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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '128.902', 'ambient_counter': 902, 'ambient_order': 128, 'ambient_tex': 'C_{16}.D_4', 'central': False, 'central_factor': True, 'centralizer_order': 16, 'characteristic': True, 'core_order': 32, 'counter': 19, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '128.902.4.h1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '4.h1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '4.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '32.15', 'subgroup_hash': 15, 'subgroup_order': 32, 'subgroup_tex': 'C_8.C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '128.902', 'aut_centralizer_order': 4, 'aut_label': '4.h1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '64.254', 'aut_weyl_index': 4, 'centralizer': '8.h1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['2.c1.a1'], 'contains': ['8.d1.a1', '8.e1.a1', '8.e1.b1'], 'core': '4.h1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [2194, 7745, 3239, 2209, 2125, 8005, 3191, 2147], 'generators': [19665, 78609, 78624, 2754, 73704], 'label': '128.902.4.h1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '4.h1.a1', 'normal_contained_in': ['2.c1.a1'], 'normal_contains': ['8.d1.a1', '8.e1.b1', '8.e1.a1'], 'normalizer': '1.a1.a1', 'old_label': '4.h1.a1', 'projective_image': '32.25', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.h1.a1', 'subgroup_fusion': None, 'weyl_group': '8.3'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 8, 'aut_gen_orders': [2, 2, 2, 8], 'aut_gens': [[1, 4], [3, 4], [1, 20], [1, 12], [21, 20]], 'aut_group': '64.254', 'aut_hash': 254, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 64, 'aut_permdeg': 10, 'aut_perms': [1, 766801, 887887, 512304], 'aut_phi_ratio': 4.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [8, 2, 2, 2], [8, 4, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_8:C_2^3', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '8.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': True, 'autcentquo_order': 8, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [4, 1, 2], [4, 2, 1], [8, 2, 4], [8, 4, 4]], 'center_label': '4.1', 'center_order': 4, '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', '2.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 15, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [8, 2, 2, 2], [8, 4, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 6, 'exponent': 8, 'exponents_of_order': [5], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 0, 4]], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '4.2', 'hash': 15, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 4], 'inner_gens': [[1, 28], [9, 4]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'D_4', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 6]], 'label': '32.15', 'linC_count': 4, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 1, 'linQ_dim': 8, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C8.C4', 'ngens': 2, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 8, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 14, 'number_divisions': 9, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 13, 'number_subgroup_classes': 15, 'number_subgroups': 18, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 4], [8, 24]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [0, 0, 28], 'outer_gens': [[1, 20], [3, 20], [23, 20]], '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': 2, 'perfect': False, 'permutation_degree': 16, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [8, 1]], 'representations': {'PC': {'code': 601866502255, 'gens': [1, 3], 'pres': [5, 2, 2, 2, 2, 2, 10, 86, 422, 42, 483, 58]}, 'GLFp': {'d': 2, 'p': 17, 'gens': [78609, 510, 9835]}, 'Perm': {'d': 16, 'gens': [15310327737600, 8402167094393, 2796019809862, 4097506710982, 1313941673647]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_8.C_4', 'transitive_degree': 16, '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': '32.36', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 8, 'aut_gen_orders': [2, 2, 2, 4, 8], 'aut_gens': [[306, 14742, 9835], [306, 14742, 73703], [306, 29484, 9835], [1377, 54054, 44219], [306, 34398, 9835], [2567, 14742, 73703]], 'aut_group': '256.53144', 'aut_hash': 53144, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 256, 'aut_permdeg': 14, 'aut_perms': [12947966641, 1, 45632568240, 13905647689, 9236062207], 'aut_phi_ratio': 4.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 2, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 2, 4, 1], [4, 4, 2, 1], [8, 1, 4, 1], [8, 2, 2, 3], [8, 4, 4, 1], [16, 1, 8, 1], [16, 2, 4, 1], [16, 2, 8, 1], [16, 4, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4^2.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.45', 'autcent_hash': 45, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '8.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': True, 'autcentquo_order': 8, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 4, 2], [4, 1, 2], [4, 2, 5], [4, 4, 2], [8, 1, 4], [8, 2, 6], [8, 4, 4], [16, 1, 8], [16, 2, 12], [16, 4, 8]], 'center_label': '16.1', 'center_order': 16, 'central_product': True, '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', '2.1', '2.1', '2.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 902, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 2], [4, 1, 2, 1], [4, 2, 1, 1], [4, 2, 2, 2], [4, 4, 1, 2], [8, 1, 4, 1], [8, 2, 2, 3], [8, 4, 2, 2], [16, 1, 8, 1], [16, 2, 4, 1], [16, 2, 8, 1], [16, 4, 4, 2]], 'element_repr_type': 'GLFp', 'elementary': 2, 'eulerian_function': 2688, 'exponent': 16, 'exponents_of_order': [7], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 0, 16]], 'familial': False, 'frattini_label': '16.5', 'frattini_quotient': '8.5', 'hash': 902, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 1, 4], 'inner_gens': [[306, 14742, 44219], [306, 14742, 9835], [3825, 14742, 9835]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': False, 'inner_tex': 'D_4', 'inner_used': [1, 3], 'irrC_degree': 2, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 4, 'irrep_stats': [[1, 32], [2, 24]], 'label': '128.902', 'linC_count': 16, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 2, 'linQ_dim': 16, 'linQ_dim_count': 2, 'linR_count': 8, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C16.D4', 'ngens': 3, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 17, 'number_characteristic_subgroups': 26, 'number_conjugacy_classes': 56, 'number_divisions': 22, 'number_normal_subgroups': 46, 'number_subgroup_autclasses': 51, 'number_subgroup_classes': 73, 'number_subgroups': 104, 'old_label': None, 'order': 128, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 11], [4, 20], [8, 32], [16, 64]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [4914, 9835, 4914, 4914], 'outer_gens': [[306, 14742, 73703], [2635, 14742, 9835], [306, 54054, 9835], [306, 24570, 73703]], 'outer_group': '32.45', 'outer_hash': 45, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 10, 'outer_perms': [120, 362880, 5040, 5049], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 32, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 8], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 5], [8, 1], [16, 2]], 'representations': {'PC': {'code': 18915408845612256302, 'gens': [1, 2, 6], 'pres': [7, 2, 2, 2, 2, 2, 2, 2, 36, 58, 80, 5045, 124, 3926]}, 'GLFp': {'d': 2, 'p': 17, 'gens': [306, 14742, 9835]}, 'Perm': {'d': 32, 'gens': [212165678664023666120575084270247743, 135955991562369039230941840932864000, 127165970120440293996769393998336000, 25199055155429228508132113875036937, 59187974339659933956082255161332976, 25199055155429228508133415361942982, 8231691320578226211046411712681647]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 8], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{16}.D_4', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '4.1', '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], [3]], '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], [2, 1, 1, 1], [4, 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], [2, 1, 1], [4, 1, 2]], 'center_label': '4.1', 'center_order': 4, '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': ['2.1', '2.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [2], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 2]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, '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, 4]], 'label': '4.1', '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': 'C4', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [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': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 1], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3]], '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': 4, 'pgroup': 2, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -2, -2, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [56, 15], 'family': 'CSOPlus'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [21]}, 'Perm': {'d': 4, 'gens': [22, 7]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
