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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1344.2815', 'ambient_counter': 2815, 'ambient_order': 1344, 'ambient_tex': '(C_6\\times D_{56}):C_2', 'central': True, 'central_factor': False, 'centralizer_order': 1344, 'characteristic': True, 'core_order': 2, 'counter': 317, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '1344.2815.672.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '672.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '672.618', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 618, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 672, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_6:D_{28}', 'simple': True, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': True, 'subgroup': '2.1', 'subgroup_hash': 1, 'subgroup_order': 2, 'subgroup_tex': 'C_2', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1344.2815', 'aut_centralizer_order': 64512, 'aut_label': '672.a1', 'aut_quo_index': 2, 'aut_stab_index': 1, 'aut_weyl_group': '1.1', 'aut_weyl_index': 64512, 'centralizer': '1.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['96.a1.a1', '224.a1.a1', '336.a1.a1', '336.b1.a1', '336.c1.a1', '336.d1.a1', '336.e1.a1', '336.e1.b1', '336.f1.a1', '336.g1.a1', '336.h1.a1', '336.h1.b1'], 'contains': ['1344.a1.a1'], 'core': '672.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [4997, 4922, 5260, 4866, 4527, 4909, 5598, 4872], 'generators': [8], 'label': '1344.2815.672.a1.a1', 'mobius_quo': -1, 'mobius_sub': 0, 'normal_closure': '672.a1.a1', 'normal_contained_in': ['96.a1.a1', '224.a1.a1', '336.a1.a1', '336.c1.a1', '336.b1.a1'], 'normal_contains': ['1344.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '672.a1.a1', 'projective_image': '672.618', 'quotient_action_image': '1.1', 'quotient_action_kernel': '672.618', 'quotient_action_kernel_order': 672, 'quotient_fusion': None, 'short_label': '672.a1.a1', 'subgroup_fusion': None, 'weyl_group': '1.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 0, 'aut_exponent': 1, 'aut_gen_orders': [], 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_hash': 1, 'aut_nilpotency_class': 0, 'aut_nilpotent': True, 'aut_order': 1, 'aut_permdeg': 1, 'aut_perms': [], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_1', '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': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 1, 'autcentquo_group': '1.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 1, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_1', 'cc_stats': [[1, 1, 1], [2, 1, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 1, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_count': 1, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 1, 'linQ_degree_count': 1, 'linQ_dim': 1, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 1, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 1, 'outer_gen_orders': [], 'outer_gen_pows': [], 'outer_gens': [], 'outer_group': '1.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 1, 'outer_permdeg': 1, 'outer_perms': [], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_1', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, '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': [6, 6, 12, 12, 2, 12, 6, 12, 12, 6], 'aut_gens': [[1, 2, 16, 224], [713, 6, 624, 224], [777, 682, 280, 224], [101, 10, 664, 224], [749, 682, 816, 224], [673, 14, 24, 224], [173, 682, 304, 1120], [837, 6, 856, 1120], [781, 2, 504, 1120], [45, 10, 728, 224], [781, 6, 600, 224]], 'aut_group': None, 'aut_hash': 5255589997166976132, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 64512, 'aut_permdeg': 76, 'aut_perms': [1190237465751964494426432441392988285538709178826699695099195529496463927522746956934700170555756795637587061147, 79307239009474276706317770753501985443513514544262354477168351027333155705951017748229446083969051884121380334, 