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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '5760.fv', 'ambient_counter': 152, 'ambient_order': 5760, 'ambient_tex': 'D_4\\times A_4\\times A_5', 'central': False, 'central_factor': False, 'centralizer_order': 40, 'characteristic': False, 'core_order': 12, 'counter': 344, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '5760.fv.96.i1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '96.i1.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': 96, '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': '60.9', 'subgroup_hash': 9, 'subgroup_order': 60, 'subgroup_tex': 'C_5\\times A_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '5760.fv', 'aut_centralizer_order': None, 'aut_label': '96.i1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '144.g1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['48.e1.a1', '48.h1.a1', '48.h1.b1', '48.k1.a1', '48.l1.a1', '48.m1.a1', '48.m1.b1'], 'contains': ['288.z1.a1', '384.a1.a1', '480.a1.a1'], 'core': '480.a1.a1', 'coset_action_label': None, 'count': 6, 'diagramx': [366, -1, 7142, -1, 16, -1, 33, -1], 'generators': [79833600, 958003273, 1520467200, 482630400], 'label': '5760.fv.96.i1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '8.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '6.e1.a1', 'old_label': '96.i1.a1', 'projective_image': '5760.fv', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '96.i1.a1', 'subgroup_fusion': None, 'weyl_group': '24.13'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '15.1', '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, 4, 4, 2, 6], 'aut_gens': [[1, 3, 6], [1, 3, 54], [34, 3, 54], [4, 3, 18], [1, 3, 42], [35, 33, 54], [31, 30, 57]], 'aut_group': '96.186', 'aut_hash': 186, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96, 'aut_permdeg': 8, 'aut_perms': [16, 5176, 16689, 18, 15960, 10936], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 4, 2, 1], [5, 1, 4, 1], [10, 3, 4, 1], [15, 4, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times S_4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 3, 1], [3, 4, 2], [5, 1, 4], [10, 3, 4], [15, 4, 8]], 'center_label': '5.1', 'center_order': 5, 'central_product': True, 'central_quotient': '12.3', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '5.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 9, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 4, 2, 1], [5, 1, 4, 1], [10, 3, 4, 1], [15, 4, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 24, 'exponent': 30, 'exponents_of_order': [2, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[3, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '60.9', 'hash': 9, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 2, 2], 'inner_gens': [[1, 30, 9], [34, 3, 6], [4, 3, 6]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'A_4', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 6, 'irrep_stats': [[1, 15], [3, 5]], 'label': '60.9', 'linC_count': 4, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 7, 'linQ_degree_count': 1, 'linQ_dim': 7, 'linQ_dim_count': 1, 'linR_count': 6, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C5*A4', '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': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 20, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 10, 'number_subgroup_classes': 10, 'number_subgroups': 20, 'old_label': None, 'order': 60, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 3], [3, 8], [5, 4], [10, 12], [15, 32]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[2, 30, 27], [1, 3, 18]], 'outer_group': '8.