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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '6048.a', 'ambient_counter': 1, 'ambient_order': 6048, 'ambient_tex': '\\SU(3,3)', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 1, 'counter': 5, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '6048.a.63.a1.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '63.a1.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': 63, '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': '96.67', 'subgroup_hash': 67, 'subgroup_order': 96, 'subgroup_tex': '\\Unitary(2,3)', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '6048.a', 'aut_centralizer_order': None, 'aut_label': '63.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1512.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['126.b1.a1', '189.a1.a1', '252.a1.a1'], 'core': '6048.a1.a1', 'coset_action_label': None, 'count': 63, 'diagramx': [7193, -1, 9112, -1, 4694, -1, 4694, -1], 'generators': [75573664, 304568424, 338630458, 254445252, 172731617, 220726122], 'label': '6048.a.63.a1.a1', 'mobius_quo': None, 'mobius_sub': -1, 'normal_closure': '1.a1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '63.a1.a1', 'old_label': '63.a1.a1', 'projective_image': '6048.a', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '63.a1.a1', 'subgroup_fusion': None, 'weyl_group': '24.12'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.1', '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, 6, 2, 2], 'aut_gens': [[4377, 362], [2193, 362], [4377, 722], [3895, 1818], [612, 766], [2193, 1818]], 'aut_group': '96.226', 'aut_hash': 226, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 96, 'aut_permdeg': 8, 'aut_perms': [16, 127, 1447, 11520, 5160], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 1], [3, 8, 1, 1], [4, 1, 2, 1], [4, 6, 1, 1], [4, 6, 4, 1], [6, 8, 1, 1], [8, 12, 2, 1], [12, 8, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2\\times 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': 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, 1], [2, 6, 1], [3, 8, 1], [4, 1, 2], [4, 6, 5], [6, 8, 1], [8, 12, 2], [12, 8, 2]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, 'central_quotient': '24.12', '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', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 67, '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], [3, 8, 1, 1], [4, 1, 2, 1], [4, 6, 1, 1], [4, 6, 2, 2], [6, 8, 1, 1], [8, 12, 2, 1], [12, 8, 2, 1]], 'element_repr_type': 'Lie', 'elementary': 1, 'eulerian_function': 36, 'exponent': 24, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 4], [4, 0, 2]], 'familial': True, 'frattini_label': '4.1', 'frattini_quotient': '24.12', 'hash': 67, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 3], 'inner_gens': [[4377, 722], [2598, 362]], 'inner_hash': 12, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': True, 'inner_tex': 'S_4', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 4], [2, 6], [3, 4], [4, 2]], 'label': '96.67', 'linC_count': 4, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 2, 'linQ_dim': 4, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'Unitary(2,3)', 'ngens': 6, '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': 10, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 16, 'number_divisions': 11, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 23, 'number_subgroup_classes': 24, 'number_subgroups': 75, 'old_label': None, 'order': 96, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 7], [3, 8], [4, 32], [6, 8], [8, 24], [12, 16]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [730, 730], 'outer_gens': [[3652, 362], [2193, 362]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [3, 2], [4, 2], [6, 1], [8, 1]], 'representations': {'PC': {'code': 49874333201553517743994568, 'gens': [1, 3, 4, 5], 'pres': [6, -2, -2, -3, -2, 2, -2, 12, 295, 146, 1443, 231, 117, 1084, 466, 202, 88]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [41537776, 1324965]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [362, 4377], 'family': 'GU'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [454, 49, 252]}, 'Perm': {'d': 24, 'gens': [80069238362132079356137, 