939162316137230345642438490134041172819218803713796960256673484864130733053466404809919951343673932636306172024, 354134836322209162441551282344407137596821852275219989988175715237278485716719734041528704981741433021662636022, 1584576473279689343198992013081729356075705833063256182368984699786348091323748794854276809679319209056150827673, 1596003148674066631396729694666440266575957024536537279846826680050932272333687987318955022374240072701364685164, 698627024017569911674416041066990838031490249935283621844285457617388706539203037377259358392128079596038724994, 715983663596667600761959479771340542729224550420407733008040344962171220703698559944886681137597944311574523778, 1591938174716915529036015386499893642386461058263954931460318768318445480207596962347855468003735817915149786338, 170184183510004116626417859280068275306501015721410424278470951020698060528617150878871683615491354175681387505], 'aut_phi_ratio': 168.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 12, 1, 1], [2, 28, 2, 1], [2, 56, 1, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 6, 2, 1], [4, 84, 2, 1], [4, 168, 1, 1], [6, 2, 1, 3], [6, 56, 2, 2], [7, 2, 3, 1], [8, 4, 2, 1], [8, 12, 2, 1], [12, 4, 1, 2], [14, 2, 3, 3], [14, 12, 6, 1], [21, 4, 3, 1], [24, 4, 4, 1], [28, 2, 6, 2], [28, 12, 6, 1], [42, 4, 3, 3], [56, 4, 12, 1], [56, 12, 12, 1], [84, 4, 6, 2], [168, 4, 24, 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, 12, 1], [2, 28, 2], [2, 56, 1], [3, 2, 1], [4, 2, 2], [4, 6, 2], [4, 84, 2], [4, 168, 1], [6, 2, 3], [6, 56, 4], [7, 2, 3], [8, 4, 2], [8, 12, 2], [12, 4, 2], [14, 2, 9], [14, 12, 6], [21, 4, 3], [24, 4, 4], [28, 2, 12], [28, 12, 6], [42, 4, 9], [56, 4, 12], [56, 12, 12], [84, 4, 12], [168, 4, 24]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '336.149', 'commutator_count': 1, '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': 2815, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 12, 1, 1], [2, 28, 1, 2], [2, 56, 1, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 6, 2, 1], [4, 84, 1, 2], [4, 168, 1, 1], [6, 2, 1, 3], [6, 56, 1, 2], [6, 56, 2, 1], [7, 2, 3, 1], [8, 4, 2, 1], [8, 12, 2, 1], [12, 4, 1, 2], [14, 2, 3, 3], [14, 12, 6, 1], [21, 4, 3, 1], [24, 4, 4, 1], [28, 2, 6, 2], [28, 12, 6, 1], [42, 4, 3, 3], [56, 4, 12, 1], [56, 12, 12, 1], [84, 4, 6, 2], [168, 4, 24, 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': 2815, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 4, 14, 3], 'inner_gens': [[1, 14, 216, 224], [5, 2, 688, 224], [41, 674, 16, 1120], [1, 2, 464, 224]], 'inner_hash': 149, 'inner_nilpotent': False, 'inner_order': 336, 'inner_split': None, 'inner_tex': 'S_3\\times D_{28}', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 66], [4, 67]], 'label': '1344.2815', 'linC_count': 96, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 76, 'linQ_dim': 14, 'linQ_dim_count': 76, 'linR_count': 342, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': '(C6*D56):C2', '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': 40, 'number_characteristic_subgroups': 60, 'number_conjugacy_classes': 141, 'number_divisions': 43, 'number_normal_subgroups': 66, 'number_subgroup_autclasses': 260, 'number_subgroup_classes': 324, 'number_subgroups': 3348, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 127], [3, 2], [4, 352], [6, 230], [7, 6], [8, 32], [12, 8], [14, 90], [21, 12], [24, 16], [28, 96], [42, 36], [56, 192], [84, 48], [168, 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, 448, 452, 1060, 0, 4], 'outer_gens': [[681, 6, 16, 224], [809, 682, 1336, 224], [101, 10, 664, 224], [837, 682, 240, 224], [673, 14, 24, 224], [197, 682, 272, 1120]], 'outer_group': '192.1543', 'outer_hash': 1543, 'outer_nilpotent': True, 'outer_order': 192, 'outer_permdeg': 192, 