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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1344.3515', 'ambient_counter': 3515, 'ambient_order': 1344, 'ambient_tex': 'D_{84}.D_4', 'central': False, 'central_factor': False, 'centralizer_order': 12, 'characteristic': False, 'core_order': 168, 'counter': 18, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '1344.3515.4.i1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '4.i1.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': 4, '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': '336.73', 'subgroup_hash': 73, 'subgroup_order': 336, 'subgroup_tex': 'C_{42}.D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.3515', 'aut_centralizer_order': 24, 'aut_label': '4.i1', 'aut_quo_index': None, 'aut_stab_index': 2, 'aut_weyl_group': '1344.11704', 'aut_weyl_index': 48, 'centralizer': '112.a1.a1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['2.b1.a1'], 'contains': ['8.a1.a1', '8.g1.a1', '8.h1.a1', '12.e1.a1', '28.e1.a1'], 'core': '8.a1.a1', 'coset_action_label': None, 'count': 2, 'diagramx': [1346, -1, 9086, -1, 1534, -1, 3869, -1], 'generators': [1, 192, 448, 8, 672, 16], 'label': '1344.3515.4.i1.a1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '2.b1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2.b1.a1', 'old_label': '4.i1.a1', 'projective_image': '336.158', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4.i1.a1', 'subgroup_fusion': None, 'weyl_group': '56.12'}
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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': 84, 'aut_gen_orders': [2, 2, 2, 6, 2, 2, 7, 2], 'aut_gens': [[1, 4, 24], [1, 4, 312], [169, 4, 312], [13, 20, 312], [13, 4, 120], [3, 4, 314], [1, 20, 314], [289, 4, 24], [15, 4, 324]], 'aut_group': '2688.ej', 'aut_hash': 3294577308875728697, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2688, 'aut_permdeg': 58, 'aut_perms': [24905401862421527613579727792241449545548669457060892630011663581117788943549, 434454695646113606345263370521912865899424452234876592824627194145212382156799, 270251405538944391131708913086276574819917794391758778901725216154836495005439, 269495495188564287478978694153236866657440276288066470208692979359799177301587, 762937453936039421773191617286666361463555100798264745335854057660027468457149, 24943490911166156304941545872729805298919719165138782214024564405634156658109, 185603548270568907323479259339572248961157854853664776165045032096251248640000, 927178833070051396450307775701665527716668712978513561394560188878045365456189], 'aut_phi_ratio': 28.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [3, 1, 2, 1], [4, 14, 4, 1], [6, 1, 2, 1], [6, 1, 4, 1], [6, 2, 4, 1], [7, 2, 3, 1], [12, 14, 8, 1], [14, 2, 3, 1], [14, 2, 6, 1], [14, 2, 12, 1], [21, 2, 6, 1], [42, 2, 6, 1], [42, 2, 12, 1], [42, 2, 24, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^3:D_4\\times F_7', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '84.7', 'autcentquo_hash': 7, 'autcentquo_nilpotent': False, 'autcentquo_order': 84, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [3, 1, 2], [4, 14, 4], [6, 1, 6], [6, 2, 4], [7, 2, 3], [12, 14, 8], [14, 2, 21], [21, 2, 6], [42, 2, 42]], 'center_label': '12.5', 'center_order': 12, 'central_product': True, 'central_quotient': '28.3', 'commutator_count': 1, 'commutator_label': '14.