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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '7680.bf', 'ambient_counter': 32, 'ambient_order': 7680, 'ambient_tex': 'C_2^7:A_5', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 64, 'counter': 18, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '7680.bf.20.c1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '20.c1', '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': 20, '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': '384.20099', 'subgroup_hash': 20099, 'subgroup_order': 384, 'subgroup_tex': 'C_2^4.S_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '7680.bf', 'aut_centralizer_order': None, 'aut_label': '20.c1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1920.b1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['10.a1'], 'contains': ['40.b1', '40.n1', '60.cp1', '80.c1', '80.g1'], 'core': '120.a1', 'coset_action_label': None, 'count': 10, 'diagramx': None, 'generators': [486426101053530383867377, 40279687, 5167, 25903229683158464681647, 122000787836933160, 40279680, 1313901393960, 432525549113620160446222], 'label': '7680.bf.20.c1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '10.a1', 'old_label': '20.c1', 'projective_image': '3840.cq', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '20.c1', 'subgroup_fusion': None, 'weyl_group': '192.1538'}
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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': 3, 'aut_exponent': 12, 'aut_gen_orders': [6, 4, 2, 2, 4], 'aut_gens': [[40325056, 806527, 806400, 518918704, 6266937607, 13412044921, 126, 7], [998328832, 127, 806400, 518918704, 121, 126, 6266937600, 13412044800], [998328953, 806527, 806400, 958003897, 121, 7, 19678982526, 13412044921], [40325063, 806527, 806400, 19160064430, 6266937607, 13412044921, 126, 7], [40325183, 806527, 806400, 19160064425, 6266937607, 13412044921, 126, 7], [6227429063, 127, 806400, 12454042030, 6266937600, 13412044800, 126, 7]], 'aut_group': '4608.dp', 'aut_hash': 5581141898365204177, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 4608, 'aut_permdeg': 48, 'aut_perms': [11624958885940817693575844047864139559988943214621063398806128, 2651167501140222953720136060168376762228542652279300602977776, 12153704126985752854142039201640667762278568338957580674418624, 5826305488051176591635386369826003929381273457324577390864048, 12150400389056263924287228078764808006793872745151790961510144], 'aut_phi_ratio': 36.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 3, 3, 2], [2, 3, 6, 1], [2, 6, 1, 2], [2, 6, 2, 1], [3, 32, 1, 1], [4, 12, 4, 1], [4, 12, 12, 1], [6, 32, 1, 1], [6, 32, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^5.D_6^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.3', 'autcent_hash': 3, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '576.8654', 'autcentquo_hash': 8654, 'autcentquo_nilpotent': False, 'autcentquo_order': 576, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': '\\POPlus(4,3)', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 3, 12], [2, 6, 4], [3, 32, 1], [4, 12, 16], [6, 32, 3]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '96.227', 'commutator_count': 1, 'commutator_label': '48.50', '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': 20099, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['192.1495', 1], ['2.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 1, 12], [2, 6, 1, 4], [3, 32, 1, 1], [4, 12, 2, 8], [6, 32, 1, 3]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 3, 'exponent': 12, 'exponents_of_order': [7, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '192.1538', 'hash': 20099, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 1, 1, 3, 2, 2, 2, 2], 'inner_gens': [[40325056, 806527, 806400, 958003776, 6266937607, 19678982526, 121, 7], [40325056, 806527, 806400, 518918704, 6266937607, 13412044921, 126, 7], [40325056, 806527, 806400, 518918704, 6266937607, 13412044921, 126, 7], [998328832, 806527, 806400, 518918704, 13412044921, 19678982526, 7, 121], [40325056, 806527, 806400, 19160064425, 6266937607, 13412044921, 126, 7], [6227429063, 806527, 806400, 7224941230, 6266937607, 13412044921, 126, 7], [40325063, 806527, 806400, 518918711, 6266937607, 13412044921, 126, 7], [40325056, 806527, 806400, 518918825, 6266937607, 13412044921, 126, 7]], 'inner_hash': 227, 'inner_nilpotent': False, 'inner_order': 96, 'inner_split': True, 'inner_tex': 'C_2^2:S_4', 'inner_used': [1, 4, 5, 7], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 4], [3, 24], [6, 4]], 'label': '384.20099', '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': 'C2^4.S4', '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': 14, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 40, 'number_divisions': 32, 'number_normal_subgroups': 33, 'number_subgroup_autclasses': 207, 'number_subgroup_classes': 847, 'number_subgroups': 4258, 'old_label': None, 'order': 384, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 63], [3, 32], [4, 192], [6, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 12], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[41005456, 806527, 806400, 518918704, 6266937600, 13412044800, 126, 7], [41005456, 127, 806400, 518918704, 6266937607, 13412044921, 126, 7], [41005583, 127, 806400, 518918704, 7, 121, 19678982400, 6266937600]], 'outer_group': '48.38', 'outer_hash': 38, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 7, 'outer_perms': [1, 