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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '256.26848', 'ambient_counter': 26848, 'ambient_order': 256, 'ambient_tex': 'D_8.D_8', 'central': False, 'central_factor': False, 'centralizer_order': 16, 'characteristic': False, 'core_order': 32, 'counter': 32, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '256.26848.8.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '8.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '8.3', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 3, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '32.11', 'subgroup_hash': 11, 'subgroup_order': 32, 'subgroup_tex': 'C_4\\wr C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '256.26848', 'aut_centralizer_order': 64, 'aut_label': '8.a1', 'aut_quo_index': 1, 'aut_stab_index': 2, 'aut_weyl_group': '32.46', 'aut_weyl_index': 128, 'centralizer': '16.k1', 'complements': [], 'conjugacy_class_count': 2, 'contained_in': ['4.e1', '4.q1'], 'contains': ['16.a1', '16.b1', '16.e1'], 'core': '8.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [4045, 7960, 4038, 8045], 'generators': [1, 64], 'label': '256.26848.8.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '8.a1', 'normal_contained_in': ['4.e1'], 'normal_contains': ['16.a1', '16.b1', '16.e1'], 'normalizer': '1.a1', 'old_label': '8.a1', 'projective_image': '128.2011', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.a1', 'subgroup_fusion': None, 'weyl_group': '16.11'}
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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': 4, 'aut_gen_orders': [2, 2, 2, 4], 'aut_gens': [[377, 120, 127], [377, 120, 503], [377, 120, 376], [253, 120, 128], [377, 65, 254]], 'aut_group': '32.46', 'aut_hash': 46, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 32, 'aut_permdeg': 8, 'aut_perms': [11527, 11536, 5160, 15120], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 2, 4, 1], [4, 4, 1, 1], [8, 4, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2\\times D_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': True, 'autcentquo_cyclic': False, 'autcentquo_exponent': 2, 'autcentquo_group': '4.2', 'autcentquo_hash': 2, 'autcentquo_nilpotent': True, 'autcentquo_order': 4, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 4, 1], [4, 1, 2], [4, 2, 5], [4, 4, 1], [8, 4, 2]], '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': 11, '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, 1], [4, 1, 2, 1], [4, 2, 1, 1], [4, 2, 2, 2], [4, 4, 1, 1], [8, 4, 2, 1]], 'element_repr_type': 'Lie', 'elementary': 2, 'eulerian_function': 12, 'exponent': 8, 'exponents_of_order': [5], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 0, 4]], 'familial': True, 'frattini_label': '8.2', 'frattini_quotient': '4.2', 'hash': 11, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 2, 4], 'inner_gens': [[377, 30, 127], [253, 120, 251], [377, 85, 127]], 'inner_hash': 3, 'inner_nilpotent': True, 'inner_order': 8, 'inner_split': True, 'inner_tex': 'D_4', 'inner_used': [1, 2, 3], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 4, 'irrep_stats': [[1, 8], [2, 6]], 'label': '32.11', 'linC_count': 4, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 2, 'linQ_dim': 4, 'linQ_dim_count': 2, 'linR_count': 2, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4wrC2', 'ngens': 3, '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': 9, 'number_characteristic_subgroups': 12, 'number_conjugacy_classes': 14, 'number_divisions': 10, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 21, 'number_subgroup_classes': 22, 'number_subgroups': 34, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 7], [4, 16], [8, 8]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [126, 126], 'outer_gens': [[377, 120, 503], [253, 120, 128]], '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': 3, 'perfect': False, 'permutation_degree': 8, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 2]], 'representations': {'PC': {'code': 2366177482, 'gens': [1, 2, 4], 'pres': [5, 2, 2, 2, 2, 2, 26, 603, 58, 504]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [25793872, 16994084]}, 'Lie': [{'d': 2, 'q': 5, 'gens': [377, 120, 127], 'family': 'COPlus'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [30, 251, 377]}, 'Perm': {'d': 8, 'gens': [28793, 23616, 16577, 16582, 5167]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4\\wr C_2', 'transitive_degree': 8, 'wreath_data': ['C_4', 'C_2', '2T1'], 'wreath_product': True}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 16, 'aut_gen_orders': [4, 8, 4, 2, 16, 16, 8], 'aut_gens': [[1, 2, 4, 64], [145, 18, 60, 112], [249, 38, 60, 112], [33, 34, 20, 240], [33, 30, 60, 224], [1, 22, 52, 240], [145, 30, 36, 96], [177, 42, 36, 240]], 'aut_group': None, 'aut_hash': 3762369782519318758, 'aut_nilpotency_class': 4, 'aut_nilpotent': True, 'aut_order': 4096, 'aut_permdeg': 24, 'aut_perms': [242104702069729617456273, 298442520804717707442455, 67445078710357078419243, 70847748908611653857133, 18076042877244264138984, 256767687316676703540902, 280384454974973357433626], 'aut_phi_ratio': 32.