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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'ambient': '96.9', 'ambient_counter': 9, 'ambient_order': 96, 'ambient_tex': 'C_{12}:C_8', 'central': False, 'central_factor': True, 'centralizer_order': 16, 'characteristic': False, 'core_order': 24, 'counter': 11, 'cyclic': False, 'direct': True, 'hall': 0, 'label': '96.9.4.c1.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '4.c1.c1', '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': True, 'standard_generators': False, 'stem': False, 'subgroup': '24.1', 'subgroup_hash': 1, 'subgroup_order': 24, 'subgroup_tex': 'C_3:C_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '96.9', 'aut_centralizer_order': 8, 'aut_label': '4.c1', 'aut_quo_index': 1, 'aut_stab_index': 4, 'aut_weyl_group': '24.14', 'aut_weyl_index': 32, 'centralizer': '6.a1.a1', 'complements': ['24.c1.a1', '24.c1.b1', '24.c1.c1', '24.c1.d1'], 'conjugacy_class_count': 1, 'contained_in': ['2.b1.b1'], 'contains': ['8.b1.b1', '12.c1.c1'], 'core': '4.c1.c1', 'coset_action_label': None, 'count': 1, 'diagramx': [9264, 9291, 269, 2948, 9449, 9349, 9074, 8919], 'generators': [9, 50, 4, 32], 'label': '96.9.4.c1.c1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '4.c1.c1', 'normal_contained_in': ['2.b1.b1'], 'normal_contains': ['8.b1.b1'], 'normalizer': '1.a1.a1', 'old_label': '4.c1.c1', 'projective_image': '24.5', 'quotient_action_image': '1.1', 'quotient_action_kernel': '4.1', 'quotient_action_kernel_order': 4, 'quotient_fusion': None, 'short_label': '4.c1.c1', 'subgroup_fusion': None, 'weyl_group': '6.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '8.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 2, 6], 'aut_gens': [[1, 8], [5, 8], [1, 16], [11, 8]], 'aut_group': '24.14', 'aut_hash': 14, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24, 'aut_permdeg': 7, 'aut_perms': [7, 136, 856], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 1], [4, 1, 2, 1], [6, 2, 1, 1], [8, 3, 4, 1], [12, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2\\times D_6', '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], [3, 2, 1], [4, 1, 2], [6, 2, 1], [8, 3, 4], [12, 2, 2]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1'], 'composition_length': 4, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 1], [4, 1, 2, 1], [6, 2, 1, 1], [8, 3, 4, 1], [12, 2, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 24, 'exponents_of_order': [3, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 2]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '6.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 16], [17, 8]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 4]], 'label': '24.1', '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': 3, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C3:C8', 'ngens': 4, 'nilpotency_class': -1, 'nilpotent': False, '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': 7, 'number_characteristic_subgroups': 7, 'number_conjugacy_classes': 12, 'number_divisions': 7, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 8, 'number_subgroups': 10, 'old_label': None, 'order': 24, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 2], [6, 2], [8, 12], [12, 4]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 8], [5, 8]], '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': 2, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 2]], 'representations': {'PC': {'code': 5358129, 'gens': [1, 4], 'pres': [4, -2, -2, -2, -3, 8, 21, 259]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [41537776, 23043043]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [539, 596]}, 'Perm': {'d': 11, 'gens': [4083265, 8443680, 12501360, 3]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [8], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_3:C_8', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '32.3', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 2, 2, 2, 2, 2, 3], 'aut_gens': [[1, 8], [75, 8], [3, 8], [7, 14], [1, 40], [49, 8], [7, 56], [1, 12], [5, 8], [33, 8]], 'aut_group': '768.196796', 'aut_hash': 196796, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 768, 'aut_permdeg': 15, 'aut_perms': [87261756024, 193077562344, 650176491600, 745, 88176211200, 88216169064, 193158120960, 88216168320, 3], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [3, 2, 1, 1], [4, 1, 4, 1], [4, 1, 8, 1], [6, 2, 1, 1], [6, 2, 2, 1], [8, 3, 16, 1], [12, 2, 4, 1], [12, 2, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3\\times C_2^4:D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.753', 'autcent_hash': 753, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4:D_4', '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, 3], [3, 2, 1], [4, 1, 12], [6, 2, 3], [8, 3, 16], [12, 2, 12]], 'center_label': '16.2', 'center_order': 16, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', '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': 9, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['24.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 2, 1, 1], [4, 1, 2, 6], [6, 2, 1, 3], [8, 3, 4, 4], [12, 2, 2, 6]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 24, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '12.4', 'hash': 9, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 40], [65, 8]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 16]], 'label': '96.9', 'linC_count': 192, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 16, 'linQ_dim': 6, 'linQ_dim_count': 16, 'linR_count': 144, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C12:C8', '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': 11, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 48, 'number_divisions': 24, 'number_normal_subgroups': 37, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 44, 'number_subgroups': 58, 'old_label': None, 'order': 96, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 12], [6, 6], [8, 48], [12, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[27, 8], [7, 8], [3, 10], [49, 8], [7, 56], [1, 12], [5, 8]], 'outer_group': '128.753', 'outer_hash': 753, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 12, 'outer_perms': [262459456, 40805400, 766201, 90842400, 40723327, 85800, 40723320], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [4, 8], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 10], [4, 10]], 'representations': {'PC': {'code': 53630121939352741569, 'gens': [1, 4], 'pres': [6, -2, -2, -2, -2, -2, -3, 12, 31, 963, 69, 2404, 88, 2309]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101742620718405, 91655870132509410]}, 'GLFp': {'d': 3, 'p': 5, 'gens': [1173753, 106892, 1692709, 518956, 1449557, 1565004]}, 'Perm': {'d': 15, 'gens': [94447180930, 13, 194123381783, 23, 288369849600, 840]}}, 'schur_multiplier': [4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [4, 8], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}:C_8', 'transitive_degree': 96, '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}
