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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1728.10970', 'ambient_counter': 10970, 'ambient_order': 1728, 'ambient_tex': 'C_{12}^2.C_{12}', 'central': True, 'central_factor': False, 'centralizer_order': 1728, 'characteristic': True, 'core_order': 1, 'counter': 102, 'cyclic': True, 'direct': True, 'hall': 1, 'label': '1728.10970.1728.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '1728.a1', 'outer_equivalence': True, 'perfect': True, 'proper': False, 'quotient': '1728.10970', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 10970, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 1728, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{12}^2.C_{12}', 'simple': False, 'solvable': True, 'special_labels': ['PC', 'U2', 'L2', 'D2', 'C9'], 'split': True, 'standard_generators': False, 'stem': True, 'subgroup': '1.1', 'subgroup_hash': 1, 'subgroup_order': 1, 'subgroup_tex': 'C_1', 'supersolvable': True, 'sylow': 1}
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gps_subgroup_data • Show schema
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{'ambient': '1728.10970', 'aut_centralizer_order': None, 'aut_label': '1728.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1.a1', 'complements': ['1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['576.a1', '576.b1', '864.a1', '864.b1'], 'contains': [], 'core': '1728.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [1699, 1999, 2579, 1934], 'generators': [], 'label': '1728.10970.1728.a1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '1728.a1', 'normal_contained_in': ['576.a1', '576.b1', '864.a1', '864.b1'], 'normal_contains': [], 'normalizer': '1.a1', 'old_label': '1728.a1', 'projective_image': '1728.10970', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '1728.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '1.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': 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]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': [], 'composition_length': 0, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1, 'exponent': 1, 'exponents_of_order': [], 'factors_of_aut_order': [], 'factors_of_order': [], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '1.1', 'hash': 1, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [], 'inner_gens': [], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 1]], 'label': '1.1', 'linC_count': 1, 'linC_degree': 0, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 0, 'linQ_degree_count': 1, 'linQ_dim': 0, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 0, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C1', 'ngens': 0, 'nilpotency_class': 0, 'nilpotent': True, 'normal_counts': [1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 1, 'number_characteristic_subgroups': 1, 'number_conjugacy_classes': 1, 'number_divisions': 1, 'number_normal_subgroups': 1, 'number_subgroup_autclasses': 1, 'number_subgroup_classes': 1, 'number_subgroups': 1, 'old_label': None, 'order': 1, 'order_factorization_type': 0, 'order_stats': [[1, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 0, 'perfect': True, 'permutation_degree': 1, 'pgroup': 1, 'primary_abelian_invariants': [], 'quasisimple': False, 'rank': 0, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 1]], 'representations': {'PC': {'code': 0, 'gens': [], 'pres': []}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_1', 'transitive_degree': 1, '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': '432.400', '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], [37, 1592, 366], [1299, 162, 1220], [867, 1326, 118], [903, 464, 956]], 'aut_group': None, 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 663552, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': 1152.0, 'aut_solvable': None, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 1, 4, 1], [3, 1, 2, 1], [3, 1, 6, 1], [4, 2, 12, 1], [4, 4, 8, 1], [6, 1, 6, 1], [6, 1, 8, 1], [6, 1, 18, 1], [6, 1, 24, 1], [9, 1, 18, 1], [12, 2, 24, 1], [12, 2, 72, 1], [12, 4, 16, 1], [12, 4, 48, 1], [18, 1, 54, 1], [18, 1, 72, 1], [36, 2, 216, 1], [36, 4, 144, 1]], '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, 1, 8], [4, 2, 12], [4, 4, 8], [6, 1, 56], [9, 1, 18], [12, 2, 96], [12, 4, 64], [18, 1, 126], [36, 2, 216], [36, 4, 144]], 'center_label': '216.114', 'center_order': 216, 'central_product': True, 'central_quotient': '8.5', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': False, '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': 10970, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['64.64', 1], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 1, 2, 4], [4, 2, 2, 6], [4, 4, 2, 4], [6, 1, 2, 28], [9, 1, 6, 3], [12, 2, 4, 24], [12, 4, 4, 16], [18, 1, 6, 21], [36, 2, 12, 18], [36, 4, 12, 12]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2184, 'exponent': 36, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '24.15', 'frattini_quotient': '72.50', 'hash': 10970, 'hyperelementary': 1, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': [2, 2, 2], 'inner_gens': [[1, 868, 936], [865, 4, 48], [889, 4, 48]], 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': 8, 'inner_split': None, 'inner_tex': 'C_2^3', 'inner_used': None, 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': None, 'irrep_stats': [[1, 432], [2, 324]], 