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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '384.1383', 'ambient_counter': 1383, 'ambient_order': 384, 'ambient_tex': 'C_2^2.(S_3\\times D_8)', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': True, 'core_order': 384, 'counter': 1, 'cyclic': False, 'direct': True, 'hall': 6, 'label': '384.1383.1.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '1.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': False, 'quotient': '1.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 1, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_1', 'simple': False, 'solvable': True, 'special_labels': ['R', 'L0', 'D0', 'C0'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '384.1383', 'subgroup_hash': 1383, 'subgroup_order': 384, 'subgroup_tex': 'C_2^2.(S_3\\times D_8)', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '384.1383', 'aut_centralizer_order': None, 'aut_label': '1.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '48.a1.a1', 'complements': ['384.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': [], 'contains': ['2.a1.a1', '2.b1.a1', '2.c1.a1', '2.d1.a1', '2.e1.a1', '2.f1.a1', '2.g1.a1', '3.a1.a1'], 'core': '1.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [2885, 6959, 3084, 3959, 3270, 5815, 3336, 5583], 'generators': [1, 8, 288, 2, 4, 128, 192, 48], 'label': '384.1383.1.a1.a1', 'mobius_quo': 0, 'mobius_sub': 1, 'normal_closure': '1.a1.a1', 'normal_contained_in': [], 'normal_contains': ['2.a1.a1', '2.b1.a1', '2.c1.a1', '2.d1.a1', '2.e1.a1', '2.f1.a1', '2.g1.a1'], 'normalizer': '1.a1.a1', 'old_label': '1.a1.a1', 'projective_image': '48.38', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '1.a1.a1', 'subgroup_fusion': None, 'weyl_group': '48.38'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [6, 4, 4, 2, 2, 2, 2, 2, 6, 2, 2, 6], 'aut_gens': [[1, 2, 8, 16], [73, 298, 8, 124], [133, 294, 8, 84], [9, 110, 8, 92], [265, 14, 8, 376], [5, 110, 8, 368], [133, 294, 8, 180], [265, 102, 8, 380], [65, 10, 8, 188], [261, 98, 8, 116], [201, 2, 8, 16], [133, 102, 8, 376], [133, 194, 8, 24]], 'aut_group': None, 'aut_hash': 5818047291243619351, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12288, 'aut_permdeg': 56, 'aut_perms': [439567502207830587558465060175601056693246064465639970433602040836010314597, 650834865730968770223028827733326541534392460101933304313034388655315488719, 482260680900315030610821916814925537697022227913569695331941081229186616051, 389094861489080764524193922948261873196582696569293754232720418218416089071, 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'autcent_group': '512.10494213', 'autcent_hash': 1718285292446712970, 'autcent_nilpotent': True, 'autcent_order': 512, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^9', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '24.14', 'autcentquo_hash': 14, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_6', 'cc_stats': [[1, 1, 1], [2, 1, 7], [3, 2, 1], [4, 2, 4], [4, 8, 4], [4, 12, 4], [4, 24, 4], [6, 2, 7], [8, 4, 4], [8, 12, 4], [12, 4, 4], [12, 8, 8], [24, 4, 8]], 'center_label': '8.5', 'center_order': 8, 'central_product': False, 'central_quotient': '48.38', 'commutator_count': 2, 'commutator_label': '48.44', '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', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 1383, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 2, 1, 1], [4, 2, 1, 4], [4, 8, 1, 2], [4, 8, 2, 1], [4, 12, 1, 2], [4, 12, 2, 1], [4, 24, 1, 2], [4, 24, 2, 1], [6, 2, 1, 7], [8, 4, 4, 1], [8, 12, 2, 2], [12, 4, 1, 4], [12, 8, 2, 2], [12, 8, 4, 1], [24, 4, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1344, 'exponent': 24, 'exponents_of_order': [7, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.10', 'frattini_quotient': '24.14', 'hash': 1383, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 1, 12], 'inner_gens': [[1, 10, 8, 276], [9, 2, 8, 120], [1, 2, 8, 16], [133, 298, 8, 16]], 'inner_hash': 38, 'inner_nilpotent': False, 'inner_order': 48, 'inner_split': None, 'inner_tex': 'S_3\\times D_4', 'inner_used': [1, 2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 38], [4, 14]], 'label': '384.1383', 'linC_count': 1280, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 272, 'linQ_dim': 12, 'linQ_dim_count': 108, 'linR_count': 48, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2.(S3*D8)', 'ngens': 8, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 35, 'number_characteristic_subgroups': 75, 'number_conjugacy_classes': 60, 'number_divisions': 40, 'number_normal_subgroups': 85, 'number_subgroup_autclasses': 196, 'number_subgroup_classes': 224, 'number_subgroups': 710, 'old_label': None, 'order': 384, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 184], [6, 14], [8, 64], [12, 80], [24, 32]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [256, 288, 288, 0, 0, 0, 0, 0], 'outer_gens': [[73, 298, 8, 124], [133, 294, 8, 84], [9, 110, 8, 92], [265, 14, 8, 376], [5, 110, 8, 368], [133, 294, 8, 180], [265, 102, 8, 380], [65, 10, 8, 188]], 'outer_group': '256.56092', 'outer_hash': 56092, 'outer_nilpotent': True, 'outer_order': 256, 'outer_permdeg': 256, 'outer_perms': 