162070578280827183738839, 432474715758828847104000, 134021272927222627062473, 79804308132549543940423, 25903229683158464681647]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\Unitary(2,3)', 'transitive_degree': 24, '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': '1.1', 'all_subgroups_known': True, 'almost_simple': True, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 168, 'aut_gen_orders': [8, 8, 2], 'aut_gens': [[258306573, 546126], [254085438, 157480260], [325831951, 94948759], [243566045, 285411522]], 'aut_group': '12096.a', 'aut_hash': 3091906055211991884, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12096, 'aut_permdeg': 56, 'aut_perms': [382434045811931101024800391082438558274389265688135943984689571554452495446, 498717635904174111494784199880809168384193113194039126092079485894139475214, 120657209303055803711710202688624959110829600214713548105582095599027565035], 'aut_phi_ratio': 7.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 63, 1, 1], [3, 56, 1, 1], [3, 672, 1, 1], [4, 63, 2, 1], [4, 378, 1, 1], [6, 504, 1, 1], [7, 864, 2, 1], [8, 756, 2, 1], [12, 504, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'G(2,2)', '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': 168, 'autcentquo_group': '12096.a', 'autcentquo_hash': 3091906055211991884, 'autcentquo_nilpotent': False, 'autcentquo_order': 12096, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'G(2,2)', 'cc_stats': [[1, 1, 1], [2, 63, 1], [3, 56, 1], [3, 672, 1], [4, 63, 2], [4, 378, 1], [6, 504, 1], [7, 864, 2], [8, 756, 2], [12, 504, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '6048.a', 'commutator_count': 1, 'commutator_label': '6048.a', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['6048.a'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 63, 1, 1], [3, 56, 1, 1], [3, 672, 1, 1], [4, 63, 2, 1], [4, 378, 1, 1], [6, 504, 1, 1], [7, 864, 2, 1], [8, 756, 2, 1], [12, 504, 2, 1]], 'element_repr_type': 'Lie', 'elementary': 1, 'eulerian_function': 2784, 'exponent': 168, 'exponents_of_order': [5, 3, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[6, -1, 1], [7, 0, 2], [7, 1, 1], [14, 1, 1], [21, 0, 2], [21, 1, 1], [27, 1, 1], [28, 0, 2], [32, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '6048.a', 'hash': 7647923293267076632, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 168, 'inner_gen_orders': [8, 8], 'inner_gens': [[258306573, 1081329], [135017367, 546126]], 'inner_hash': 7647923293267076632, 'inner_nilpotent': False, 'inner_order': 6048, 'inner_split': True, 'inner_tex': '\\SU(3,3)', 'inner_used': [1, 2], 'irrC_degree': 6, 'irrQ_degree': 6, 'irrQ_dim': 7, 'irrR_degree': 7, 'irrep_stats': [[1, 1], [6, 1], [7, 3], [14, 1], [21, 3], [27, 1], [28, 2], [32, 2]], 'label': '6048.a', 'linC_count': 1, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 7, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 7, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'SU(3,3)', 'ngens': 2, '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': 10, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 14, 'number_divisions': 10, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 36, 'number_subgroup_classes': 36, 'number_subgroups': 5150, 'old_label': None, 'order': 6048, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 63], [3, 728], [4, 504], [6, 504], [7, 1728], [8, 1512], [12, 1008]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [225385976], 'outer_gens': [[106730078, 120279390]], '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': None, 'perfect': True, 'permutation_degree': 28, 'pgroup': 0, 'primary_abelian_invariants': [], 'quasisimple': True, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [6, 1], [7, 1], [14, 2], [21, 1], [27, 1], [42, 1], [56, 1], [64, 1]], 'representations': {'Lie': [{'d': 3, 'q': 3, 'gens': [546126, 258306573], 'family': 'SU'}, {'d': 3, 'q': 3, 'family': 'PSU'}, {'d': 3, 'q': 3, 'gens': [546126, 258286890], 'family': 'PGU'}, {'d': 3, 'q': 3, 'family': 'CSU'}], 'Perm': {'d': 28, 'gens': [61843509759306133516009798498, 202759273640950069082290682359]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\SU(3,3)', 'transitive_degree': 28, 'wreath_data': None, 'wreath_product': False}
