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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1920.240593', 'ambient_counter': 240593, 'ambient_order': 1920, 'ambient_tex': 'C_2^2\\times C_4:S_5', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': False, 'core_order': 16, 'counter': 11, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1920.240593.6.a1', 'maximal': True, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '6.a1', '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': 6, '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': '320.1591', 'subgroup_hash': 1591, 'subgroup_order': 320, 'subgroup_tex': 'D_{10}.C_2^4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1920.240593', 'aut_centralizer_order': 1, 'aut_label': '6.a1', 'aut_quo_index': None, 'aut_stab_index': 6, 'aut_weyl_group': '61440.em', 'aut_weyl_index': 6, 'centralizer': '240.b1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['1.a1'], 'contains': ['12.a1', '12.b1', '12.c1', '30.e1'], 'core': '120.a1', 'coset_action_label': None, 'count': 6, 'diagramx': [2208, -1, 4142, -1], 'generators': [80213176, 5896, 11520, 7, 16, 127370880, 1045100296], 'label': '1920.240593.6.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '6.a1', 'old_label': '6.a1', 'projective_image': '240.189', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.a1', 'subgroup_fusion': None, 'weyl_group': '40.12'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '32.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 120, 'aut_gen_orders': [2, 4, 2, 5, 4, 2, 2, 2, 2, 2, 2], 'aut_gens': [[1, 2, 8, 16], [169, 10, 8, 144], [1, 2, 8, 272], [161, 170, 8, 144], [1, 130, 8, 16], [1, 251, 8, 144], [161, 171, 168, 304], [1, 2, 8, 153], [1, 162, 8, 144], [1, 162, 8, 304], [8, 162, 1, 144], [1, 11, 8, 144]], 'aut_group': '61440.em', 'aut_hash': 5907412864111550192, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 61440, 'aut_permdeg': 80, 'aut_perms': [5594800199753698094584207757572722539964085204190530852146612122702993900178460505002435004828549318359225558596089842, 68851923750974802716263448293933692482687165113097724521852473440832781786562662607499268713163381834348783286854346240, 16184585276369329221734354900823932319009179047434760936316596063352870419590493377055970504387433429858417645929561709, 7248716704173495525314976137649241214375934829555037409689330589946464425793483170907107476565415864401443898517346560, 57072023638460676751995659883046236043340967401980937427582340289689476813037694038493097496811142305636239422206519039, 6457170821260400442273872841380054963878165564677821450180403488789815231254640259303354268493948146972463521618669539, 7247834141831606432759521472568323921526638270522544616928246172250319522724345875139126121127181799469640669128293984, 18120909197443868393909271234554392357764915068178928901208705677137537843013037787396944592051082728239856414218852784, 18120909197443868393909271234554392357764914661757175663780033243188217570986443300482379334642907049625111905747913264, 17226292869860394149717470855359069007437667227009993108073820756755492136473795238953235334628878451447029793167818008, 6331006711149799453743905293012873782238369273040049414709876767695017073289031784481156777171792160092036880328294399], 'aut_phi_ratio': 480.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [2, 5, 1, 2], [2, 5, 6, 1], [4, 2, 4, 1], [4, 10, 4, 1], [4, 10, 8, 2], [5, 4, 1, 1], [10, 4, 1, 1], [10, 4, 6, 1], [20, 4, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_5\\times C_2^6:(C_2\\times S_4)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '1536.408640869', 'autcent_hash': 4599629986281706675, 'autcent_nilpotent': False, 'autcent_order': 1536, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^6:S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '40.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 40, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 5, 8], [4, 2, 4], [4, 10, 20], [5, 4, 1], [10, 4, 7], [20, 4, 8]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '40.12', 'commutator_count': 1, 'commutator_label': '10.