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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '960.10975', 'ambient_counter': 10975, 'ambient_order': 960, 'ambient_tex': '\\GL(2,3):D_{10}', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 2, 'counter': 191, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '960.10975.60.o1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '60.o1.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': 60, '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': '16.8', 'subgroup_hash': 8, 'subgroup_order': 16, 'subgroup_tex': '\\SD_{16}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '960.10975', 'aut_centralizer_order': 16, 'aut_label': '60.o1', 'aut_quo_index': None, 'aut_stab_index': 15, 'aut_weyl_group': '16.11', 'aut_weyl_index': 240, 'centralizer': '240.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['12.q1.a1', '30.e1.a1', '30.h1.a1', '30.k1.a1'], 'contains': ['120.h1.a1', '120.ba1.a1', '120.bd1.a1'], 'core': '480.a1.a1', 'coset_action_label': None, 'count': 15, 'diagramx': [8927, -1, 2675, -1, 5607, -1, 7794, -1], 'generators': [3, 744], 'label': '960.10975.60.o1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.c1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '15.a1.a1', 'old_label': '60.o1.a1', 'projective_image': '480.1193', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '60.o1.a1', 'subgroup_fusion': None, 'weyl_group': '16.11'}
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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, 2, 2, 2], 'aut_gens': [[1, 2], [1, 14], [5, 6], [9, 10], [9, 2]], 'aut_group': '16.11', 'aut_hash': 11, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 16, 'aut_permdeg': 6, 'aut_perms': [24, 288, 415, 414], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 1, 1], [4, 2, 1, 1], [4, 4, 1, 1], [8, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times 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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': True, 'autcentquo_cyclic': False, 'autcentquo_exponent': 2, 'autcentquo_group': '4.2', 'autcentquo_hash': 2, 'autcentquo_nilpotent': True, 'autcentquo_order': 4, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 4, 1], [4, 2, 1], [4, 4, 1], [8, 2, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '8.3', 'commutator_count': 1, 'commutator_label': '4.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 8, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 4, 1, 1], [4, 2, 1, 1], [4, 4, 1, 1], [8, 2, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 6, 'exponent': 8, 'exponents_of_order': [4], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 0, 2]], 'familial': True, 'frattini_label': '4.1', 'frattini_quotient': '4.2', 'hash': 8, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4], 'inner_gens': [[1, 6], [13, 2]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': True, 'inner_tex': 'D_4', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 3]], 'label': '16.8', 'linC_count': 2, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'SD16', 'ngens': 2, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 7, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 10, 'number_subgroup_classes': 10, 'number_subgroups': 15, 'old_label': None, 'order': 16, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 5], [4, 6], [8, 4]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[1, 10]], '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': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [4, 1]], 'representations': {'PC': {'code': 2368542, 'gens': [1, 2], 'pres': [4, -2, 2, -2, -2, 49, 21, 146, 34]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [20182740, 25793872]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [67, 64, 41], 'family': 'COMinus'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 43]}, 'Perm': {'d': 8, 'gens': [28776, 143, 16577, 5167]}}, 'schur_multiplier': [], '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': '\\SD_{16}', 'transitive_degree': 8, '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': 3, 'aut_exponent': 60, 'aut_gen_orders': [12, 2, 2, 4, 2, 2, 2, 4, 10, 10], 'aut_gens': [[1, 2, 12, 24, 48], [5, 770, 252, 264, 396], [253, 722, 12, 24, 528], [733, 254, 492, 504, 48], [481, 398, 492, 24, 336], [253, 158, 492, 504, 432], [253, 254, 492, 504, 48], [253, 242, 12, 24, 528], [733, 338, 12, 24, 144], [5, 778, 492, 504, 540], [253, 542, 492, 504, 48]], 'aut_group': '3840.ho', 'aut_hash': 4584141657853489991, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 3840, 'aut_permdeg': 24, 'aut_perms': [378931551016160721645154, 421649936607912673531952, 84403268332408750553653, 570158345723735210005360, 12418109160875681364245, 12418109161637312319493, 421649936608849623362612, 582369376471527101555830, 106884774239164672317368, 12418109161136254607881], 'aut_phi_ratio': 15.