'outer_perms': [192856457807972654461525809511547182257644013047636191245343549110120255302316709977635194397344582362198861758740205207508583751864269175752641803326797572264976845501722713894376781358982624963296810482203659682287229876532426996746642738515510056683475010187169000443801670133946583316787637554079731725984799431964673116739226696452788580162314850195977, 195109878434246615609310258102660247542340479280547906862050326139028530372688460718398003861641824411325635017475893501996662824416672890964738647765151523229471575965448779249288547937138689058333603278930431148549004646989019915115113601669597848117890444488518438546174848405299665761019939394748436873649876369775531139360194734173173085015713865900414, 5732432406382706196555240534149233799440772930120618541658879306995116970040654150692969101588506235356606261733958330532470864992840179742343931997122190333313848408268009079772046194710667067489457607405170611092030747007125546091243347328874768065190405911585882756460396405419334904337429580189148618226888989879234091737602136273383709673683301633209, 160390799042662248762254048076179927249498609320867076463634029993930475640974273391401146036038327397674657544106469898899818581976528600886475905147237100030048761172514786641484770413079118187495420103224349370066504029404814934819496734169046225312915457938838722776253114441496791772494286217070986643095765736682997093115402873953704140465871803430110, 9449790627551812003865161323805554620269660547183579088240691569824335008112914850350341813164116970735451668971028547844334840757179286781360715026104464930757522653415738350669638832720773270331967962134127107645173793340504663870378757141304903062861933988962412307323417442018810702150604213557566774410250422308895640574933606765157064397371199116808, 200203177639687665450433336024921987118163433188311297999300699124989690967427349963443127252932324236844393953263284246919086856587519888429849399618657085743322030306592187054815235248963929725504688984727050146101046751261437520523842709813831553775872306123157169453686073354419671190383904352674173166200194831277781203295641381741239765853423489721992], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^5\\times C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 22, '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, 6], [24, 4], [48, 3]], 'representations': {'PC': {'code': 1111473967584376499147818642917873441641995015861, 'gens': [1, 2, 5, 7], 'pres': [8, -2, -2, -2, -2, 2, -7, -2, -3, 225, 41, 290, 66, 8644, 13772, 116, 9221, 3958, 166, 3623]}, 'Perm': {'d': 22, 'gens': [2439325304713761120, 6326137567, 121620365, 183053656, 217728840, 183053640, 93405312000, 55969571858264064000]}}, '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': '(C_6\\times D_{56}):C_2', 'transitive_degree': 672, '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': 42, 'aut_gen_orders': [6, 6, 14, 6, 6, 6, 6, 6, 6], 'aut_gens': [[1, 2, 4, 8], [193, 230, 4, 140], [5, 230, 4, 188], [49, 562, 4, 236], [481, 454, 4, 492], [53, 226, 4, 140], [97, 338, 4, 376], [101, 454, 4, 296], [293, 2, 4, 380], [293, 114, 4, 424]], 'aut_group': None, 'aut_hash': 6441793362254340223, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 16128, 'aut_permdeg': 44, 'aut_perms': [60783079811553384806627579338456935816877052721722865, 60783059411330235134485962755766911690893919538279231, 727829478267204612023151655513487386545128225012100181, 118419356647870992605865956731218197341724381095821855, 240423864235401180701157773296178262102724441045831401, 1701304772742060474852916143709644339293477197570935142, 118419091462670006990535249137215472456675650439886969, 2539383123234995847356505533139817910275415275500512264, 325949507434847946953045452864660909642935107282897687], 'aut_phi_ratio': 84.