1', 'subgroup_fusion': None, 'weyl_group': '4.2'}
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gps_groups • Show schema
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{'Agroup': False, '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': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 6], [3, 2]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [5, 9], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [4, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [4, 2, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', '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': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [4, 2, 1, 1]], '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': [[2, 1, 1]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 6], [5, 2]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 1]], 'label': '8.3', 'linC_count': 1, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D4', 'ngens': 2, 'nilpotency_class': 2, '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': 5, 'number_divisions': 5, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 10, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 5], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [2], 'outer_gens': [[3, 2]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', '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], [2, 1]], 'representations': {'PC': {'code': 294, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 37, 16]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [29, 56, 24], 'family': 'COPlus'}, {'d': 1, 'q': 4, 'gens': [7, 16, 1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [12, 55, 56]}, 'Perm': {'d': 4, 'gens': [6, 16, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4', 'transitive_degree': 4, 'wreath_data': ['C_2', 'C_2', '2T1'], 'wreath_product': True}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [3, 3, 6, 12, 2, 2, 2, 6, 60], 'aut_gens': [[11, 840, 3670571, 8025137, 7, 5, 23, 124, 50804050], [11, 840, 3670571, 8025130, 23, 6, 16, 128, 50804049], [11, 840, 3670571, 8025137, 7, 5, 23, 1564, 50804050], [11, 840, 164063531, 123420970, 23, 6, 16, 728, 80237649], [8, 840, 131364008, 160031529, 16, 6, 23, 731, 127774930], [20, 840, 3670580, 8025142, 7, 14, 23, 128, 50804050], [19, 840, 3670579, 8025133, 23, 5, 7, 123, 50804062], [12, 840, 3670572, 8025137, 7, 14, 23, 135, 50804053], [12, 1440, 3669972, 8024538, 16, 2, 7, 124, 50804657], [4, 840, 131364004, 160031537, 7, 2, 16, 1572, 50804058]], 'aut_group': '2880.hl', 'aut_hash': 799193401477229059, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2880, 'aut_permdeg': 14, 'aut_perms': [164133512, 4354969080, 4069535740, 4094543449, 123747120, 124108712, 126288152, 84838496, 86180832618], 'aut_phi_ratio': 3.75, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 2], [2, 5, 1, 1], [2, 6, 1, 1], [2, 9, 1, 1], [2, 15, 1, 2], [2, 18, 1, 1], [2, 30, 1, 1], [2, 45, 1, 1], [2, 90, 1, 1], [3, 2, 1, 1], [3, 8, 1, 1], [3, 16, 1, 1], [4, 5, 1, 2], [4, 6, 1, 1], [4, 15, 1, 4], [4, 18, 1, 1], [4, 30, 1, 5], [4, 45, 1, 2], [4, 90, 1, 5], [5, 4, 1, 1], [6, 6, 1, 1], [6, 10, 1, 1], [6, 12, 1, 1], [6, 24, 1, 1], [6, 30, 1, 1], [6, 40, 1, 1], [6, 60, 1, 1], [6, 80, 1, 1], [6, 120, 1, 1], [10, 12, 1, 2], [10, 24, 1, 1], [10, 36, 1, 1], [10, 72, 1, 1], [12, 10, 1, 2], [12, 12, 1, 1], [12, 30, 1, 2], [12, 40, 1, 2], [12, 60, 1, 5], [12, 80, 1, 2], [12, 120, 1, 2], [15, 8, 1, 1], [15, 32, 1, 1], [15, 64, 1, 1], [20, 24, 1, 1], [20, 72, 1, 1], [30, 24, 1, 1], [30, 48, 1, 1], [30, 96, 1, 1], [60, 48, 1, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_3\\times F_5\\times S_4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 1, 'autcent_group': '1.