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 1], [2, 10, 1, 1], [2, 12, 1, 1], [2, 30, 1, 1], [2, 60, 1, 1], [3, 8, 1, 1], [4, 1, 2, 1], [4, 6, 1, 1], [4, 10, 1, 1], [4, 12, 1, 1], [4, 30, 1, 1], [4, 60, 1, 1], [5, 2, 2, 1], [6, 8, 1, 1], [6, 40, 2, 1], [8, 12, 1, 2], [8, 60, 1, 2], [10, 2, 2, 1], [10, 12, 2, 1], [10, 24, 2, 1], [12, 8, 2, 1], [12, 40, 2, 1], [15, 16, 2, 1], [20, 2, 4, 1], [20, 12, 2, 1], [20, 24, 2, 1], [30, 16, 2, 1], [40, 12, 4, 2], [60, 16, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^3\\times F_5\\times S_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': 60, 'autcentquo_group': '480.1189', 'autcentquo_hash': 1189, 'autcentquo_nilpotent': False, 'autcentquo_order': 480, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\times S_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 6, 1], [2, 10, 1], [2, 12, 1], [2, 30, 1], [2, 60, 1], [3, 8, 1], [4, 1, 2], [4, 6, 1], [4, 10, 1], [4, 12, 1], [4, 30, 1], [4, 60, 1], [5, 2, 2], [6, 8, 1], [6, 40, 2], [8, 12, 2], [8, 60, 2], [10, 2, 2], [10, 12, 2], [10, 24, 2], [12, 8, 2], [12, 40, 2], [15, 16, 2], [20, 2, 4], [20, 12, 2], [20, 24, 2], [30, 16, 2], [40, 12, 8], [60, 16, 4]], 'center_label': '4.1', 'center_order': 4, 'central_product': True, 'central_quotient': '240.194', 'commutator_count': 1, 'commutator_label': '120.15', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 10975, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 1], [2, 10, 1, 1], [2, 12, 1, 1], [2, 30, 1, 1], [2, 60, 1, 1], [3, 8, 1, 1], [4, 1, 2, 1], [4, 6, 1, 1], [4, 10, 1, 1], [4, 12, 1, 1], [4, 30, 1, 1], [4, 60, 1, 1], [5, 2, 2, 1], [6, 8, 1, 1], [6, 40, 2, 1], [8, 12, 1, 2], [8, 60, 1, 2], [10, 2, 2, 1], [10, 12, 2, 1], [10, 24, 2, 1], [12, 8, 2, 1], [12, 40, 2, 1], [15, 16, 2, 1], [20, 2, 4, 1], [20, 12, 2, 1], [20, 24, 2, 1], [30, 16, 2, 1], [40, 12, 4, 2], [60, 16, 4, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 60480, 'exponent': 120, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[4, 0, 8], [8, 0, 4]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '480.1193', 'hash': 10975, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 6, 2, 2, 10], 'inner_gens': [[1, 490, 720, 276, 300], [485, 2, 252, 264, 204], [733, 722, 12, 24, 528], [733, 722, 12, 24, 528], [253, 350, 492, 504, 48]], 'inner_hash': 194, 'inner_nilpotent': False, 'inner_order': 240, 'inner_split': True, 'inner_tex': 'D_5\\times S_4', 'inner_used': [1, 2, 3, 5], 'irrC_degree': 4, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 12], [3, 8], [4, 18], [6, 8], [8, 4]], 'label': '960.10975', 'linC_count': 8, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 4, 'linQ_dim': 12, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'GL(2,3):D10', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 34, 'number_characteristic_subgroups': 37, 'number_conjugacy_classes': 58, 'number_divisions': 34, 'number_normal_subgroups': 37, 'number_subgroup_autclasses': 286, 'number_subgroup_classes': 288, 'number_subgroups': 2622, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 119], [3, 8], [4, 120], [5, 4], [6, 88], [8, 144], [10, 76], [12, 96], [15, 32], [20, 80], [30, 32], [40, 96], [60, 64]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 10, 720, 276, 300], [1, 2, 12, 504, 48], [1, 2, 12, 24, 624]], 'outer_group': '16.10', 'outer_hash': 10, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 37, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [3, 8], [4, 4], [8, 3], [12, 4], [16, 1], [32, 2]], 'representations': {'PC': {'code': 7567724842510842343758350768390661213255008832298086275421778351883267, 'gens': [1, 2, 4, 5, 6], 'pres': [8, -2, -2, -3, -2, 2, 2, -2, -5, 7841, 41, 194, 23043, 4043, 1939, 1307, 11044, 5292, 7580, 14405, 4909, 741, 2141, 1093, 141, 24206, 166, 24591]}, 'Perm': {'d': 37, 'gens': [373514256714495354806664002824821700047367, 52050725004556402597407233298883483393920, 807229158230279671557109135065742684067280, 11849385358973077387273156485114926640, 37, 1190711863117819449816118715305434268038240, 1573392232681143023500314908354406774167760, 1914691532983209994902123734568870426582480]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\GL(2,3):D_{10}', 'transitive_degree': 160, 'wreath_data': None, 'wreath_product': False}
