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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '1344.2538', 'ambient_counter': 2538, 'ambient_order': 1344, 'ambient_tex': 'C_{168}.D_4', 'central': True, 'central_factor': False, 'centralizer_order': 1344, 'characteristic': True, 'core_order': 1, 'counter': 220, 'cyclic': True, 'direct': True, 'hall': 1, 'label': '1344.2538.1344.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '1344.a1.a1', 'outer_equivalence': False, 'perfect': True, 'proper': False, 'quotient': '1344.2538', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 2538, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 1344, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_{168}.D_4', 'simple': False, 'solvable': True, 'special_labels': ['PC', 'U3', 'D2', 'C8'], 'split': True, 'standard_generators': False, 'stem': True, 'subgroup': '1.1', 'subgroup_hash': 1, 'subgroup_order': 1, 'subgroup_tex': 'C_1', 'supersolvable': True, 'sylow': 1}
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gps_subgroup_data • Show schema
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{'ambient': '1344.2538', 'aut_centralizer_order': None, 'aut_label': '1344.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1.a1.a1', 'complements': ['1.a1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['192.a1.a1', '448.a1.a1', '672.a1.a1', '672.b1.a1', '672.b1.b1', '672.c1.a1'], 'contains': [], 'core': '1344.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [7695, 8261, 1615, 6879, 2025, 6060, 1731, 3051], 'generators': [], 'label': '1344.2538.1344.a1.a1', 'mobius_quo': 1, 'mobius_sub': 0, 'normal_closure': '1344.a1.a1', 'normal_contained_in': ['192.a1.a1', '448.a1.a1', '672.b1.a1', '672.b1.b1', '672.a1.a1'], 'normal_contains': [], 'normalizer': '1.a1.a1', 'old_label': '1344.a1.a1', 'projective_image': '1344.2538', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '1344.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': '1.1', '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': [], '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]], '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]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': [], 'composition_length': 0, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 0, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1, 'exponent': 1, 'exponents_of_order': [], 'factors_of_aut_order': [], 'factors_of_order': [], 'faithful_reps': [[1, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '1.1', 'hash': 1, 'hyperelementary': 1, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [], 'inner_gens': [], '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, 1]], 'label': '1.1', 'linC_count': 1, 'linC_degree': 0, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 0, 'linQ_degree_count': 1, 'linQ_dim': 0, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 0, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C1', 'ngens': 0, 'nilpotency_class': 0, 'nilpotent': True, 'normal_counts': [1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 1, 'number_characteristic_subgroups': 1, 'number_conjugacy_classes': 1, 'number_divisions': 1, 'number_normal_subgroups': 1, 'number_subgroup_autclasses': 1, 'number_subgroup_classes': 1, 'number_subgroups': 1, 'old_label': None, 'order': 1, 'order_factorization_type': 0, 'order_stats': [[1, 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': 0, 'perfect': True, 'permutation_degree': 1, 'pgroup': 1, 'primary_abelian_invariants': [], 'quasisimple': False, 'rank': 0, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 1]], 'representations': {'PC': {'code': 0, 'gens': [], 'pres': []}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_1', 'transitive_degree': 1, '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': 168, 'aut_gen_orders': [12, 56, 12, 12, 2, 6, 12, 24], 'aut_gens': [[1, 2, 8], [453, 942, 380], [673, 890, 232], [1121, 650, 156], [677, 414, 1148], [1, 894, 1112], [225, 102, 248], [673, 750, 184], [673, 366, 136]], 'aut_group': None, 'aut_hash': 7015646718572698385, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 129024, 'aut_permdeg': 124, 'aut_perms': [1217636558367327916741163416274798244814631847198364702855527246345963438326030590280388698721115147298952291518542342576274924256017179888777381950707572588718942334485778004915847025848048286881283763125787, 915675048604424742046926936602849371928912180497736804263689667349357176189734508039327989745384084544664380250793494408840190667738093412398560042464124653320680025895918504968217633403934050700967438659775, 776974668180851251827859903762584546532233257774597610779460868552326427094262361831347760914576386297838253472606474629564536024732090722713605861119908278644934692432501644907797513853117479448181271331911, 868959799663229004419708047519028835990768404359441016094125378001197610892529929865495269510512268350089796332146984075577867206965751410967662295377012327667602194753025625819927968596563245466708316891167, 522224739425571649163372615552394360542354863305913502410505784073344425908906717020580400594033475430217083730443308207179531387789903237129233585794281080924580670018768748559296247967201594919780876520441, 393460043031861981439280444729263481910498146811378069775443310802837543699376177441498451680282176600430636800375751252674208718738082367485403992017860928564977611410799063442327625398233080237816308892090, 378135476444826151389509029461915450001391227146863044354076068469425607919197163188459994519982277673412203790534961519978699169164039448281641248779947268420479205188301090238477842061918331708565795569830, 1437154281923740186897834084857997439764759847132470583272232352149878030878083579831405989402200011941530403197273843999187059900549471998956342939147194341421075896829922815907353099799057628053430414512591], 'aut_phi_ratio': 336.