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [3, 1], [4, 1], [8, 1], [12, 1]], 'representations': {'PC': {'code': 4852423747, 'gens': [1, 2, 3], 'pres': [4, -3, -2, 2, -5, 241, 110, 34]}, 'GLFp': {'d': 3, 'p': 11, 'gens': [656967451, 1691722594, 857494092, 1907820797]}, 'Perm': {'d': 9, 'gens': [5760, 33, 41040, 90720]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [15], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times A_4', 'transitive_degree': 20, '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': '12.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [2, 2, 2, 2, 6, 2, 4, 3, 2], 'aut_gens': [[43545600, 482630400, 482631120, 1037877840, 1037927520, 79924346, 958134971, 90720], [562464000, 482630400, 482631120, 1037877840, 1037968560, 1440765386, 1477012331, 131760], [562464000, 482630400, 482762160, 1037877840, 1037877120, 1440673946, 1476922331, 40320], [1476921600, 482630400, 482721120, 1037877840, 1037837520, 1001549546, 562504331, 720], [562464000, 482630400, 482670720, 1037877840, 1037968560, 1440765386, 1477012331, 131760], [958003200, 1520467200, 1520507520, 482671440, 482762160, 519050267, 1477012394, 131760], [43545600, 482630400, 482762160, 1037877840, 1037877120, 79873980, 958003939, 40320], [1001548800, 1520467200, 1520598960, 1037877840, 1037877120, 1476961946, 79834331, 40320], [1476921600, 1037836800, 1037837520, 1520508240, 1520557920, 1001639546, 43677371, 90720], [1476921600, 482630400, 482762160, 1037877840, 1037877120, 1001589146, 562464731, 40320]], 'aut_group': '23040.ch', 'aut_hash': 4448622648006636622, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 23040, 'aut_permdeg': 27, 'aut_perms': [403292434483114520192726550, 403292435701366285813284774, 2374509988355920784746087854, 403292436436216512260386854, 63382298663454176464646916, 6871124859217728413218325, 4747725386297074090774748208, 357405258060747636088216470, 2374509989334104553117813294], 'aut_phi_ratio': 15.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [2, 3, 1, 2], [2, 6, 2, 1], [2, 15, 1, 2], [2, 30, 2, 1], [2, 45, 1, 2], [2, 90, 2, 1], [3, 4, 2, 1], [3, 20, 1, 1], [3, 80, 2, 1], [4, 2, 1, 1], [4, 6, 1, 1], [4, 30, 1, 1], [4, 90, 1, 1], [5, 12, 2, 1], [6, 4, 2, 1], [6, 8, 4, 1], [6, 20, 1, 1], [6, 40, 2, 1], [6, 60, 1, 2], [6, 60, 2, 2], [6, 80, 2, 1], [6, 120, 2, 1], [6, 120, 4, 1], [6, 160, 4, 1], [10, 12, 2, 1], [10, 24, 4, 1], [10, 36, 2, 2], [10, 72, 4, 1], [12, 8, 2, 1], [12, 40, 1, 1], [12, 120, 1, 1], [12, 120, 2, 1], [12, 160, 2, 1], [15, 48, 4, 1], [20, 24, 2, 1], [20, 72, 2, 1], [30, 48, 4, 1], [30, 96, 8, 1], [60, 96, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'D_4\\times S_4\\times S_5', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '5760.co', 'autcentquo_hash': 6537139331229567448, 'autcentquo_nilpotent': False, 'autcentquo_order': 5760, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2\\times S_4\\times S_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [2, 3, 2], [2, 6, 2], [2, 15, 2], [2, 30, 2], [2, 45, 2], [2, 90, 2], [3, 4, 2], [3, 20, 1], [3, 80, 2], [4, 2, 1], [4, 6, 1], [4, 30, 1], [4, 90, 1], [5, 12, 2], [6, 4, 2], [6, 8, 4], [6, 20, 1], [6, 40, 2], [6, 60, 6], [6, 80, 2], [6, 120, 6], [6, 160, 4], [10, 12, 2], [10, 24, 4], [10, 36, 4], [10, 72, 4], [12, 8, 2], [12, 40, 1], [12, 120, 3], [12, 160, 2], [15, 48, 4], [20, 24, 2], [20, 72, 2], [30, 48, 4], [30, 96, 8], [60, 96, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '2880.gn', 'commutator_count': 1, 'commutator_label': '480.1187', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '60.5'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 152, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['60.5', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [2, 3, 