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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '960.6254', 'ambient_counter': 6254, 'ambient_order': 960, 'ambient_tex': 'C_2.(C_{20}\\times S_4)', 'central': False, 'central_factor': False, 'centralizer_order': 40, 'characteristic': False, 'core_order': 2, 'counter': 116, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '960.6254.120.j1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '120.j1.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': 120, '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': '8.4', 'subgroup_hash': 4, 'subgroup_order': 8, 'subgroup_tex': 'Q_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '960.6254', 'aut_centralizer_order': 32, 'aut_label': '120.j1', 'aut_quo_index': None, 'aut_stab_index': 6, 'aut_weyl_group': '8.3', 'aut_weyl_index': 192, 'centralizer': '24.d1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['24.j1.a1', '60.e1.a1'], 'contains': ['240.d1.a1', '240.e1.a1', '240.e1.b1'], 'core': '480.b1.a1', 'coset_action_label': None, 'count': 6, 'diagramx': [5455, -1, 6094, -1, 6915, -1, 5621, -1], 'generators': [1, 36], 'label': '960.6254.120.j1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '10.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '6.b1.a1', 'old_label': '120.j1.a1', 'projective_image': '480.1014', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '120.j1.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': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 3, 2, 2], 'aut_gens': [[1, 2], [2, 1], [3, 5], [5, 2], [1, 6]], 'aut_group': '24.12', 'aut_hash': 12, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24, 'aut_permdeg': 4, 'aut_perms': [2, 4, 16, 7], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 2, 3]], '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': 4, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 2, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [[2, -1, 1]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 4, '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': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 1]], 'label': '8.4', 'linC_count': 1, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, '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': 'Q8', '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': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 5, 'number_divisions': 5, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 1], [4, 6]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [2, 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': 8, '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': 2343, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 12, 37, 16]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [25793872, 22766861]}, 'Lie': [{'d': 2, 'q': 5, 'gens': [90, 377, 504, 253], 'family': 'CSOPlus'}, {'d': 2, 'q': 3, 'gens': [56, 67, 15], 'family': 'CSOMinus'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [67, 21]}, 'Perm': {'d': 8, 'gens': [28776, 16577, 5167]}}, 'schur_multiplier': [], 'semidirect_product': False, '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': 'Q_8', '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': '40.9', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [4, 4, 2, 4, 2, 4, 2], 'aut_gens': [[1, 2, 24, 48], [481, 14, 24, 816], [489, 514, 744, 624], [493, 250, 504, 456], [245, 250, 240, 888], [25, 254, 504, 912], [481, 502, 504, 168], [493, 262, 504, 456]], 'aut_group': '1536.408640917', 'aut_hash': 5803646388267248920, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1536, 'aut_permdeg': 32, 'aut_perms': [72492346837544948525812709140295040, 17714254334117046740566889572428778, 228438120960164784153576475574475013, 38322722589341942860973755379070386, 68633375664754202388549720999804170, 195775235401432989449168757903641875, 228438120954580697325004279729840695], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 8, 1, 1], [4, 2, 2, 1], [4, 6, 1, 2], [4, 6, 2, 1], [4, 12, 2, 2], [5, 1, 4, 1], [6, 8, 1, 3], [8, 12, 2, 2], [10, 1, 4, 3], [12, 8, 4, 1], [15, 8, 4, 1], [20, 2, 8, 1], [20, 6, 4, 2], [20, 6, 8, 1], [20, 12, 8, 2], [30, 8, 4, 3], [40, 12, 8, 2], [60, 8, 16, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4\\times C_4\\times S_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '64.260', 'autcent_hash': 260, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4\\times 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, 1, 3], [3, 8, 1], [4, 2, 2], [4, 6, 4], [4, 12, 4], [5, 1, 4], [6, 8, 3], [8, 12, 4], [10, 1, 12], [12, 8, 4], [15, 8, 4], [20, 2, 8], [20, 6, 16], [20, 12, 16], [30, 8, 12], [40, 12, 16], [60, 8, 16]], 'center_label': '20.5', 'center_order': 20, 'central_product': True, 'central_quotient': '48.48', 'commutator_count': 1, 'commutator_label': '24.3', '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': 6254, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [['192.947', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 8, 1, 1], [4, 2, 2, 1], [4, 6, 1, 2], [4, 6, 2, 1], [4, 12, 1, 2], [4, 12, 2, 1], [5, 1, 4, 1], [6, 8, 1, 3], [8, 12, 1, 2], [8, 12, 2, 1], [10, 1, 4, 3], [12, 8, 4, 1], [15, 8, 4, 1], [20, 2, 8, 1], [20, 6, 4, 2], [20, 6, 8, 1], [20, 12, 4, 2], [20, 12, 8, 1], [30, 8, 4, 3], [40, 12, 4, 2], [40, 12, 8, 1], [60, 8, 16, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 108, 'exponent': 120, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '240.196', 'hash': 6254, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 6, 2, 2], 'inner_gens': [[1, 490, 504, 552], [497, 2, 240, 72], [481, 266, 24, 528], [505, 26, 504, 48]], 'inner_hash': 48, 'inner_nilpotent': False, 'inner_order': 48, 'inner_split': True, 'inner_tex': 'C_2\\times S_4', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 40], [2, 20], [3, 40], [4, 30]], 'label': '960.6254', 'linC_count': 576, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 16, 'linQ_dim': 14, 'linQ_dim_count': 24, 'linR_count': 160, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'C2.(C20*S4)', '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': 30, 'number_conjugacy_classes': 130, 'number_divisions': 38, 'number_normal_subgroups': 34, 'number_subgroup_autclasses': 120, 'number_subgroup_classes': 134, 'number_subgroups': 446, 'old_label': None, 'order': 960, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 3], [3, 8], [4, 76], [5, 4], [6, 24], [8, 48], [10, 12], [12, 32], [15, 32], [20, 304], [30, 96], [40, 192], [60, 128]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 10, 504, 552], [13, 2, 24, 48], [1, 14, 24, 48], [1, 2, 24, 816]], 'outer_group': '32.45', 'outer_hash': 45, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 32, 'outer_perms': [8250311271909446485476156660344041, 17508267730465192346769804442694884, 26047898994165116246942772090139938, 34546977905640644393140055043261696], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 25, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [3, 4], [4, 7], [6, 2], [8, 6], [12, 4], [16, 3], [24, 2], [32, 2]], 'representations': {'PC': {'code': 7906472554701253088876788589678947255936198011654646889137866998019, 'gens': [1, 2, 5, 6], 'pres': [8, -2, -2, -2, -3, -2, 2, -2, -5, 3840, 7841, 41, 482, 66, 515, 20164, 4812, 2660, 1228, 836, 26501, 1741, 9525, 461, 1093, 141, 166]}, 'Perm': {'d': 25, 'gens': [705164489033053638758407, 104585624273058431328017, 46200, 7, 1586389556817030067200, 1374006611855239004966400, 2024501140208834040902400, 2594704629037433343379200]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 20], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2.(C_{20}\\times S_4)', 'transitive_degree': 320, 'wreath_data': None, 'wreath_product': False}
