Formats: - HTML - YAML - JSON - 2026-07-24T03:16:24.598877
  • gps_subgroup_search • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '12130560.c', 'ambient_counter': 3, 'ambient_order': 12130560, 'ambient_tex': '\\PGL(4,3)', 'central': False, 'central_factor': False, 'centralizer_order': 1, 'characteristic': False, 'core_order': 1, 'counter': 236, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '12130560.c.18720.Y', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '18720.x2', 'outer_equivalence': True, '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': 18720, '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': '648.730', 'subgroup_hash': 730, 'subgroup_order': 648, 'subgroup_tex': 'C_3^4:Q_8', 'supersolvable': False, 'sylow': 0}
  • gps_subgroup_data • Show schema
    {'ambient': '12130560.c', 'aut_centralizer_order': None, 'aut_label': '18720.Y', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '12130560.a1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['9360.E', '9360.G', '9360.Q', '9360.V', '9360.W', '9360.BC', '9360.BD'], 'contains': ['37440.W', '37440.X', '168480.BE'], 'core': '12130560.a1', 'coset_action_label': None, 'count': 2340, 'diagramx': None, 'generators': [10464457, 29075747, 22749994, 41950063, 30987847, 6272156, 14615915], 'label': '12130560.c.18720.Y', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '1170.a1', 'old_label': '18720.x2', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '18720.Y', 'subgroup_fusion': None, 'weyl_group': None}
  • gps_groups • Show schema
    {'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': 5, 'aut_exponent': 72, 'aut_gen_orders': [6, 24], 'aut_gens': [[11445301, 24577758, 24507118, 14850479, 2939606, 36731395, 7355200], [42961672, 24577758, 14481270, 2793523, 32864349, 14582400, 20258849], [24734080, 14633333, 2721277, 14634003, 14686477, 6973453, 32703882]], 'aut_group': '93312.jq', 'aut_hash': 4164635616307602250, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 93312, 'aut_permdeg': 486, 'aut_perms': [5437964660965527585979868445704484624832474997271145096148891209700100478376184576002795419461316555390849901037825186759105777071549839590085117592194816475757418798367695604316141897118558141162596806297221485843253356115885107937381030258848521611473345481693221798260630503776151587480845045243553131787257518117875170737019066223168012784351183610493649370229479820121881448464577765734517223924906958819366969569678769001616790702829234004784940839608963844625215951610902000412492153578174584950344287138620583164860370809385327847571885845775519412080808887555478910110485668491891620559393102016701008506746686647802795064390841591650024235219026942795086518419254772525711387960052227199831433463686208762975604450593557139925892295926662917731731660161922955295834036863534252400980534072706359582048863890101314086042085805665568744042377895891536995302071140808896763805714162406763315529342434886837120502260096792717374925110349083685154209584621083444423565648883784302658210130068930176485080500757198955401448076983927584406283993653086861600484142388618949629829279954550580109, 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'aut_phi_ratio': 432.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [3, 8, 4, 1], [3, 8, 6, 1], [4, 162, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4:\\GL(2,3):S_4', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 72, 'autcentquo_group': '93312.jq', 'autcentquo_hash': 4164635616307602250, 'autcentquo_nilpotent': False, 'autcentquo_order': 93312, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^4:\\GL(2,3):S_4', 'cc_stats': [[1, 1, 1], [2, 81, 1], [3, 8, 10], [4, 162, 3]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '648.730', 'commutator_count': 1, 'commutator_label': '162.54', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 730, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 81, 1, 1], [3, 8, 1, 10], [4, 162, 1, 3]], 'element_repr_type': 'GLFp', 'elementary': 1, 'eulerian_function': 1, 'exponent': 12, 'exponents_of_order': [4, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 1, 6]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '648.730', 'hash': 730, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [4, 3, 3, 4, 3, 3, 2], 'inner_gens': [[11445301, 11495158, 11225629, 20327188, 43014227, 24507118, 20207646], [24508308, 24577758, 24507118, 11169773, 2939606, 36731395, 11534549], [24734080, 24577758, 24507118, 11659160, 2939606, 36731395, 11607619], [42746117, 32864349, 32479219, 14850479, 24577758, 6973453, 11501225], [14358460, 24577758, 24507118, 6970741, 2939606, 36731395, 20477175], [20481904, 24577758, 24507118, 3156995, 2939606, 36731395, 32870443], [42652886, 20201582, 20254942, 24511696, 32864349, 11225629, 7355200]], 'inner_hash': 730, 