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 73, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['112.18', 1], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [3, 1, 2, 1], [4, 14, 2, 2], [6, 1, 2, 3], [6, 2, 2, 2], [7, 2, 3, 1], [12, 14, 4, 2], [14, 2, 3, 3], [14, 2, 6, 2], [21, 2, 6, 1], [42, 2, 6, 3], [42, 2, 12, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 12, 'exponent': 84, 'exponents_of_order': [4, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '84.12', 'hash': 73, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 14, 'inner_gen_orders': [2, 1, 14], 'inner_gens': [[1, 4, 324], [1, 4, 24], [61, 4, 24]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 28, 'inner_split': True, 'inner_tex': 'D_{14}', 'inner_used': [1, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 24], [2, 78]], 'label': '336.73', 'linC_count': 384, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 20, 'linQ_dim': 12, 'linQ_dim_count': 60, 'linR_count': 168, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C42.D4', 'ngens': 6, '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': 18, 'number_characteristic_subgroups': 18, 'number_conjugacy_classes': 102, 'number_divisions': 28, 'number_normal_subgroups': 38, 'number_subgroup_autclasses': 44, 'number_subgroup_classes': 68, 'number_subgroups': 176, 'old_label': None, 'order': 336, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 7], [3, 2], [4, 56], [6, 14], [7, 6], [12, 112], [14, 42], [21, 12], [42, 84]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 12], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[1, 4, 36], [3, 20, 36], [169, 4, 24], [181, 4, 122]], 'outer_group': '96.221', 'outer_hash': 221, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 11, 'outer_perms': [2424, 1680, 41064, 11614588], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{12}:C_2^3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 18, '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], [6, 4], [12, 6], [24, 2]], 'representations': {'PC': {'code': 3232765270731628065641, 'gens': [1, 3, 5], 'pres': [6, 2, 2, 2, 3, 2, 7, 12, 50, 9724, 88, 10373]}, 'GLZN': {'d': 2, 'p': 28, 'gens': [548825, 22345, 329295, 22065, 22049, 33335]}, 'Perm': {'d': 18, 'gens': [21010451213544, 264, 3, 7983360, 1680, 397707843456000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 12], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{42}.D_4', 'transitive_degree': 168, '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': 84, 'aut_gen_orders': [12, 14, 6, 6, 6, 42, 6, 6, 12, 12], 'aut_gens': [[1, 2, 16, 32], [1173, 922, 16, 608], [589, 6, 16, 32], [17, 474, 16, 432], [1157, 462, 16, 376], [777, 14, 16, 376], [1177, 926, 16, 40], [201, 898, 16, 1000], [409, 10, 16, 808], [781, 458, 16, 1208], [989, 898, 16, 160]], 'aut_group': None, 'aut_hash': 5255589997166976132, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 64512, 'aut_permdeg': 84, 'aut_perms': [3196763172354862037923256995942856024892637839319051535992101385083249878809406751646790103183867535036496097942923295607814747, 1263389245182034523009822749166305446705105391749463610870798676764814482822241108995979822476453284531277590527510298762303712, 2503484258021997428828734831765139059659725807241993356351364463076884485014873363523498234639217747572243568041247338971794549, 1639297303131424514906676895096139820044982104578809656987110900450560011441961749120303380613570341103275499324494180408464278, 810967897833873179410141950820197652651995407430486712151295598393940524122495822576535108183663190780707524896838247865446532, 2176872326991379256628765661723703002510439395627958099511353592854334848516326729733709562915613670723236419735402871565383595, 3005327552102926812076439705720255444319301521249755323125155535564986943498822273837709670063051696511669636315621463161226576, 1782377834265763963086861421920517236116012272691131434338158983793301302778053940077758471165754641090978459410403516949326687, 1639384185700787472084494065558146600496439265760991427927203198031382332267435403771664131096056923071255364351259278275649422, 592342676954997508887588074543603544307593135441330595737013676515937843169579018493568271929299266037497778124745361568907432], 'aut_phi_ratio': 168.