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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '384.1654', 'ambient_counter': 1654, 'ambient_order': 384, 'ambient_tex': 'D_{48}:C_4', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 32, 'counter': 29, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '384.1654.6.c1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': False, 'old_label': '6.c1.b1', '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': 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': '64.38', 'subgroup_hash': 38, 'subgroup_order': 64, 'subgroup_tex': 'D_8:C_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '384.1654', 'aut_centralizer_order': 8, 'aut_label': '6.c1', 'aut_quo_index': None, 'aut_stab_index': 6, 'aut_weyl_group': '256.53142', 'aut_weyl_index': 48, 'centralizer': '96.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.b1.b1', '3.a1.a1'], 'contains': ['12.c1.a1', '12.f1.b1', '12.g1.a1'], 'core': '12.c1.a1', 'coset_action_label': None, 'count': 3, 'diagramx': [7097, -1, 7818, -1, 6624, -1, 7526, -1], 'generators': [9, 222], 'label': '384.1654.6.c1.b1', 'mobius_quo': None, 'mobius_sub': 1, 'normal_closure': '2.b1.b1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '3.a1.a1', 'old_label': '6.c1.b1', 'projective_image': '96.110', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '6.c1.b1', 'subgroup_fusion': None, 'weyl_group': '32.39'}
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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': 2, 'aut_exponent': 8, 'aut_gen_orders': [2, 2, 4, 2, 8], 'aut_gens': [[1, 2, 32], [49, 34, 32], [49, 2, 32], [1, 42, 32], [1, 30, 32], [37, 18, 32]], 'aut_group': '256.53142', 'aut_hash': 53142, 'aut_nilpotency_class': 3, 'aut_nilpotent': True, 'aut_order': 256, 'aut_permdeg': 16, 'aut_perms': [4296764102423, 4296764102407, 3152832091920, 9709968804503, 6744390065303], 'aut_phi_ratio': 8.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 8, 2, 1], [4, 2, 1, 2], [4, 8, 2, 1], [8, 2, 2, 2], [16, 2, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4^2.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '16.11', 'autcentquo_hash': 11, 'autcentquo_nilpotent': True, 'autcentquo_order': 16, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 8, 2], [4, 2, 2], [4, 8, 2], [8, 2, 4], [16, 2, 8]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '16.7', 'commutator_count': 1, 'commutator_label': '8.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 38, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 8, 1, 2], [4, 2, 1, 2], [4, 8, 2, 1], [8, 2, 2, 2], [16, 2, 8, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 6, 'exponent': 16, 'exponents_of_order': [6], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.5', 'frattini_quotient': '4.2', 'hash': 38, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 8, 'inner_gen_orders': [2, 8, 1], 'inner_gens': [[1, 46, 32], [53, 2, 32], [1, 2, 32]], 'inner_hash': 7, 'inner_nilpotent': True, 'inner_order': 16, 'inner_split': True, 'inner_tex': 'D_8', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 14]], 'label': '64.38', 'linC_count': 32, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 6, 'linQ_dim': 10, 'linQ_dim_count': 6, 'linR_count': 12, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D8:C4', 'ngens': 2, 'nilpotency_class': 4, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 11, 'number_characteristic_subgroups': 15, 'number_conjugacy_classes': 22, 'number_divisions': 12, 'number_normal_subgroups': 17, 'number_subgroup_autclasses': 29, 'number_subgroup_classes': 33, 'number_subgroups': 89, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 19], [4, 20], [8, 8], [16, 16]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4], 'outer_gen_pows': [4, 0, 0], 'outer_gens': [[5, 2, 32], [1, 14, 32], [1, 6, 32]], 'outer_group': '16.10', 'outer_hash': 10, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 20, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 2], [8, 2]], 'representations': {'PC': {'code': 2360935209500910, 'gens': [1, 2, 6], 'pres': [6, -2, 2, -2, -2, -2, 2, 553, 31, 506, 50, 579, 69]}, 'GLZN': {'d': 2, 'p': 21, 'gens': [102661, 120406, 110805, 163489, 74096, 38503]}, 'GLZq': {'d': 2, 'q': 16, 'gens': [12339, 37015, 4129, 36873, 4161, 4225]}, 'Perm': {'d': 20, 'gens': [135611728994337129, 28149026307670936, 278324232825800280, 406560631538446576, 534038018458453320, 406560631538446560]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_8:C_4', 'transitive_degree': 32, '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': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 48, 'aut_gen_orders': [6, 8, 4, 2, 24, 4, 24], 'aut_gens': [[1, 2, 8], [373, 2, 248], [113, 198, 44], [73, 98, 188], [101, 194, 376], [281, 294, 348], [357, 194, 92], [233, 290, 348]], 'aut_group': None, 'aut_hash': 1940817208993041345, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12288, 