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 12, 1, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 12, 1, 1], [4, 56, 2, 1], [4, 168, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [7, 2, 3, 1], [8, 4, 2, 1], [8, 12, 2, 1], [12, 4, 1, 2], [12, 56, 4, 1], [14, 2, 3, 1], [14, 2, 6, 1], [14, 12, 6, 1], [21, 4, 3, 1], [24, 4, 4, 1], [28, 2, 6, 2], [28, 12, 6, 1], [42, 4, 3, 1], [42, 4, 6, 1], [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_2^4', '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': 84, 'autcentquo_group': '2016.eo', 'autcentquo_hash': 2236184988062812809, 'autcentquo_nilpotent': False, 'autcentquo_order': 2016, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times D_4\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 12, 1], [3, 2, 1], [4, 2, 2], [4, 12, 1], [4, 56, 2], [4, 168, 2], [6, 2, 3], [7, 2, 3], [8, 4, 2], [8, 12, 2], [12, 4, 2], [12, 56, 4], [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': 2538, '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], [3, 2, 1, 1], [4, 2, 1, 2], [4, 12, 1, 1], [4, 56, 1, 2], [4, 168, 1, 2], [6, 2, 1, 3], [7, 2, 3, 1], [8, 4, 1, 2], [8, 12, 2, 1], [12, 4, 1, 2], [12, 56, 2, 2], [14, 2, 3, 3], [14, 12, 6, 1], [21, 4, 3, 1], [24, 4, 2, 2], [28, 2, 6, 2], [28, 12, 6, 1], [42, 4, 3, 3], [56, 4, 6, 2], [56, 12, 12, 1], [84, 4, 6, 2], [168, 4, 12, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 5376, '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': 2538, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 2, 84], 'inner_gens': [[1, 6, 232], [5, 2, 1112], [1121, 242, 8]], 'inner_hash': 149, 'inner_nilpotent': False, 'inner_order': 336, 'inner_split': False, 'inner_tex': 'S_3\\times D_{28}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 62], [4, 68]], 'label': '1344.2538', 'linC_count': 912, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 88, 'linQ_dim': 18, 'linQ_dim_count': 80, 'linR_count': 192, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C168.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': 33, 'number_characteristic_subgroups': 42, 'number_conjugacy_classes': 138, 'number_divisions': 44, 'number_normal_subgroups': 68, 'number_subgroup_autclasses': 152, 'number_subgroup_classes': 220, 'number_subgroups': 1780, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 15], [3, 2], [4, 464], [6, 6], [7, 6], [8, 32], [12, 232], [14, 90], [21, 12], [24, 16], [28, 96], [42, 36], [56, 192], [84, 48], [168, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 2, 2, 12], 'outer_gen_pows': [0, 0, 0, 0, 0, 1012], 'outer_gens': [[677, 2, 904], [1, 2, 680], [673, 2, 12], [677, 2, 1240], [1, 2, 904], [1, 170, 968]], 'outer_group': '384.19891', 'outer_hash': 19891, 'outer_nilpotent': True, 'outer_order': 384, 'outer_permdeg': 15, 'outer_perms': [25430670720, 1680, 31658379384, 744, 294111336960, 108424915204], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^6:C_6', 'pc_rank': 3, '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, 8], [4, 7], [6, 4], [8, 2], [12, 8], [24, 5], [48, 2]], 'representations': {'PC': {'code': 24348791756442289127121050441909756686148252420783953563619351307, 'gens': [1, 2, 4], 'pres': [8, -2, -2, -2, -2, -2, -2, -3, -7, 5376, 97, 41, 7427, 17803, 91, 18564, 17612, 116, 44549, 9997, 141, 28678, 23310, 222, 36879]}, 'Perm': {'d': 30, 'gens': [305292256521391761153460514649, 51367855497544223287, 406269022549189680, 531589335404370480, 661079113471536616, 661079113471536600, 1175091669949317120000, 9452345886333784620057231360000]}}, 'schur_multiplier': [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_{168}.D_4', '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': 168, 'aut_gen_orders': [12, 56, 12, 12, 2, 6, 12, 24], 'aut_gens': [[1, 2, 8], [453, 942, 380], [673, 890, 232], [1121, 650, 156], [677, 414, 1148], [1, 894, 1112], [225, 102, 248], [673, 750, 184], [673, 366, 136]], 'aut_group': None, 'aut_hash': 7015646718572698385, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 129024, 'aut_permdeg': 124, 'aut_perms': [1217636558367327916741163416274798244814631847198364702855527246345963438326030590280388698721115147298952291518542342576274924256017179888777381950707572588718942334485778004915847025848048286881283763125787, 915675048604424742046926936602849371928912180497736804263689667349357176189734508039327989745384084544664380250793494408840190667738093412398560042464124653320680025895918504968217633403934050700967438659775, 