1680, 2403], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3\\times D_4', 'pc_rank': 6, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 6], [3, 12], [6, 10]], 'representations': {'PC': {'code': 104257088864392800241437219595958650968933598480, 'gens': [1, 3, 5, 6, 7, 8], 'pres': [8, -2, -2, -2, -3, -2, 2, -2, 2, 16, 482, 66, 515, 500, 388, 3461, 885, 173, 2710, 2046, 18439, 4631, 799]}, 'GLFq': {'d': 4, 'q': 4, 'gens': [1079969729, 1083246081, 1074808321, 1087472897, 1074873345, 1074840833, 47179457, 1087506177]}, 'Perm': {'d': 14, 'gens': [40325056, 806527, 806400, 518918704, 6266937607, 13412044921, 126, 7]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^4.S_4', 'transitive_degree': 24, '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': 120, 'aut_gen_orders': [10, 6, 8, 4], 'aut_gens': [[378308841549345263697982, 566494961291888556194639], [620397310421371508138281, 25853422241304320579153], [162122696269716436531201, 350208309664173976254857], [242141100914435219510263, 51860750276451518760], [378307558425799428652678, 108120220398415119674743]], 'aut_group': None, 'aut_hash': 8093570508868592544, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 30720, 'aut_permdeg': 144, 'aut_perms': [1484949149426121940435755031434406427512256169783641754094189229917146534642765143909952540993276053777779816436398247737802985873194748150255672074277275167224186100664173025599778780478543446515449345794844965155937250784699320185751842750533117194, 3276221270369845148473191305344824928718771933270277217002238797541830315015335577012673526632113330775485435013396980418451467955831188962075408021795576189803788395332120278398774544646950660827672102825166681015777071756483682997384488682214401357, 2552355279045212634936772323056497844694259634597522492536112769949984178312992995945587234184732882491553374115088783421148343913339146673132017103182135444476848639362756361103853565934109432051459196321898195268142291360077943084447491546572313946, 2487166738999069595321069793861259919166927805310792926556618024429859713716482642229647492703060090400774960932949517292928656644888406613778802710092583747866297362197005075005438017996967736820025518371783690494106572213423409378162992787844817872], 'aut_phi_ratio': 15.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 2, 1], [2, 10, 2, 1], [2, 12, 2, 1], [2, 15, 1, 2], [2, 20, 2, 1], [2, 120, 1, 2], [3, 320, 1, 1], [4, 120, 1, 2], [4, 120, 2, 6], [5, 192, 2, 1], [6, 320, 1, 1], [6, 320, 2, 1], [6, 320, 4, 1], [10, 192, 2, 1], [10, 192, 4, 3]], 'aut_supersolvable': False, 'aut_tex': 'C_2^8.S_5', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': '7680.x', 'autcentquo_hash': 4264655218017036024, 'autcentquo_nilpotent': False, 'autcentquo_order': 7680, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2^6:S_5', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 6, 2], [2, 10, 2], [2, 12, 2], [2, 15, 2], [2, 20, 2], [2, 120, 2], [3, 320, 1], [4, 120, 14], [5, 192, 2], [6, 320, 7], [10, 192, 14]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '3840.cq', 'commutator_count': 1, 'commutator_label': '1920.240998', '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', '60.5'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 32, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 2], [2, 10, 1, 2], [2, 12, 1, 2], [2, 15, 1, 2], [2, 20, 1, 2], [2, 120, 1, 2], [3, 320, 1, 1], [4, 120, 1, 14], [5, 192, 2, 1], [6, 320, 1, 3], [6, 320, 2, 2], [10, 192, 2, 3], [10, 192, 4, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 456, 'exponent': 60, 'exponents_of_order': [9, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[12, 0, 8], [12, 1, 2], [20, 0, 4], [20, 1, 2]], 'familial': False, 'frattini_label': '32.51', 'frattini_quotient': '240.190', 'hash': 5060586257443296289, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [6, 10], 'inner_gens': [[378308841549345263697982, 52496445892500731518], [242141100914435219505336, 566494961291888556194639]], 'inner_hash': 2248252827435658470, 'inner_nilpotent': False, 'inner_order': 3840, 'inner_split': True, 'inner_tex': 'C_2^6.A_5', 'inner_used': [1, 2], 'irrC_degree': 12, 'irrQ_degree': 12, 'irrQ_dim': 12, 'irrR_degree': 12, 'irrep_stats': [[1, 4], [3, 8], [4, 4], [5, 4], [12, 10], [15, 16], [20, 6]], 'label': '7680.bf', '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': 'C2^7:A5', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 27, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 52, 'number_divisions': 40, 'number_normal_subgroups': 12, 'number_subgroup_autclasses': 1463, 'number_subgroup_classes': 2911, 'number_subgroups': 122040, 'old_label': None, 'order': 7680, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 367], [3, 320], [4, 1680], [5, 384], [6, 2240], [10, 2688]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 242141492972754395210286], 'outer_gens': [[404212071232491274327697, 566494961291888556194639], [162123724018085014926697, 270239970422663918208120]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [6, 17], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 24, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [4, 4], [5, 4], [6, 4], [12, 2], [15, 16], [20, 2], [40, 2], [48, 2]], 'representations': {'Perm': {'d': 24, 'gens': [566494961291888556194639, 378308841549345263697982]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^7:A_5', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