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27044740141713833825445852306040680129587906895586203655329951247052902857751765733416747168327116201926620081966793000724706670348383309738656139918459778556255288101797195640508363573363383925918760372303545576468079252200862096559940505644270968765511980135113859607400477663054601924723381930400910429165477405755767073182522755459397508208676535095053841080277575242493859918333413915639556704447337722858274036168155690987158347216324540386148609106730227377159846040918425346696438595841233820960688], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^8', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 23, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 12], [4, 13], [8, 7]], 'representations': {'PC': {'code': 962112208136386945114210160837576384901889, 'gens': [1, 2, 4, 5], 'pres': [8, -2, -2, -2, 2, 2, -2, -2, -3, 32, 161, 41, 11044, 2412, 116, 7685, 5389, 141, 17926, 166, 16391]}, 'Perm': {'d': 23, 'gens': [1231432558559728446361, 2401761594782566211785, 207302479228226483544, 16511160, 3639884278164422783160, 186827135160, 2401761594769625088000, 3]}}, 'schur_multiplier': [2, 2], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2.(S_3\\times D_8)', 'transitive_degree': 384, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 12, 'aut_gen_orders': [6, 4, 4, 2, 2, 2, 2, 2, 6, 2, 2, 6], 'aut_gens': [[1, 2, 8, 16], [73, 298, 8, 124], [133, 294, 8, 84], [9, 110, 8, 92], [265, 14, 8, 376], [5, 110, 8, 368], [133, 294, 8, 180], [265, 102, 8, 380], [65, 10, 8, 188], [261, 98, 8, 116], [201, 2, 8, 16], [133, 102, 8, 376], [133, 194, 8, 24]], 'aut_group': None, 'aut_hash': 5818047291243619351, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12288, 'aut_permdeg': 56, 'aut_perms': [439567502207830587558465060175601056693246064465639970433602040836010314597, 650834865730968770223028827733326541534392460101933304313034388655315488719, 482260680900315030610821916814925537697022227913569695331941081229186616051, 389094861489080764524193922948261873196582696569293754232720418218416089071, 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'autcent_group': '512.10494213', 'autcent_hash': 1718285292446712970, 'autcent_nilpotent': True, 'autcent_order': 512, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^9', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '24.14', 'autcentquo_hash': 14, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_6', 'cc_stats': [[1, 1, 1], [2, 1, 7], [3, 2, 1], [4, 2, 4], [4, 8, 4], [4, 12, 4], [4, 24, 4], [6, 2, 7], [8, 4, 4], [8, 12, 4], [12, 4, 4], [12, 8, 8], [24, 4, 8]], 'center_label': '8.5', 'center_order': 8, 'central_product': False, 'central_quotient': '48.38', 'commutator_count': 2, 'commutator_label': '48.44', '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', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 1383, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 2, 1, 1], [4, 2, 1, 4], [4, 8, 1, 2], [4, 8, 2, 1], [4, 12, 1, 2], [4, 12, 2, 1], [4, 24, 1, 2], [4, 24, 2, 1], [6, 2, 1, 7], [8, 4, 4, 1], [8, 12, 2, 2], [12, 4, 1, 4], [12, 8, 2, 2], [12, 8, 4, 1], [24, 4, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1344, 'exponent': 24, 'exponents_of_order': [7, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.10', 'frattini_quotient': '24.14', 'hash': 1383, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 1, 12], 'inner_gens': [[1, 10, 8, 276], [9, 2, 8, 120], [1, 2, 8, 16], [133, 298, 8, 16]], 'inner_hash': 38, 'inner_nilpotent': False, 'inner_order': 48, 'inner_split': None, 'inner_tex': 'S_3\\times D_4', 'inner_used': [1, 2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 38], [4, 14]], 'label': '384.1383', 'linC_count': 1280, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 272, 'linQ_dim': 12, 'linQ_dim_count': 108, 'linR_count': 48, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2.(S3*D8)', 'ngens': 8, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 35, 'number_characteristic_subgroups': 75, 'number_conjugacy_classes': 60, 'number_divisions': 40, 'number_normal_subgroups': 85, 'number_subgroup_autclasses': 196, 'number_subgroup_classes': 224, 'number_subgroups': 710, 'old_label': None, 'order': 384, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 184], [6, 14], [8, 64], [12, 80], [24, 32]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [256, 288, 288, 0, 0, 0, 0, 0], 'outer_gens': [[73, 298, 8, 124], [133, 294, 8, 84], [9, 110, 8, 92], [265, 14, 8, 376], [5, 110, 8, 368], [133, 294, 8, 180], [265, 102, 8, 380], [65, 10, 8, 188]], 'outer_group': '256.56092', 'outer_hash': 56092, 'outer_nilpotent': True, 'outer_order': 256, 'outer_permdeg': 256, 'outer_perms': 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2412, 116, 7685, 5389, 141, 17926, 166, 16391]}, 'Perm': {'d': 23, 'gens': [1231432558559728446361, 2401761594782566211785, 207302479228226483544, 16511160, 3639884278164422783160, 186827135160, 2401761594769625088000, 3]}}, 'schur_multiplier': [2, 2], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2.(S_3\\times D_8)', 'transitive_degree': 384, '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': '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}