'inner_nilpotent': False, 'inner_order': 648, 'inner_split': False, 'inner_tex': 'C_3^4:Q_8', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 4], [2, 1], [8, 10]], 'label': '648.730', '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': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^4:Q8', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 5, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 15, 'number_divisions': 15, 'number_normal_subgroups': 11, 'number_subgroup_autclasses': 29, 'number_subgroup_classes': 157, 'number_subgroups': 3298, 'old_label': None, 'order': 648, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 81], [3, 80], [4, 486]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 3, 3, 2, 2], 'outer_gen_pows': [14408200, 2790694, 36905383, 14408200, 2790694, 2790694], 'outer_gens': [[42894928, 20201582, 20254942, 32752764, 11495158, 32479219, 11660736], [20695177, 32864349, 14800134, 24511696, 24577758, 14582400, 37006991], [20695177, 32864349, 14633333, 24582579, 24577758, 14853281, 36902914], [42894928, 11495158, 11225629, 14413702, 2939606, 6973453, 20477175], [11445301, 20254942, 24577758, 20206376, 32479219, 37043725, 24782714], [11445301, 14481270, 24454001, 24781216, 14633333, 37010401, 32703882]], 'outer_group': '144.183', 'outer_hash': 183, 'outer_nilpotent': False, 'outer_order': 144, 'outer_permdeg': 7, 'outer_perms': [1, 120, 144, 3, 744, 1680], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'S_3\\times S_4', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1], [8, 10]], 'representations': {'PC': {'code': 39726950185964376922945666064459919893945921282447, 'gens': [1, 2, 4, 5, 6, 7], 'pres': [7, -2, -2, -2, -3, 3, -3, 3, 28, 85, 36, 1795, 346, 129, 1404, 291, 438, 21173, 6060, 1531, 28230, 12361, 5312]}, 'GLFp': {'d': 4, 'p': 3, 'gens': [11445301, 24577758, 24507118, 14850479, 2939606, 36731395, 7355200]}, 'Perm': {'d': 12, 'gens': [267361200, 135354691, 3633865, 43545748, 43590963, 45507, 79879200]}}, 'schur_multiplier': [3, 3, 3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:Q_8', 'transitive_degree': 12, 'wreath_data': None, 'wreath_product': False}
  • gps_groups • Show schema
    {'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': True, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 4680, 'aut_gen_orders': [8, 8, 8], 'aut_gens': [[14408201, 9605495], [23793960, 32856903], [19251676, 29495479], [21546876, 29607407]], 'aut_group': None, 'aut_hash': 977983066756978958, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24261120, 'aut_permdeg': 1080, 'aut_perms': 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['2.1', '6065280.a'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1080, 1, 1], [2, 2106, 1, 1], [2, 5265, 1, 1], [3, 1040, 1, 1], [3, 6240, 1, 1], [3, 74880, 1, 1], [4, 4212, 1, 1], [4, 63180, 1, 1], [4, 189540, 1, 1], [4, 379080, 1, 1], [5, 303264, 1, 1], [6, 84240, 1, 1], [6, 112320, 1, 1], [6, 168480, 1, 1], [6, 336960, 1, 1], [6, 673920, 1, 1], [8, 63180, 2, 1], [8, 151632, 2, 1], [8, 379080, 1, 1], [8, 758160, 1, 1], [9, 449280, 1, 1], [10, 303264, 1, 1], [12, 336960, 1, 1], [12, 505440, 1, 1], [13, 466560, 4, 1], [20, 303264, 2, 1], [24, 505440, 2, 1], [26, 466560, 4, 1], [40, 303264, 4, 1]], 'element_repr_type': 'Lie', 'elementary': 1, 'eulerian_function': None, 'exponent': 4680, 'exponents_of_order': [8, 6, 1, 1], 'factors_of_aut_order': [2, 3, 5, 13], 'factors_of_order': [2, 3, 5, 13], 'faithful_reps': [[39, 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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': False, 'metacyclic': False, 'monomial': False, 'name': 'PGL(4,3)', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, '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, 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, 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, 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, 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, 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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': None, 'perfect': False, 'permutation_degree': 40, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [39, 2], [52, 1], [90, 2], [104, 1], [130, 1], [260, 2], [351, 2], [390, 2], [416, 2], [468, 1], [520, 1], [729, 2], [780, 2], [832, 1], [936, 1], [1170, 1], [1664, 1], [2080, 1], [2560, 2]], 'representations': {'Lie': [{'d': 4, 'q': 3, 'gens': [14408201, 9605495], 'family': 'PGL'}, {'d': 4, 'q': 3, 'family': 'PGammaL'}], 'Perm': {'d': 40, 'gens': [612318134292892751059980015582271582240530785125, 341668435346154550866383681368759885266699329565, 65971900822765570518592262312207230347168537455]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\PGL(4,3)', 'transitive_degree': 40, 'wreath_data': None, 'wreath_product': False}