1, 2], [2, 6, 1, 2], [2, 15, 1, 2], [2, 30, 1, 2], [2, 45, 1, 2], [2, 90, 1, 2], [3, 4, 2, 1], [3, 20, 1, 1], [3, 80, 2, 1], [4, 2, 1, 1], [4, 6, 1, 1], [4, 30, 1, 1], [4, 90, 1, 1], [5, 12, 2, 1], [6, 4, 2, 1], [6, 8, 2, 2], [6, 20, 1, 1], [6, 40, 1, 2], [6, 60, 1, 2], [6, 60, 2, 2], [6, 80, 2, 1], [6, 120, 1, 2], [6, 120, 2, 2], [6, 160, 2, 2], [10, 12, 2, 1], [10, 24, 2, 2], [10, 36, 2, 2], [10, 72, 2, 2], [12, 8, 2, 1], [12, 40, 1, 1], [12, 120, 1, 1], [12, 120, 2, 1], [12, 160, 2, 1], [15, 48, 4, 1], [20, 24, 2, 1], [20, 72, 2, 1], [30, 48, 4, 1], [30, 96, 4, 2], [60, 96, 4, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 228, 'exponent': 60, 'exponents_of_order': [7, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[18, 1, 2], [24, 1, 1], [30, 1, 1]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '2880.gn', 'hash': 5932148942766767589, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [3, 2, 2, 2, 2, 6, 6, 2], 'inner_gens': [[43545600, 1037836800, 1037837520, 1520508240, 1520557920, 79924346, 1477053371, 90720], [1476921600, 482630400, 482631120, 1037877840, 1037927520, 1001639546, 562595771, 90720], [1476921600, 482630400, 482631120, 1037877840, 1037968560, 1001680586, 562554731, 131760], [562464000, 482630400, 482631120, 1037877840, 1037927520, 1440724346, 1477053371, 90720], [562464000, 482630400, 482670720, 1037877840, 1037927520, 1440724346, 1477053371, 90720], [43545600, 1520467200, 1520507520, 482671440, 482721120, 79924346, 562595841, 90720], [562464000, 1037836800, 1037877120, 1520508240, 1520557920, 1440724375, 958134971, 90720], [43545600, 482630400, 482670720, 1037877840, 1037927520, 79924346, 958134971, 90720]], 'inner_hash': 5094570882729245637, 'inner_nilpotent': False, 'inner_order': 2880, 'inner_split': True, 'inner_tex': 'C_2^4:\\GL(2,4)', 'inner_used': [1, 2, 3, 5, 6, 7], 'irrC_degree': 18, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 18, 'irrep_stats': [[1, 12], [2, 3], [3, 28], [4, 12], [5, 12], [6, 7], [8, 3], [9, 8], [10, 3], [12, 4], [15, 4], [18, 2], [24, 1], [30, 1]], 'label': '5760.fv', 'linC_count': 288, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 9, 'linQ_degree_count': 16, 'linQ_dim': 9, 'linQ_dim_count': 16, 'linR_count': 32, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'D4*A4*A5', 'ngens': 8, '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, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 48, 'number_characteristic_subgroups': 24, 'number_conjugacy_classes': 100, 'number_divisions': 60, 'number_normal_subgroups': 36, 'number_subgroup_autclasses': 926, 'number_subgroup_classes': 1646, 'number_subgroups': 61958, 'old_label': None, 'order': 5760, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 383], [3, 188], [4, 128], [5, 24], [6, 2020], [10, 552], [12, 736], [15, 192], [20, 192], [30, 960], [60, 384]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [111, 0, 0], 'outer_gens': [[43545600, 482630400, 482631120, 1037877840, 1037927520, 79924375, 958135059, 90720], [43545600, 482630400, 482721120, 1037877840, 1037837520, 79834346, 958043531, 720], [79833600, 1037836800, 1037837520, 482671440, 482721120, 43636346, 519050171, 90720]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': None, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 5], [3, 4], [4, 5], [5, 4], [6, 5], [8, 5], [10, 5], [12, 9], [15, 4], [16, 1], [18, 4], [20, 1], [24, 2], [30, 1], [36, 1]], 'representations': {'Perm': {'d': 13, 'gens': [43545600, 482630400, 482631120, 1037877840, 1037927520, 79924346, 958134971, 90720]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'D_4\\times A_4\\times A_5', 'transitive_degree': 60, 'wreath_data': None, 'wreath_product': False}