776974668180851251827859903762584546532233257774597610779460868552326427094262361831347760914576386297838253472606474629564536024732090722713605861119908278644934692432501644907797513853117479448181271331911, 868959799663229004419708047519028835990768404359441016094125378001197610892529929865495269510512268350089796332146984075577867206965751410967662295377012327667602194753025625819927968596563245466708316891167, 522224739425571649163372615552394360542354863305913502410505784073344425908906717020580400594033475430217083730443308207179531387789903237129233585794281080924580670018768748559296247967201594919780876520441, 393460043031861981439280444729263481910498146811378069775443310802837543699376177441498451680282176600430636800375751252674208718738082367485403992017860928564977611410799063442327625398233080237816308892090, 378135476444826151389509029461915450001391227146863044354076068469425607919197163188459994519982277673412203790534961519978699169164039448281641248779947268420479205188301090238477842061918331708565795569830, 1437154281923740186897834084857997439764759847132470583272232352149878030878083579831405989402200011941530403197273843999187059900549471998956342939147194341421075896829922815907353099799057628053430414512591], 'aut_phi_ratio': 336.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 12, 1, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 12, 1, 1], [4, 56, 2, 1], [4, 168, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [7, 2, 3, 1], [8, 4, 2, 1], [8, 12, 2, 1], [12, 4, 1, 2], [12, 56, 4, 1], [14, 2, 3, 1], [14, 2, 6, 1], [14, 12, 6, 1], [21, 4, 3, 1], [24, 4, 4, 1], [28, 2, 6, 2], [28, 12, 6, 1], [42, 4, 3, 1], [42, 4, 6, 1], [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_2^4', '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': 84, 'autcentquo_group': '2016.eo', 'autcentquo_hash': 2236184988062812809, 'autcentquo_nilpotent': False, 'autcentquo_order': 2016, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times D_4\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 12, 1], [3, 2, 1], [4, 2, 2], [4, 12, 1], [4, 56, 2], [4, 168, 2], [6, 2, 3], [7, 2, 3], [8, 4, 2], [8, 12, 2], [12, 4, 2], [12, 56, 4], [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': 2538, '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], [3, 2, 1, 1], [4, 2, 1, 2], [4, 12, 1, 1], [4, 56, 1, 2], [4, 168, 1, 2], [6, 2, 1, 3], [7, 2, 3, 1], [8, 4, 1, 2], [8, 12, 2, 1], [12, 4, 1, 2], [12, 56, 2, 2], [14, 2, 3, 3], [14, 12, 6, 1], [21, 4, 3, 1], [24, 4, 2, 2], [28, 2, 6, 2], [28, 12, 6, 1], [42, 4, 3, 3], [56, 4, 6, 2], [56, 12, 12, 1], [84, 4, 6, 2], [168, 4, 12, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 5376, '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': 2538, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [2, 2, 84], 'inner_gens': [[1, 6, 232], [5, 2, 1112], [1121, 242, 8]], 'inner_hash': 149, 'inner_nilpotent': False, 'inner_order': 336, 'inner_split': False, 'inner_tex': 'S_3\\times D_{28}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 62], [4, 68]], 'label': '1344.2538', 'linC_count': 912, 'linC_degree': 6, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 88, 'linQ_dim': 18, 'linQ_dim_count': 80, 'linR_count': 192, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C168.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': 33, 'number_characteristic_subgroups': 42, 'number_conjugacy_classes': 138, 'number_divisions': 44, 'number_normal_subgroups': 68, 'number_subgroup_autclasses': 152, 'number_subgroup_classes': 220, 'number_subgroups': 1780, 'old_label': None, 'order': 1344, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 15], [3, 2], [4, 464], [6, 6], [7, 6], [8, 32], [12, 232], [14, 90], [21, 12], [24, 16], [28, 96], [42, 36], [56, 192], [84, 48], [168, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 2, 2, 12], 'outer_gen_pows': [0, 0, 0, 0, 0, 1012], 'outer_gens': [[677, 2, 904], [1, 2, 680], [673, 2, 12], [677, 2, 1240], [1, 2, 904], [1, 170, 968]], 'outer_group': '384.19891', 'outer_hash': 19891, 'outer_nilpotent': True, 'outer_order': 384, 'outer_permdeg': 15, 'outer_perms': [25430670720, 1680, 31658379384, 744, 294111336960, 108424915204], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^6:C_6', 'pc_rank': 3, '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, 8], [4, 7], [6, 4], [8, 2], [12, 8], [24, 5], [48, 2]], 'representations': {'PC': {'code': 24348791756442289127121050441909756686148252420783953563619351307, 'gens': [1, 2, 4], 'pres': [8, -2, -2, -2, -2, -2, -2, -3, -7, 5376, 97, 41, 7427, 17803, 91, 18564, 17612, 116, 44549, 9997, 141, 28678, 23310, 222, 36879]}, 'Perm': {'d': 30, 'gens': [305292256521391761153460514649, 51367855497544223287, 406269022549189680, 531589335404370480, 661079113471536616, 661079113471536600, 1175091669949317120000, 9452345886333784620057231360000]}}, 'schur_multiplier': [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_{168}.D_4', 'transitive_degree': 672, 'wreath_data': None, 'wreath_product': False}