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1591, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['80.31', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 5, 1, 8], [4, 2, 1, 4], [4, 10, 1, 4], [4, 10, 2, 8], [5, 4, 1, 1], [10, 4, 1, 7], [20, 4, 2, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 52080, 'exponent': 20, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '160.236', 'hash': 1591, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [1, 4, 1, 10], 'inner_gens': [[1, 2, 8, 16], [1, 2, 8, 48], [1, 2, 8, 16], [1, 290, 8, 16]], 'inner_hash': 12, 'inner_nilpotent': False, 'inner_order': 40, 'inner_split': False, 'inner_tex': 'C_2\\times F_5', 'inner_used': [2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 8], [4, 16]], 'label': '320.1591', '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': 'D10.C2^4', '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': 14, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 56, 'number_divisions': 44, 'number_normal_subgroups': 196, 'number_subgroup_autclasses': 78, 'number_subgroup_classes': 418, 'number_subgroups': 1386, 'old_label': None, 'order': 320, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 47], [4, 208], [5, 4], [10, 28], [20, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 3, 2, 2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 80, 0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[9, 10, 8, 176], [8, 11, 9, 17], [1, 90, 8, 16], [1, 163, 168, 176], [161, 10, 8, 16], [161, 11, 168, 25], [161, 10, 8, 184], [1, 2, 8, 176], [1, 3, 8, 16], [1, 10, 8, 16]], 'outer_group': '1536.408632661', 'outer_hash': 6321312703651895429, 'outer_nilpotent': False, 'outer_order': 1536, 'outer_permdeg': 16, 'outer_perms': [4296764107855, 262017867456, 4302991486080, 7, 126, 2877842004342, 5413204742273, 5167, 2878807236480, 1321086775680], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^6:S_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 4], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 16], [4, 8], [8, 4]], 'representations': {'PC': {'code': 728382543465111153222064060268547, 'gens': [1, 2, 4, 5], 'pres': [7, -2, -2, -2, -2, -2, -2, -5, 36, 851, 1278, 102, 2028, 3043, 124, 4717, 3156]}, 'GLZN': {'d': 2, 'p': 20, 'gens': [8009, 8201, 28219, 72009, 24069, 8081, 152019]}, 'Perm': {'d': 13, 'gens': [482671449, 1037927520, 1037968560, 1437136560, 16, 1037836800, 34]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 4], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{10}.C_2^4', 'transitive_degree': 80, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 120, 'aut_gen_orders': [4, 20, 24, 4], 'aut_gens': [[559209616, 3633967, 80213176, 1045100296], [961994896, 1480555583, 961648680, 1916012287], [559209616, 123758776, 562843583, 526191127], [1916017927, 83473223, 119761200, 526191143], [2039397136, 3645496, 80213183, 1556761087]], 'aut_group': None, 'aut_hash': 3860228924967480141, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 368640, 'aut_permdeg': 160, 'aut_perms': [122753485820965292842545554837186657819303363462971654614597211771217753175025793112436709637044517837774046091998034652606787542901030635693749808063698210667415969769968700251650393021115982481977897559382909572503571273941521324460416589105473156603737258213882772603870392616500338, 435339050004505918538942363835029130138502511159712463284346629886879221150895452809870179556130220702836635482758452823963993455035720715331108009639090014863419705133072708970008522529494994744182656523266362267844901961569025957434641252598968263278541303126874610811203590806879849, 76654116026626735335687037920743217239569047396663215761926374510206852563794859866498655740594018013919271224966300558571826401251072596263529800286079947781978194547586803608056060872968450817322151121408008257538431771445111771848527736240471161073772562679954320733475906317399701, 240963925252761363042331015129013922505238545033368637853737166715172231224078919469897967832645883818939227460599009474419204289172571129630296687697228718678902677217587209676839662066165079223804349855857426357433222561428285613603789416143722542868140924466765952353101252878266798], 'aut_phi_ratio': 720.