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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '192.65', 'ambient_counter': 65, 'ambient_order': 192, 'ambient_tex': 'C_{48}.C_4', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 96, 'counter': 2, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '192.65.2.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '2.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '2.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 2, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2', 'simple': False, 'solvable': True, 'special_labels': ['C1'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '96.26', 'subgroup_hash': 26, 'subgroup_order': 96, 'subgroup_tex': 'C_{24}.C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '192.65', 'aut_centralizer_order': 2, 'aut_label': '2.a1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '384.19764', 'aut_weyl_index': 4, 'centralizer': '48.b1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['4.a1.a1', '4.d1.a1', '6.b1.a1'], 'core': '2.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [5843, 4850, 3995, 3613, 3682, 3856, 6038, 4379], 'generators': [1, 64, 96, 72, 2, 48], 'label': '192.65.2.a1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '2.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['4.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '2.a1.a1', 'projective_image': '48.7', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '2.a1.a1', 'subgroup_fusion': None, 'weyl_group': '48.7'}
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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': 24, 'aut_gen_orders': [2, 2, 2, 2, 24], 'aut_gens': [[1, 4], [51, 52], [51, 4], [1, 68], [51, 44], [45, 4]], 'aut_group': '384.19764', 'aut_hash': 19764, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 384, 'aut_permdeg': 13, 'aut_perms': [766800, 1, 7, 480140647, 1612406910], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [3, 2, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [6, 2, 1, 1], [6, 2, 2, 1], [8, 2, 2, 2], [8, 12, 4, 1], [12, 2, 2, 2], [24, 2, 4, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_{24}:C_2^4', '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': 12, 'autcentquo_group': '48.38', 'autcentquo_hash': 38, 'autcentquo_nilpotent': False, 'autcentquo_order': 48, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times D_4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [3, 2, 1], [4, 1, 2], [4, 2, 1], [6, 2, 3], [8, 2, 4], [8, 12, 4], [12, 2, 4], [24, 2, 8]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, 'central_quotient': '24.6', 'commutator_count': 1, 'commutator_label': '12.2', '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': 26, '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], [3, 2, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [6, 2, 1, 1], [6, 2, 2, 1], [8, 2, 2, 2], [8, 12, 2, 2], [12, 2, 2, 2], [24, 2, 4, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 24, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 8]], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '12.4', 'hash': 26, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 12], 'inner_gens': [[1, 92], [9, 4]], 'inner_hash': 6, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': False, 'inner_tex': 'D_{12}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 22]], 'label': '96.26', 'linC_count': 8, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 4, 'linQ_dim': 10, 'linQ_dim_count': 2, 'linR_count': 4, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C24.C4', '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': 15, 'number_characteristic_subgroups': 19, 'number_conjugacy_classes': 30, 'number_divisions': 16, 'number_normal_subgroups': 21, 'number_subgroup_autclasses': 26, 'number_subgroup_classes': 30, 'number_subgroups': 50, 'old_label': None, 'order': 96, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 4], [6, 6], [8, 56], [12, 8], [24, 16]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 12], 'outer_gens': [[1, 52], [3, 4], [1, 28], [13, 4]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 2], [8, 1], [16, 1]], 'representations': {'PC': {'code': 494470783665631792265153, 'gens': [1, 3], 'pres': [6, -2, -2, -2, -2, -2, -3, 12, 295, 1658, 50, 2115, 69, 2404, 88, 2309]}, 'GLFp': {'d': 2, 'p': 73, 'gens': [25675174, 7300, 28009225]}, 'Perm': {'d': 19, 'gens': [7267562766059065, 14655762618185784, 21513873212260584, 27932200545167304, 34716404770367160, 3]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{24}.C_4', 'transitive_degree': 48, '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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 48, 'aut_gen_orders': [2, 2, 2, 4, 48], 'aut_gens': [[1, 4], [3, 100], [1, 68], [109, 188], [169, 44], [77, 4]], 'aut_group': '1536.230683028', 'aut_hash': 8779664174755546961, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 1536, 'aut_permdeg': 21, 'aut_perms': [1, 6, 17566011496111166646, 15000853136486968806, 10508941326955237230], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [3, 2, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [6, 2, 1, 1], [6, 2, 2, 1], [8, 2, 2, 2], [8, 24, 4, 1], [12, 2, 2, 2], [16, 2, 4, 2], [24, 2, 4, 2], [48, 2, 8, 2]], 'aut_supersolvable': True, 'aut_tex': 'D_6\\times D_{16}:C_4', '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': 24, 'autcentquo_group': '192.1331', 'autcentquo_hash': 1331, 'autcentquo_nilpotent': False, 'autcentquo_order': 192, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_8:D_6', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [3, 2, 1], [4, 1, 2], [4, 2, 1], [6, 2, 3], [8, 2, 4], [8, 24, 4], [12, 2, 4], [16, 2, 8], [24, 2, 8], [48, 2, 16]], 'center_label': '4.1', 'center_order': 4, 'central_product': False, 'central_quotient': '48.7', 'commutator_count': 1, 'commutator_label': '24.2', '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'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 65, '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], [3, 2, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [6, 2, 1, 1], [6, 2, 2, 1], [8, 2, 2, 2], [8, 24, 2, 2], [12, 2, 2, 2], [16, 2, 4, 2], [24, 2, 4, 2], [48, 2, 8, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 48, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 0, 16]], 'familial': False, 'frattini_label': '16.5', 'frattini_quotient': '12.4', 'hash': 65, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [2, 24], 'inner_gens': [[1, 92], [105, 4]], 'inner_hash': 7, 'inner_nilpotent': False, 'inner_order': 48, 'inner_split': False, 'inner_tex': 'D_{24}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 46]], 'label': '192.65', 'linC_count': 16, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 18, 'linQ_degree_count': 4, 'linQ_dim': 18, 'linQ_dim_count': 2, 'linR_count': 8, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C48.C4', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 19, 'number_characteristic_subgroups': 25, 'number_conjugacy_classes': 54, 'number_divisions': 20, 'number_normal_subgroups': 27, 'number_subgroup_autclasses': 34, 'number_subgroup_classes': 40, 'number_subgroups': 88, 'old_label': None, 'order': 192, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 4], [6, 6], [8, 104], [12, 8], [16, 16], [24, 16], [48, 32]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [0, 0, 60, 0], 'outer_gens': [[3, 4], [1, 28], [13, 4], [1, 148]], 'outer_group': '32.45', 'outer_hash': 45, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 10, 'outer_perms': [362880, 5040, 120, 17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 35, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 4], [8, 2], [16, 1], [32, 1]], 'representations': {'PC': {'code': 1181929598271331148122060380168065, 'gens': [1, 3], 'pres': [7, -2, -2, -2, -2, -2, -2, -3, 14, 680, 1934, 58, 5155, 80, 6164, 102, 6725, 124, 6278]}, 'GLFp': {'d': 2, 'p': 97, 'gens': [11543, 87616609, 60236443]}, 'Perm': {'d': 35, 'gens': [315020797296331646918615236130521602241, 645281784668631175692030067221227990880, 949420623217391539478097555887055738984, 1253671306282987188392545345199804029344, 1540370342827916708849695444873770235704, 1853275604215155380388220644227032279224, 3]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{48}.C_4', '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': '2.1', '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': [[1]], '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], [2, 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], [2, 1, 1]], 'center_label': '2.1', 'center_order': 2, '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'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', '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': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', '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': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 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': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, 'wreath_data': None, 'wreath_product': False}