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [2, 84, 2, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 2, 2, 1], [4, 56, 1, 2], [4, 84, 2, 1], [6, 2, 1, 3], [6, 4, 2, 1], [7, 2, 3, 1], [8, 12, 2, 2], [12, 4, 1, 2], [12, 4, 2, 1], [12, 56, 2, 2], [14, 2, 3, 3], [14, 4, 6, 1], [21, 4, 3, 1], [28, 2, 6, 2], [28, 4, 6, 1], [42, 4, 3, 3], [42, 4, 12, 1], [56, 12, 12, 2], [84, 4, 6, 2], [84, 4, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{42}.(C_2^5\\times C_6).C_2^3', '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': 42, 'autcentquo_group': '1008.906', 'autcentquo_hash': 906, 'autcentquo_nilpotent': False, 'autcentquo_order': 1008, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times D_6\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 4, 1], [2, 84, 2], [3, 2, 1], [4, 2, 4], [4, 56, 2], [4, 84, 2], [6, 2, 3], [6, 4, 2], [7, 2, 3], [8, 12, 4], [12, 4, 4], [12, 56, 4], [14, 2, 9], [14, 4, 6], [21, 4, 3], [28, 2, 12], [28, 4, 6], [42, 4, 21], [56, 12, 24], [84, 4, 24]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '336.158', 'commutator_count': 2, 'commutator_label': '168.39', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 3515, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 4, 1, 1], [2, 84, 1, 2], [3, 2, 1, 1], [4, 2, 1, 2], [4, 2, 2, 1], [4, 56, 1, 2], [4, 84, 1, 2], [6, 2, 1, 3], [6, 4, 2, 1], [7, 2, 3, 1], [8, 12, 2, 2], [12, 4, 1, 2], [12, 4, 2, 1], [12, 56, 2, 2], [14, 2, 3, 3], [14, 4, 6, 1], [21, 4, 3, 1], [28, 2, 6, 2], [28, 4, 6, 1], [42, 4, 3, 3], [42, 4, 12, 1], [56, 12, 12, 2], [84, 4, 6, 2], [84, 4, 12, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 10752, 'exponent': 168, 'exponents_of_order': [6, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '168.50', 'hash': 3515, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 4, 1, 42], 'inner_gens': [[1, 6, 16, 440], [13, 2, 16, 944], [1, 2, 16, 32], [985, 466, 16, 32]], 'inner_hash': 158, 'inner_nilpotent': False, 'inner_order': 336, 'inner_split': True, 'inner_tex': 'C_6:D_{28}', 'inner_used': [1, 2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 66], [4, 67]], 'label': '1344.3515', 'linC_count': 1632, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 72, 'linQ_dim': 16, 'linQ_dim_count': 32, 'linR_count': 384, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D84.D4', '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, 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': 42, 'number_characteristic_subgroups': 64, 'number_conjugacy_classes': 141, 'number_divisions': 44, 'number_normal_subgroups': 68, 'number_subgroup_autclasses': 244, 'number_subgroup_classes': 304, 'number_subgroups': 3092, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 175], [3, 2], [4, 288], [6, 14], [7, 6], [8, 48], [12, 240], [14, 42], [21, 12], [28, 48], [42, 84], [56, 288], [84, 96]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 2, 2, 6], 'outer_gen_pows': [0, 0, 0, 0, 0, 0], 'outer_gens': [[1, 2, 16, 928], [1, 10, 16, 32], [1, 2, 16, 40], [1, 2, 16, 56], [17, 2, 16, 32], [1, 2, 16, 992]], 'outer_group': '192.1543', 'outer_hash': 1543, 'outer_nilpotent': True, 'outer_order': 192, 'outer_permdeg': 15, 'outer_perms': [3669864, 87178291200, 40320, 744, 479001600, 27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5\\times C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 30, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [4, 5], [6, 4], [8, 3], [12, 8], [24, 3], [48, 3]], 'representations': {'PC': {'code': 13490975788599759148173783241647978458445981811296011, 'gens': [1, 2, 5, 6], 'pres': [8, -2, -2, -2, -2, 2, 2, -3, -7, 64, 97, 41, 290, 66, 21125, 22669, 141, 46598, 14350, 222, 73735]}, 'Perm': {'d': 30, 'gens': [305292256521392060635254156125, 51367467554395056845, 406278934966084447, 16, 528199954665556440, 656686659640798800, 1175091669949317120000, 9452345886333784620057231360000]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, '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': 'D_{84}.D_4', 'transitive_degree': 672, 'wreath_data': None, 'wreath_product': False}