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 6, 2, 1], [2, 14, 2, 1], [2, 28, 1, 1], [2, 42, 2, 1], [3, 2, 1, 1], [4, 4, 1, 1], [4, 12, 1, 1], [4, 84, 1, 1], [6, 2, 1, 3], [6, 28, 2, 2], [7, 2, 3, 1], [12, 4, 2, 1], [14, 2, 3, 3], [14, 12, 6, 1], [21, 4, 3, 1], [28, 4, 6, 1], [28, 12, 6, 1], [42, 4, 3, 3], [84, 4, 12, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^6\\times S_3\\times F_7', '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': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^6', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 42, 'autcentquo_group': '252.26', 'autcentquo_hash': 26, 'autcentquo_nilpotent': False, 'autcentquo_order': 252, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 6, 2], [2, 14, 2], [2, 28, 1], [2, 42, 2], [3, 2, 1], [4, 4, 1], [4, 12, 1], [4, 84, 1], [6, 2, 3], [6, 28, 4], [7, 2, 3], [12, 4, 2], [14, 2, 9], [14, 12, 6], [21, 4, 3], [28, 4, 6], [28, 12, 6], [42, 4, 9], [84, 4, 12]], 'center_label': '4.2', 'center_order': 4, 'central_product': False, 'central_quotient': '168.50', 'commutator_count': 1, 'commutator_label': '84.15', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 618, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 6, 1, 2], [2, 14, 1, 2], [2, 28, 1, 1], [2, 42, 1, 2], [3, 2, 1, 1], [4, 4, 1, 1], [4, 12, 1, 1], [4, 84, 1, 1], [6, 2, 1, 3], [6, 28, 1, 2], [6, 28, 2, 1], [7, 2, 3, 1], [12, 4, 2, 1], [14, 2, 3, 3], [14, 12, 3, 2], [21, 4, 3, 1], [28, 4, 6, 1], [28, 12, 6, 1], [42, 4, 3, 3], [84, 4, 12, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 5376, 'exponent': 84, 'exponents_of_order': [5, 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': '168.50', 'hash': 618, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 42, 'inner_gen_orders': [2, 2, 1, 42], 'inner_gens': [[1, 2, 4, 440], [1, 2, 4, 236], [1, 2, 4, 8], [241, 454, 4, 8]], 'inner_hash': 50, 'inner_nilpotent': False, 'inner_order': 168, 'inner_split': True, 'inner_tex': 'S_3\\times D_{14}', 'inner_used': [1, 2, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 38], [4, 32]], 'label': '672.618', 'linC_count': 48, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 256, 'linQ_dim': 12, 'linQ_dim_count': 256, 'linR_count': 252, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'D6:D28', '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, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 30, 'number_characteristic_subgroups': 44, 'number_conjugacy_classes': 78, 'number_divisions': 35, 'number_normal_subgroups': 54, 'number_subgroup_autclasses': 184, 'number_subgroup_classes': 260, 'number_subgroups': 2376, 'old_label': None, 'order': 672, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 155], [3, 2], [4, 100], [6, 118], [7, 6], [12, 8], [14, 90], [21, 12], [28, 96], [42, 36], [84, 48]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 2, 2, 6], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[1, 2, 4, 232], [5, 2, 4, 8], [337, 2, 4, 8], [1, 338, 4, 8], [1, 2, 4, 488]], 'outer_group': '96.231', 'outer_hash': 231, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 13, 'outer_perms': [720, 479001600, 3628800, 40320, 27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4\\times C_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 18, '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, 4], [6, 4], [12, 6], [24, 3]], 'representations': {'PC': {'code': 9478402235132424014463575342871288591371, 'gens': [1, 2, 3, 4], 'pres': [7, -2, -2, -2, 2, -2, -3, -7, 12323, 3314, 80, 7284, 8131, 102, 17477, 5388, 166, 28230]}, 'GLZN': {'d': 2, 'p': 84, 'gens': [17188445, 1042546, 36897650, 41391699, 24300865, 16498173, 593713]}, 'Perm': {'d': 18, 'gens': [21010447947624, 1681, 4032024, 7983360, 744, 3, 397707843456000]}}, 'schur_multiplier': [2, 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_6:D_{28}', 'transitive_degree': 168, 'wreath_data': None, 'wreath_product': False}