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [2, 15, 1, 2], [2, 15, 6, 1], [2, 20, 8, 1], [3, 20, 1, 1], [4, 2, 4, 1], [4, 30, 4, 1], [4, 60, 8, 1], [5, 24, 1, 1], [6, 20, 1, 1], [6, 20, 6, 1], [6, 40, 8, 1], [10, 24, 1, 1], [10, 24, 6, 1], [12, 40, 4, 1], [20, 24, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4.C_2^4.D_6.S_5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '1536.408640869', 'autcent_hash': 4599629986281706675, 'autcent_nilpotent': False, 'autcent_order': 1536, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^6:S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '240.189', 'autcentquo_hash': 189, 'autcentquo_nilpotent': False, 'autcentquo_order': 240, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2\\times S_5', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 15, 8], [2, 20, 8], [3, 20, 1], [4, 2, 4], [4, 30, 4], [4, 60, 8], [5, 24, 1], [6, 20, 7], [6, 40, 8], [10, 24, 7], [12, 40, 4], [20, 24, 8]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '240.189', 'commutator_count': 1, 'commutator_label': '120.35', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '60.5'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 240593, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['480.944', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 15, 1, 8], [2, 20, 1, 8], [3, 20, 1, 1], [4, 2, 1, 4], [4, 30, 1, 4], [4, 60, 1, 8], [5, 24, 1, 1], [6, 20, 1, 7], [6, 40, 1, 8], [10, 24, 1, 7], [12, 40, 1, 4], [20, 24, 2, 4]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 60, 'exponents_of_order': [7, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '960.11355', 'hash': 240593, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [5, 2, 4, 10], 'inner_gens': [[559209616, 1959920047, 1477301176, 962000776], [526187536, 3633967, 47191096, 1517216416], [1437379216, 123747247, 80213176, 522925216], [606021136, 1959931567, 602385976, 1045100296]], 'inner_hash': 189, 'inner_nilpotent': False, 'inner_order': 240, 'inner_split': False, 'inner_tex': 'C_2\\times S_5', 'inner_used': [1, 2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 4], [4, 16], [5, 16], [6, 16], [8, 4], [10, 4]], 'label': '1920.240593', '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': False, 'name': 'C2^2*C4:S5', 'ngens': 4, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 19, 'number_characteristic_subgroups': 9, 'number_conjugacy_classes': 76, 'number_divisions': 72, 'number_normal_subgroups': 105, 'number_subgroup_autclasses': 262, 'number_subgroup_classes': 2181, 'number_subgroups': 35094, 'old_label': None, 'order': 1920, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 287], [3, 20], [4, 608], [5, 24], [6, 460], [10, 168], [12, 160], [20, 192]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 3, 2, 2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 0, 15127, 0, 0, 0, 0, 0, 0, 0], 'outer_gens': [[559209623, 3633967, 80213183, 1045109543], [559209623, 3633967, 80213160, 1045100303], [559209616, 3630367, 80207296, 1045100296], [559198096, 3633976, 80213167, 1045109536], [559198096, 3633960, 80201663, 1045109536], [559209616, 3633967, 80213176, 1045100280], [559209616, 3633967, 80213176, 1045100303], [559209616, 3633967, 80213176, 1045109536], [559209616, 3633976, 80213167, 1045100296], [559209616, 3633960, 80213183, 1045100296]], 'outer_group': '1536.408632661', 'outer_hash': 6321312703651895429, 'outer_nilpotent': False, 'outer_order': 1536, 'outer_permdeg': 16, 'outer_perms': [5693416651915, 93409304176, 5780595300480, 1314028777444, 2789745611130, 1313941708511, 4103734104930, 5455, 1314028759680, 2789745598080], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^6:S_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 4], [4, 16], [5, 16], [6, 8], [8, 4], [10, 4], [12, 4]], 'representations': {'Perm': {'d': 13, 'gens': [559209616, 3633967, 80213176, 1045100296]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2\\times C_4:S_5', 'transitive_degree': 80, 'wreath_data': None, 'wreath_product': False}
