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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': 24, 'characteristic': False, 'core_order': 24, 'counter': 45, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '384.1383.8.k1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '8.k1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 8, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '48.22', 'subgroup_hash': 22, 'subgroup_order': 48, 'subgroup_tex': 'C_4:C_{12}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '384.1383', 'aut_centralizer_order': 192, 'aut_label': '8.k1', 'aut_quo_index': None, 'aut_stab_index': 2, 'aut_weyl_group': '32.46', 'aut_weyl_index': 384, 'centralizer': '16.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['4.b1.a1', '4.l1.a1', '4.l1.b1'], 'contains': ['16.b1.a1', '16.i1.a1', '24.i1.a1'], 'core': '16.b1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [4158, -1, 1723, -1, 3058, -1, 3611, -1], 'generators': [2, 192, 96, 128, 4], 'label': '384.1383.8.k1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '4.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.e1.a1', 'old_label': '8.k1.a1', 'projective_image': '96.91', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.k1.a1', 'subgroup_fusion': None, 'weyl_group': '8.3'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '24.9', '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, 2, 4], 'aut_gens': [[1, 4], [1, 6], [27, 28], [1, 20], [3, 4], [13, 4]], 'aut_group': '64.202', 'aut_hash': 202, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 64, 'aut_permdeg': 10, 'aut_perms': [368045, 5, 136, 806416, 85689], 'aut_phi_ratio': 4.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [4, 2, 2, 1], [4, 2, 4, 1], [6, 1, 2, 3], [12, 2, 4, 1], [12, 2, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^3:D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '32.51', 'autcent_hash': 51, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^5', '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, 3], [3, 1, 2], [4, 2, 6], [6, 1, 6], [12, 2, 12]], 'center_label': '12.5', 'center_order': 12, 'central_product': True, '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', '2.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 22, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['16.4', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [3, 1, 2, 1], [4, 2, 1, 2], [4, 2, 2, 2], [6, 1, 2, 3], [12, 2, 2, 2], [12, 2, 4, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 12, 'exponent': 12, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '12.5', 'hash': 22, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 28], [25, 4]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 24], [2, 6]], 'label': '48.22', 'linC_count': 64, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 30, 'linQ_dim': 6, 'linQ_dim_count': 15, 'linR_count': 4, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4:C12', 'ngens': 5, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 12, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 30, 'number_divisions': 16, 'number_normal_subgroups': 22, 'number_subgroup_autclasses': 20, 'number_subgroup_classes': 26, 'number_subgroups': 30, 'old_label': None, 'order': 48, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 3], [3, 2], [4, 12], [6, 6], [12, 24]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 4, 0, 0], 'outer_gens': [[1, 22], [13, 20], [3, 20], [3, 4]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [126, 55, 289, 288], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 4, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 8], [4, 4]], 'representations': {'PC': {'code': 233217983905, 'gens': [1, 3], 'pres': [5, -2, -2, -2, -2, -3, 10, 422, 42, 58]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101739150919715, 91655858505667262]}, 'GLFp': {'d': 3, 'p': 13, 'gens': [3384305574, 7341833547, 5431593440, 1172775617, 8157592830]}, 'Perm': {'d': 11, 'gens': [766944, 4033680, 4, 7983360, 1680]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 12], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4:C_{12}', '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': '8.5', '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, 279305828651175719296840218754644317098605558552748175132391063734133308447, 3062545019491378506810224616460009108431724803561202805356237086945427154, 361915153574177202027350802376628694577893818939373700162703842630824710766, 565392036540904164237774387168305038790599350918425040274945661825047046807, 626836494989952762334642466416719663268200453833945571611565705100492554128, 3536278893541393257184484750059090448664415185047133362736431688593724, 272837768925130377810374503062352546749622468524093516863885832917401033091, 168465505090958284457519465314007316137747844489288948910477507580086894697], 'aut_phi_ratio': 96.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [3, 2, 1, 1], [4, 2, 1, 4], [4, 8, 2, 2], [4, 12, 2, 2], [4, 24, 2, 2], [6, 2, 1, 7], [8, 4, 4, 1], [8, 12, 4, 1], [12, 4, 1, 4], [12, 8, 4, 2], [24, 4, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_3:(C_2^7.C_2^5)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, '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': [3351422023934656266081931436769490705747323103423106298832007889524734196034050997041161100857526185207567190034432856690730327456509652072958911691731125375506629686841874116080660909106369844379400344255087916490182804052223232667305373595404020781005504786110187377720655396626265263717074325402070353830433447564819521644993727568233444500510194519274134864307660991032367399123295468207374970146421619898965611711750238972900029482047565794932069782979254483161945817863261940465137841340416216442996, 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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}