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 1, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_1', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '2880.hl', 'autcentquo_hash': 799193401477229059, 'autcentquo_nilpotent': False, 'autcentquo_order': 2880, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\times F_5\\times S_4', 'cc_stats': [[1, 1, 1], [2, 3, 2], [2, 5, 1], [2, 6, 1], [2, 9, 1], [2, 15, 2], [2, 18, 1], [2, 30, 1], [2, 45, 1], [2, 90, 1], [3, 2, 1], [3, 8, 1], [3, 16, 1], [4, 5, 2], [4, 6, 1], [4, 15, 4], [4, 18, 1], [4, 30, 5], [4, 45, 2], [4, 90, 5], [5, 4, 1], [6, 6, 1], [6, 10, 1], [6, 12, 1], [6, 24, 1], [6, 30, 1], [6, 40, 1], [6, 60, 1], [6, 80, 1], [6, 120, 1], [10, 12, 2], [10, 24, 1], [10, 36, 1], [10, 72, 1], [12, 10, 2], [12, 12, 1], [12, 30, 2], [12, 40, 2], [12, 60, 5], [12, 80, 2], [12, 120, 2], [15, 8, 1], [15, 32, 1], [15, 64, 1], [20, 24, 1], [20, 72, 1], [30, 24, 1], [30, 48, 1], [30, 96, 1], [60, 48, 1]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '2880.hl', 'commutator_count': 1, 'commutator_label': '180.31', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '5.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 194, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['20.3', 1], ['24.12', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 2], [2, 5, 1, 1], [2, 6, 1, 1], [2, 9, 1, 1], [2, 15, 1, 2], [2, 18, 1, 1], [2, 30, 1, 1], [2, 45, 1, 1], [2, 90, 1, 1], [3, 2, 1, 1], [3, 8, 1, 1], [3, 16, 1, 1], [4, 5, 2, 1], [4, 6, 1, 1], [4, 15, 2, 2], [4, 18, 1, 1], [4, 30, 1, 1], [4, 30, 2, 2], [4, 45, 2, 1], [4, 90, 1, 1], [4, 90, 2, 2], [5, 4, 1, 1], [6, 6, 1, 1], [6, 10, 1, 1], [6, 12, 1, 1], [6, 24, 1, 1], [6, 30, 1, 1], [6, 40, 1, 1], [6, 60, 1, 1], [6, 80, 1, 1], [6, 120, 1, 1], [10, 12, 1, 2], [10, 24, 1, 1], [10, 36, 1, 1], [10, 72, 1, 1], [12, 10, 2, 1], [12, 12, 1, 1], [12, 30, 2, 1], [12, 40, 2, 1], [12, 60, 1, 1], [12, 60, 2, 2], [12, 80, 2, 1], [12, 120, 2, 1], [15, 8, 1, 1], [15, 32, 1, 1], [15, 64, 1, 1], [20, 24, 1, 1], [20, 72, 1, 1], [30, 24, 1, 1], [30, 48, 1, 1], [30, 96, 1, 1], [60, 48, 1, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 1935360, 'exponent': 60, 'exponents_of_order': [6, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[24, 1, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '2880.hl', 'hash': 799193401477229059, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [3, 3, 6, 12, 2, 2, 2, 6, 60], 'inner_gens': [[11, 840, 3670571, 8025129, 16, 21, 7, 135, 50804057], [11, 840, 3670571, 8025137, 7, 5, 23, 1564, 50804050], [11, 840, 3670571, 8025129, 16, 21, 7, 735, 80237657], [15, 840, 3670575, 8025137, 7, 21, 16, 731, 127774929], [12, 840, 3670572, 8025137, 7, 14, 23, 135, 50804053], [19, 840, 3670579, 8025133, 23, 5, 7, 123, 50804062], [20, 840, 3670580, 8025142, 7, 14, 23, 128, 50804050], [20, 1440, 3669980, 8024530, 23, 1, 16, 124, 50804658], [4, 840, 164063524, 123420978, 16, 1, 23, 1580, 50804050]], 'inner_hash': 799193401477229059, 'inner_nilpotent': False, 'inner_order': 2880, 'inner_split': True, 'inner_tex': 'S_3\\times F_5\\times S_4', 'inner_used': [1, 2, 3, 4, 6, 8, 9], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 16], [2, 16], [3, 16], [4, 8], [6, 8], [8, 4], [12, 4], [16, 1], [24, 2]], 'label': '2880.hl', 'linC_count': 512, 'linC_degree': 9, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 9, 'linQ_degree_count': 128, 'linQ_dim': 9, 'linQ_dim_count': 128, 'linR_count': 128, 'linR_degree': 9, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'S3*F5*S4', 'ngens': 9, '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, 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': 75, 'number_characteristic_subgroups': 70, 'number_conjugacy_classes': 75, 'number_divisions': 60, 'number_normal_subgroups': 70, 'number_subgroup_autclasses': 964, 'number_subgroup_classes': 964, 'number_subgroups': 15912, 'old_label': None, 'order': 2880, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 239], [3, 26], [4, 784], [5, 4], [6, 382], [10, 156], [12, 872], [15, 104], [20, 96], [30, 168], [60, 48]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [3, 8], [4, 10], [6, 8], [8, 5], [12, 6], [16, 1], [24, 2]], 'representations': {'PC': {'code': '21522185185716292696711620568826601672798636284756590689059687996771809905710317640448294086764609405212', 'gens': [1, 2, 3, 6, 9], 'pres': [9, 2, 2, 2, 2, 3, 2, 3, 5, 2, 26245, 39422, 6761, 74, 79203, 732, 102, 66244, 733, 72590, 20759, 15908, 7493, 158, 33279, 3048, 3057, 249, 10393, 10402, 174968, 29177, 43766, 7325, 10979]}, 'Perm': {'d': 12, 'gens': [11, 840, 3670571, 8025137, 7, 5, 23, 124, 50804050]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'S_3\\times F_5\\times S_4', 'transitive_degree': 60, 'wreath_data': None, 'wreath_product': False}
