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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1728.22767', 'ambient_counter': 22767, 'ambient_order': 1728, 'ambient_tex': 'C_2^3.S_3^3', 'central': False, 'central_factor': False, 'centralizer_order': 864, 'characteristic': True, 'core_order': 3, 'counter': 1094, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '1728.22767.576.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '576.c1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '576.3123', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 3123, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 576, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_6.(S_3\\times C_4^2)', 'simple': True, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '3.1', 'subgroup_hash': 1, 'subgroup_order': 3, 'subgroup_tex': 'C_3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1728.22767', 'aut_centralizer_order': None, 'aut_label': '576.c1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '2.a1.a1', 'complements': ['3.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['192.b1.a1', '192.c1.a1', '192.f1.a1', '288.c1.a1', '288.f1.a1', '288.i1.a1', '288.l1.a1', '288.o1.a1', '288.r1.a1', '288.u1.a1'], 'contains': ['1728.a1.a1'], 'core': '576.c1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [6332, 2170, 4034, 1614, 5065, 2261, 6092, 3259], 'generators': [192], 'label': '1728.22767.576.c1.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '576.c1.a1', 'normal_contained_in': ['192.c1.a1', '192.b1.a1', '288.r1.a1', '288.u1.a1', '288.c1.a1', '288.l1.a1', '288.i1.a1', '288.f1.a1', '288.o1.a1'], 'normal_contains': ['1728.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '576.c1.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '576.c1.a1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '3.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2], 'aut_gens': [[1], [2]], 'aut_group': '2.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 2, 'aut_permdeg': 2, 'aut_perms': [1], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [3, 1, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [3, 1, 2]], 'center_label': '3.1', 'center_order': 3, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [3, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 3, 'eulerian_function': 1, 'exponent': 3, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [3], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '3.1', 'hash': 1, 'hyperelementary': 3, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 3]], 'label': '3.1', 'linC_count': 2, 'linC_degree': 1, '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': 'C3', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 3, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 3, 'order_factorization_type': 1, 'order_stats': [[1, 1], [3, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[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': 1, 'perfect': False, 'permutation_degree': 3, 'pgroup': 3, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -3]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [7]}, 'Lie': [{'d': 1, 'q': 3, 'gens': [3], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [7]}, 'Perm': {'d': 3, 'gens': [4]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [3], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3', 'transitive_degree': 3, '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': '32.21', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': None, 'aut_cyclic': None, 'aut_derived_length': None, 'aut_exponent': None, 'aut_gen_orders': None, 'aut_gens': [[1, 4, 48, 576], [3, 20, 240, 576], [25, 20, 48, 576], [289, 20, 48, 1152], [1, 6, 48, 1152], [1, 28, 48, 576], [1, 308, 240, 576], [1, 20, 242, 1152], [1, 20, 264, 1152], [1, 20, 528, 1152], [1, 20, 48, 576], [1, 4, 240, 576], [1, 4, 48, 1152], [17, 4, 48, 576], [1, 196, 48, 576], [1, 4, 624, 576]], 'aut_group': '13824.kv', 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 110592, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': 192.0, 'aut_solvable': None, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 2, 1, 3], [3, 4, 1, 3], [3, 8, 1, 1], [4, 6, 4, 3], [4, 9, 8, 1], [4, 18, 4, 2], [4, 54, 4, 1], [6, 2, 1, 21], [6, 4, 1, 21], [6, 8, 1, 7], [12, 6, 8, 2], [12, 12, 4, 4], [12, 12, 8, 3], [12, 18, 8, 1], [12, 36, 4, 2]], 'aut_supersolvable': None, 'aut_tex': 'C_6^2:(D_4\\times \\GL(2,3))', 'autcent_abelian': None, 'autcent_cyclic': None, 'autcent_exponent': None, 'autcent_group': None, 'autcent_hash': None, 'autcent_nilpotent': None, 'autcent_order': None, 'autcent_solvable': None, 'autcent_split': None, 'autcent_supersolvable': None, 'autcent_tex': None, 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': None, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 1, 7], [3, 2, 3], [3, 4, 3], [3, 8, 1], [4, 6, 12], [4, 9, 8], [4, 18, 8], [4, 54, 4], [6, 2, 21], [6, 4, 21], [6, 8, 7], [12, 6, 16], [12, 12, 40], [12, 18, 8], [12, 36, 8]], 'center_label': '8.5', 'center_order': 8, 'central_product': False, 'central_quotient': '216.162', 'commutator_count': 1, 'commutator_label': '54.15', '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', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 22767, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 2, 1, 3], [3, 4, 1, 3], [3, 8, 1, 1], [4, 6, 1, 4], [4, 6, 2, 4], [4, 9, 2, 4], [4, 18, 2, 4], [4, 54, 2, 2], [6, 2, 1, 21], [6, 4, 1, 21], [6, 8, 1, 7], [12, 6, 2, 4], [12, 6, 4, 2], [12, 12, 1, 4], [12, 12, 2, 14], [12, 12, 4, 2], [12, 18, 2, 4], [12, 36, 2, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 10752, 'exponent': 12, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '216.162', 'hash': 22767, 'hyperelementary': 1, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': [2, 6, 6, 3], 'inner_gens': [[1, 308, 48, 576], [321, 4, 528, 