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 2, 1], [2, 8, 2, 1], [4, 2, 1, 2], [4, 4, 2, 2], [4, 8, 2, 1], [4, 16, 4, 1], [8, 2, 2, 2], [8, 4, 1, 2], [8, 8, 2, 1], [8, 16, 2, 1], [16, 2, 4, 1], [16, 4, 2, 1], [16, 4, 4, 1], [16, 8, 4, 1]], 'aut_supersolvable': True, 'aut_tex': '(C_2\\times C_4\\times C_8).C_2^6', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 8, 'autcentquo_group': '256.54053', 'autcentquo_hash': 54053, 'autcentquo_nilpotent': True, 'autcentquo_order': 256, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_4^2:C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 4, 2], [2, 8, 2], [4, 2, 2], [4, 4, 4], [4, 8, 2], [4, 16, 4], [8, 2, 4], [8, 4, 2], [8, 8, 2], [8, 16, 2], [16, 2, 4], [16, 4, 6], [16, 8, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '128.2011', 'commutator_count': 1, 'commutator_label': '16.5', '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', '2.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 26848, '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], [2, 8, 1, 2], [4, 2, 1, 2], [4, 4, 1, 4], [4, 8, 1, 2], [4, 16, 1, 4], [8, 2, 2, 2], [8, 4, 1, 2], [8, 8, 1, 2], [8, 16, 1, 2], [16, 2, 4, 1], [16, 4, 2, 1], [16, 4, 4, 1], [16, 8, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 322560, 'exponent': 16, 'exponents_of_order': [8], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[4, -1, 8]], 'familial': False, 'frattini_label': '16.5', 'frattini_quotient': '16.14', 'hash': 26848, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 8, 'inner_gen_orders': [2, 2, 8, 4], 'inner_gens': [[1, 2, 4, 240], [1, 2, 60, 80], [1, 10, 4, 64], [145, 50, 4, 64]], 'inner_hash': 2011, 'inner_nilpotent': True, 'inner_order': 128, 'inner_split': False, 'inner_tex': 'D_4\\times D_8', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 4, 'irrQ_degree': 16, 'irrQ_dim': 32, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 16], [4, 11]], 'label': '256.26848', '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': 'D8.D8', 'ngens': 4, 'nilpotency_class': 4, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 21, 'number_characteristic_subgroups': 33, 'number_conjugacy_classes': 43, 'number_divisions': 32, 'number_normal_subgroups': 105, 'number_subgroup_autclasses': 139, 'number_subgroup_classes': 318, 'number_subgroups': 855, 'old_label': None, 'order': 256, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 27], [4, 100], [8, 64], [16, 64]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4], 'outer_gen_pows': [0, 72, 4, 0], 'outer_gens': [[1, 2, 4, 96], [105, 2, 4, 64], [1, 6, 4, 64], [1, 2, 20, 64]], 'outer_group': '32.45', 'outer_hash': 45, 'outer_nilpotent': True, 'outer_order': 32, 'outer_permdeg': 10, 'outer_perms': [120, 362880, 5040, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 64, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 8], [4, 5], [8, 1], [16, 2]], 'representations': {'PC': {'code': 42868259576438274226198885829673087069, 'gens': [1, 2, 3, 7], 'pres': [8, 2, 2, 2, 2, 2, 2, 2, 2, 256, 730, 66, 907, 91, 972, 116, 13446, 2254, 166, 10247, 5135]}, 'Perm': {'d': 64, 'gens': [50333379976919853828055995732539039199481743370587753166185702980189296658275073052060118, 66450134624445933678799560050331607746391532584685435254529204790382042061733888000000000, 34216625329393954566794504615428400130324458319121250581748252894554539835113394874537647, 16116754647525886103169130832883889877597823742414793216380478825221577993660115377372856, 12087040917006856825820464386751442449456779291427783372556125414865848445276683553449656, 4028664439651445760304855308394900109385475549335344194762832808937879643333428506006982, 6010765313484600225807159681127960740186976874672868603891438014304108462838676947041862, 1983116034341975454789449189498620955194254460449807567074683265838922666583379392681647]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_8.D_8', 'transitive_degree': 64, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 6], [3, 2]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [5, 9], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [4, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', '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': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [4, 2, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [4, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 6], [5, 2]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 1]], 'label': '8.3', 'linC_count': 1, 'linC_degree': 2, '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': 'D4', 'ngens': 2, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 5, 'number_divisions': 5, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 10, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 5], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [2], 'outer_gens': [[3, 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': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1]], 'representations': {'PC': {'code': 294, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 37, 16]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [29, 56, 24], 'family': 'COPlus'}, {'d': 1, 'q': 4, 'gens': [7, 16, 1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [12, 55, 56]}, 'Perm': {'d': 4, 'gens': [6, 16, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4', 'transitive_degree': 4, 'wreath_data': ['C_2', 'C_2', '2T1'], 'wreath_product': True}