'aut_permdeg': 104, 'aut_perms': [5496095809581499797581062217882651092513002652207371696085827237819971511979873713988901937948867716880579425736896294769441058163833169458257843894282664103339274397, 3702807978845317318720927332770923661455501452264486661684285880385226142979246910345472550700152895516922103138974925010277982173443528022847998949316171313624801849, 8000808424957830110619024304044923166489080158526901880963413081348971337024013431854964543596523615096375149196873257493792154957338035589832610883134348612038217513, 4901874383910816485868306058549136141182827931627480786478031891914537933663305097386063108263110644686261434391752995210322873830155421662668700620982122220364233599, 6097283526054852769573422244875380344349317563470094810031876511984642510381003552101590897892976099953330602149647045951150283478975072737291017871754397604572539568, 5197764308830759408422181180594668910759445939316214651220591271232032684587992605194397854743542206381991831576366730153338975283113157117794999635053063031087238466, 8700704432723342798650914245004409874354549821579627676006772625223317577411881366211293741995459019206619092513829758878420852499839055797390921306468792269120978813], 'aut_phi_ratio': 96.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 24, 4, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 2, 4, 1], [4, 24, 4, 1], [6, 2, 1, 1], [6, 2, 2, 1], [8, 2, 2, 2], [8, 4, 2, 1], [12, 2, 2, 2], [12, 4, 4, 1], [16, 4, 4, 2], [24, 2, 4, 2], [24, 4, 4, 1], [48, 4, 8, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_3:((C_2^5\\times C_8).C_2^4)', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '64.267', 'autcent_hash': 267, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '192.1331', 'autcentquo_hash': 1331, 'autcentquo_nilpotent': False, 'autcentquo_order': 192, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_8:D_6', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 24, 4], [3, 2, 1], [4, 2, 6], [4, 24, 4], [6, 2, 3], [8, 2, 4], [8, 4, 2], [12, 2, 4], [12, 4, 4], [16, 4, 8], [24, 2, 8], [24, 4, 4], [48, 4, 16]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '96.110', 'commutator_count': 1, 'commutator_label': '24.2', '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': 1654, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 24, 1, 4], [3, 2, 1, 1], [4, 2, 1, 2], [4, 2, 2, 2], [4, 24, 2, 2], [6, 2, 1, 3], [8, 2, 2, 2], [8, 4, 2, 1], [12, 2, 2, 2], [12, 4, 2, 2], [16, 4, 2, 2], [16, 4, 4, 1], [24, 2, 4, 2], [24, 4, 4, 1], [48, 4, 4, 2], [48, 4, 8, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1344, 'exponent': 48, 'exponents_of_order': [7, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.5', 'frattini_quotient': '24.14', 'hash': 1654, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [2, 2, 24], 'inner_gens': [[1, 2, 376], [1, 2, 200], [17, 194, 8]], 'inner_hash': 110, 'inner_nilpotent': False, 'inner_order': 96, 'inner_split': False, 'inner_tex': 'C_2\\times D_{24}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 44], [4, 12]], 'label': '384.1654', 'linC_count': 64, 'linC_degree': 5, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 36, 'linQ_dim': 12, 'linQ_dim_count': 36, 'linR_count': 32, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D48:C4', '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': 24, 'number_characteristic_subgroups': 33, 'number_conjugacy_classes': 72, 'number_divisions': 34, 'number_normal_subgroups': 61, 'number_subgroup_autclasses': 94, 'number_subgroup_classes': 172, 'number_subgroups': 918, 'old_label': None, 'order': 384, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 99], [3, 2], [4, 108], [6, 6], [8, 16], [12, 24], [16, 32], [24, 32], [48, 64]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 4], 'outer_gen_pows': [2, 0, 2, 0, 0, 0], 'outer_gens': [[1, 2, 296], [1, 2, 248], [1, 6, 348], [1, 6, 248], [5, 194, 8], [25, 102, 248]], 'outer_group': '128.2163', 'outer_hash': 2163, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 12, 'outer_perms': [40284840, 11527, 23, 362903, 7, 127008022], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 23, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [4, 7], [8, 6], [16, 3]], 'representations': {'PC': {'code': 2454404743784893813247212185732299782657, 'gens': [1, 2, 4], 'pres': [8, -2, -2, -2, -2, -2, -2, -2, -3, 41, 12035, 3211, 91, 14724, 116, 16901, 141, 17926, 166, 16391]}, 'GLZN': {'d': 2, 'p': 48, 'gens': [111169, 2764825, 111361, 111745, 110737, 1216813, 1218419, 110881]}, 'Perm': {'d': 23, 'gens': [1187674301142637056129, 207814739863639547656, 2299861438581917145618, 3645384031221691852800, 2555318406835139904016, 4874091960115956211200, 2555318406835139904000, 840]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [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_{48}:C_4', 'transitive_degree': 192, 'wreath_data': None, 'wreath_product': False}