'label': '1728.10970', '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': 'C12^2.C12', 'ngens': 9, 'nilpotency_class': 2, 'nilpotent': True, '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': 20, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 756, 'number_divisions': 144, 'number_normal_subgroups': 410, 'number_subgroup_autclasses': 102, 'number_subgroup_classes': 690, 'number_subgroups': 1050, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 7], [3, 8], [4, 56], [6, 56], [9, 18], [12, 448], [18, 126], [36, 1008]], 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': True, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': None, 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 82944, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': None, 'pc_rank': 3, 'perfect': False, 'permutation_degree': 32, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4, 3, 9], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 36], [4, 22], [6, 24], [8, 24], [12, 12], [24, 18]], 'representations': {'PC': {'code': 65060350797320707318235475908970458503699, 'gens': [1, 3, 6], 'pres': [9, 2, 2, 2, 2, 3, 2, 2, 3, 3, 18, 23438, 74, 102, 50549, 158, 186, 286]}, 'Perm': {'d': 32, 'gens': [8241131970392862943371758537548800, 17260056661189508751733190717875200, 1070190771706677315147650073676800, 507587063266483200, 25764927116365842750596579328000000, 53952542520388435123200, 357120, 79833600, 80884]}}, 'schur_multiplier': [2, 2, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6, 36], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}^2.C_{12}', 'transitive_degree': 1728, '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': '432.400', '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], [37, 1592, 366], [1299, 162, 1220], [867, 1326, 118], [903, 464, 956]], 'aut_group': None, 'aut_hash': None, 'aut_nilpotency_class': None, 'aut_nilpotent': None, 'aut_order': 663552, 'aut_permdeg': None, 'aut_perms': None, 'aut_phi_ratio': 1152.0, 'aut_solvable': None, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 1, 4, 1], [3, 1, 2, 1], [3, 1, 6, 1], [4, 2, 12, 1], [4, 4, 8, 1], [6, 1, 6, 1], [6, 1, 8, 1], [6, 1, 18, 1], [6, 1, 24, 1], [9, 1, 18, 1], [12, 2, 24, 1], [12, 2, 72, 1], [12, 4, 16, 1], [12, 4, 48, 1], [18, 1, 54, 1], [18, 1, 72, 1], [36, 2, 216, 1], [36, 4, 144, 1]], '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, 1, 8], [4, 2, 12], [4, 4, 8], [6, 1, 56], [9, 1, 18], [12, 2, 96], [12, 4, 64], [18, 1, 126], [36, 2, 216], [36, 4, 144]], 'center_label': '216.114', 'center_order': 216, 'central_product': True, 'central_quotient': '8.5', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': False, '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': 10970, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['64.64', 1], ['9.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 1, 2, 4], [4, 2, 2, 6], [4, 4, 2, 4], [6, 1, 2, 28], [9, 1, 6, 3], [12, 2, 4, 24], [12, 4, 4, 16], [18, 1, 6, 21], [36, 2, 12, 18], [36, 4, 12, 12]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 2184, 'exponent': 36, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '24.15', 'frattini_quotient': '72.50', 'hash': 10970, 'hyperelementary': 1, 'inner_abelian': None, 'inner_cyclic': None, 'inner_exponent': None, 'inner_gen_orders': [2, 2, 2], 'inner_gens': [[1, 868, 936], [865, 4, 48], [889, 4, 48]], 'inner_hash': None, 'inner_nilpotent': None, 'inner_order': 8, 'inner_split': None, 'inner_tex': 'C_2^3', 'inner_used': None, 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': None, 'irrep_stats': [[1, 432], [2, 324]], 'label': '1728.10970', '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': 'C12^2.C12', 'ngens': 9, 'nilpotency_class': 2, 'nilpotent': True, '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': 20, 'number_characteristic_subgroups': 20, 'number_conjugacy_classes': 756, 'number_divisions': 144, 'number_normal_subgroups': 410, 'number_subgroup_autclasses': 102, 'number_subgroup_classes': 690, 'number_subgroups': 1050, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 7], [3, 8], [4, 56], [6, 56], [9, 18], [12, 448], [18, 126], [36, 1008]], 'outer_abelian': None, 'outer_cyclic': None, 'outer_equivalence': True, 'outer_exponent': None, 'outer_gen_orders': None, 'outer_gen_pows': None, 'outer_gens': None, 'outer_group': None, 'outer_hash': None, 'outer_nilpotent': None, 'outer_order': 82944, 'outer_permdeg': None, 'outer_perms': None, 'outer_solvable': None, 'outer_supersolvable': None, 'outer_tex': None, 'pc_rank': 3, 'perfect': False, 'permutation_degree': 32, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4, 3, 9], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 36], [4, 22], [6, 24], [8, 24], [12, 12], [24, 18]], 'representations': {'PC': {'code': 65060350797320707318235475908970458503699, 'gens': [1, 3, 6], 'pres': [9, 2, 2, 2, 2, 3, 2, 2, 3, 3, 18, 23438, 74, 102, 50549, 158, 186, 286]}, 'Perm': {'d': 32, 'gens': [8241131970392862943371758537548800, 17260056661189508751733190717875200, 1070190771706677315147650073676800, 507587063266483200, 25764927116365842750596579328000000, 53952542520388435123200, 357120, 79833600, 80884]}}, 'schur_multiplier': [2, 2, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 6, 36], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}^2.C_{12}', 'transitive_degree': 1728, 'wreath_data': None, 'wreath_product': False}