576], [1, 100, 48, 1152], [1, 4, 1200, 576]], 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': 216, 'inner_split': None, 'inner_tex': 'S_3^3', 'inner_used': None, 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 72], [4, 56], [8, 8]], 'label': '1728.22767', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3.S3^3', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 83, 'number_characteristic_subgroups': 158, 'number_conjugacy_classes': 168, 'number_divisions': 116, 'number_normal_subgroups': 264, 'number_subgroup_autclasses': 750, 'number_subgroup_classes': 1106, 'number_subgroups': 6420, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 7], [3, 26], [4, 504], [6, 182], [12, 1008]], 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': False, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': '512.10494213', 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 512, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': 'C_2^9', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 21, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 36], [4, 48], [8, 24]], 'representations': {'PC': {'code': 3701051189634223562405558612405258637906625792104979, 'gens': [1, 3, 6, 9], 'pres': [9, -2, -2, -2, -2, -3, -2, -2, -3, -3, 18, 8318, 74, 1443, 102, 1444, 7151, 158, 7584, 186, 6937, 1997]}, 'Perm': {'d': 21, 'gens': [2446418218101580809, 2452841614229534769, 374525, 5160, 87657292816, 16, 5243564987781120000, 3991680, 7664017935974400000]}}, 'schur_multiplier': [2, 2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3.S_3^3', 'transitive_degree': 192, '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': '32.21', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': None, 'aut_cyclic': None, 'aut_derived_length': None, 'aut_exponent': None, 'aut_gen_orders': None, 'aut_gens': [[1, 4, 48, 192], [3, 20, 48, 192], [25, 4, 48, 384], [97, 4, 48, 192], [1, 22, 48, 192], [1, 44, 48, 192], [1, 100, 48, 192], [1, 4, 50, 384], [1, 4, 72, 384], [1, 4, 144, 384], [1, 20, 48, 192], [1, 4, 48, 384], [17, 4, 48, 192], [1, 4, 240, 192]], 'aut_group': None, 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 18432, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': 96.0, 'aut_solvable': None, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 2, 1, 2], [3, 4, 1, 1], [4, 2, 4, 1], [4, 6, 4, 4], [4, 9, 8, 1], [4, 18, 4, 1], [6, 2, 1, 14], [6, 4, 1, 7], [12, 4, 4, 2], [12, 4, 8, 1], [12, 12, 4, 4]], 'aut_supersolvable': None, 'aut_tex': None, 'autcent_abelian': None, 'autcent_cyclic': None, 'autcent_exponent': None, 'autcent_group': None, 'autcent_hash': None, 'autcent_nilpotent': None, 'autcent_order': None, 'autcent_solvable': None, 'autcent_split': None, 'autcent_supersolvable': None, 'autcent_tex': None, 'autcentquo_abelian': None, 'autcentquo_cyclic': None, 'autcentquo_exponent': None, 'autcentquo_group': None, 'autcentquo_hash': None, 'autcentquo_nilpotent': None, 'autcentquo_order': None, 'autcentquo_solvable': None, 'autcentquo_supersolvable': None, 'autcentquo_tex': None, 'cc_stats': [[1, 1, 1], [2, 1, 7], [3, 2, 2], [3, 4, 1], [4, 2, 4], [4, 6, 16], [4, 9, 8], [4, 18, 4], [6, 2, 14], [6, 4, 7], [12, 4, 16], [12, 12, 16]], 'center_label': '8.5', 'center_order': 8, 'central_product': False, 'central_quotient': '72.46', 'commutator_count': 1, 'commutator_label': '18.5', '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', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 3123, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 2, 1, 2], [3, 4, 1, 1], [4, 2, 2, 2], [4, 6, 1, 4], [4, 6, 2, 6], [4, 9, 2, 4], [4, 18, 2, 2], [6, 2, 1, 14], [6, 4, 1, 7], [12, 4, 2, 8], [12, 12, 1, 4], [12, 12, 2, 6]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2688, 'exponent': 12, 'exponents_of_order': [6, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.5', 'frattini_quotient': '72.46', 'hash': 3123, 'hyperelementary': 1, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': [2, 6, 2, 3], 'inner_gens': [[1, 116, 48, 192], [129, 4, 144, 192], [1, 100, 48, 384], [1, 4, 432, 192]], 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': 72, 'inner_split': None, 'inner_tex': 'S_3\\times D_6', 'inner_used': None, 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 40], [4, 24]], 'label': '576.3123', 'linC_count': 2304, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 1328, 'linQ_dim': 8, 'linQ_dim_count': 96, 'linR_count': 96, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6.(S3*C4^2)', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 46, 'number_characteristic_subgroups': 88, 'number_conjugacy_classes': 96, 'number_divisions': 68, 'number_normal_subgroups': 156, 'number_subgroup_autclasses': 289, 'number_subgroup_classes': 442, 'number_subgroups': 1428, 'old_label': None, 'order': 576, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 7], [3, 8], [4, 248], [6, 56], [12, 256]], 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': False, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': '256.56092', 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 256, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': 'C_2^8', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 18, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 28], [4, 28], [8, 4]], 'representations': {'PC': {'code': 107137483660527644010560408965819657, 'gens': [1, 3, 6, 8], 'pres': [8, -2, -2, -2, -2, -3, -2, -2, -3, 16, 2786, 66, 1283, 91, 1284, 1749, 141, 559]}, 'GLZN': {'d': 2, 'p': 60, 'gens': [1519553, 8856041, 10584049, 6858916, 5114443, 3875533, 346333, 11594413]}, 'Perm': {'d': 18, 'gens': [87660962785, 21110087538144, 87665316744, 87660963600, 3669120, 1680, 376610217984000, 3]}}, 'schur_multiplier': [2, 2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6.(S_3\\times C_4^2)', 'transitive_degree': 192, 'wreath_data': None, 'wreath_product': False}
