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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '800.814', 'ambient_counter': 814, 'ambient_order': 800, 'ambient_tex': 'C_2\\times C_{20}.C_{20}', 'central': False, 'central_factor': True, 'centralizer_order': 40, 'characteristic': True, 'core_order': 160, 'counter': 20, 'cyclic': False, 'direct': True, 'hall': 0, 'label': '800.814.5.a1.a1', 'maximal': True, 'maximal_normal': True, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '5.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '5.1', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': True, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 5, 'quotient_simple': True, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_5', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '160.142', 'subgroup_hash': 142, 'subgroup_order': 160, 'subgroup_tex': 'C_{10}:\\OD_{16}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '800.814', 'aut_centralizer_order': 4, 'aut_label': '5.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '2560.dp', 'aut_weyl_index': 4, 'centralizer': '20.c1.a1', 'complements': ['160.a1.a1', '160.c1.b1', '160.c1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['1.a1.a1'], 'contains': ['10.b1.a1', '10.c1.a1', '10.c1.b1', '10.d1.a1', '10.d1.b1', '10.d1.c1', '10.d1.d1', '25.a1.a1'], 'core': '5.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [7995, 9315, 9339, 9100, 8516, 8040, 7680, 1119], 'generators': [1, 160, 400, 20, 10, 40], 'label': '800.814.5.a1.a1', 'mobius_quo': 0, 'mobius_sub': -1, 'normal_closure': '5.a1.a1', 'normal_contained_in': ['1.a1.a1'], 'normal_contains': ['10.b1.a1', '10.c1.a1', '10.c1.b1', '10.d1.a1', '10.d1.b1', '10.d1.c1', '10.d1.d1'], 'normalizer': '1.a1.a1', 'old_label': '5.a1.a1', 'projective_image': '100.14', 'quotient_action_image': '1.1', 'quotient_action_kernel': '5.1', 'quotient_action_kernel_order': 5, 'quotient_fusion': None, 'short_label': '5.a1.a1', 'subgroup_fusion': None, 'weyl_group': '20.4'}
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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': 20, 'aut_gen_orders': [2, 2, 4, 2, 20, 2, 2, 2, 4, 2], 'aut_gens': [[1, 2, 16], [1, 3, 16], [9, 87, 16], [1, 86, 24], [9, 14, 16], [89, 46, 24], [9, 86, 16], [1, 10, 144], [81, 82, 16], [9, 83, 48], [9, 15, 24]], 'aut_group': '2560.dp', 'aut_hash': 7799020297190345341, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2560, 'aut_permdeg': 15, 'aut_perms': [561960322560, 288369849601, 361327599360, 1, 1118397174, 361806641281, 88216168734, 193077561600, 456978412968, 649138654080], 'aut_phi_ratio': 40.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [4, 1, 2, 2], [4, 2, 2, 1], [5, 2, 2, 1], [8, 10, 8, 1], [10, 2, 2, 1], [10, 2, 4, 1], [10, 2, 8, 1], [20, 2, 4, 2], [20, 2, 8, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^4:D_4\\times F_5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '64.202', 'autcent_hash': 202, 'autcent_nilpotent': True, 'autcent_order': 64, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3:D_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '40.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 40, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [4, 1, 4], [4, 2, 2], [5, 2, 2], [8, 10, 8], [10, 2, 14], [20, 2, 16]], 'center_label': '8.2', 'center_order': 8, 'central_product': True, 'central_quotient': '20.4', 'commutator_count': 1, 'commutator_label': '10.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 142, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['80.10', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [4, 1, 2, 2], [4, 2, 1, 2], [5, 2, 2, 1], [8, 10, 2, 4], [10, 2, 2, 3], [10, 2, 4, 2], [20, 2, 4, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 504, 'exponent': 40, 'exponents_of_order': [5, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '40.13', 'hash': 142, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 2, 5], 'inner_gens': [[1, 10, 16], [9, 2, 144], [1, 34, 16]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 20, 'inner_split': False, 'inner_tex': 'D_{10}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 36]], 'label': '160.142', 'linC_count': 128, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 8, 'linQ_degree_count': 8, 'linQ_dim': 8, 'linQ_dim_count': 4, 'linR_count': 32, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C10:OD16', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 15, 'number_characteristic_subgroups': 19, 'number_conjugacy_classes': 52, 'number_divisions': 24, 'number_normal_subgroups': 49, 'number_subgroup_autclasses': 40, 'number_subgroup_classes': 68, 'number_subgroups': 120, 'old_label': None, 'order': 160, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 7], [4, 8], [5, 4], [8, 80], [10, 28], [20, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4, 4], 'outer_gen_pows': [0, 1, 0, 0, 0], 'outer_gens': [[9, 6, 16], [1, 95, 16], [1, 6, 24], [81, 83, 120], [1, 82, 112]], 'outer_group': '128.2167', 'outer_hash': 2167, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 14, 'outer_perms': [362880, 6227410495, 26815, 13451988342, 388017], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4^2:C_2^3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 10], [16, 2]], 'representations': {'PC': {'code': 1362514926049079549955, 'gens': [1, 2, 5], 'pres': [6, -2, -2, -2, -2, -2, -5, 121, 31, 50, 2170, 88, 2315]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [122109387504, 130454516622, 19018892858, 47766553017, 58658021008, 14933835741]}, 'Perm': {'d': 15, 'gens': [6267307263, 375196, 2309, 18550, 12316, 99672249600]}}, 'schur_multiplier': [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': 'C_{10}:\\OD_{16}', 'transitive_degree': 80, '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': '80.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [10, 4, 4, 20, 10, 4, 20], 'aut_gens': [[1, 2, 80], [1, 619, 80], [441, 418, 120], [1, 426, 600], [41, 618, 120], [441, 262, 80], [1, 794, 600], [401, 95, 120]], 'aut_group': None, 'aut_hash': 2289281346496416815, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 10240, 'aut_permdeg': 84, 'aut_perms': [1347635516957511032442421268629574640229450836498191687471829589793502616451392700141104747767200085229965448031460968535491693, 666739281598761533674890887413466970161444746562331456558217283577046640398646313232667279540078274837698376845107673970050439, 203422261355303525598846536474177135799530469611131942604233228338666962885635294703862182624903718541306207327733643320775585, 1417502530639987264289953662869845891774790328130498098054812233004273857843691604961833886063885525555602022186621547801162405, 3022105977780417866242186246949704877490180194736197669597178349306028966060841865775391280472022241370036772471022997713296471, 1282292564721173261917732548038008375042210286929881603692222099766543343058012419467132721661626710103212260849873943240697701, 1669581855064167704327156672180468187367963565387684735896154898663223781693575777101281069209543886181065869660363582993384152], 'aut_phi_ratio': 32.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [4, 1, 2, 2], [4, 2, 2, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 8, 1], [8, 10, 8, 1], [10, 1, 4, 1], [10, 1, 8, 1], [10, 2, 2, 1], [10, 2, 4, 1], [10, 2, 8, 3], [10, 2, 16, 1], [10, 2, 32, 1], [20, 1, 8, 2], [20, 2, 4, 2], [20, 2, 8, 2], [20, 2, 16, 2], [20, 2, 32, 1], [40, 10, 32, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_5:(C_2\\times C_4^2\\times C_2\\wr C_2^2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '256.27015', 'autcent_hash': 27015, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3.C_2^5', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '40.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 40, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\times F_5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [4, 1, 4], [4, 2, 2], [5, 1, 4], [5, 2, 10], [8, 10, 8], [10, 1, 12], [10, 2, 78], [20, 1, 16], [20, 2, 88], [40, 10, 32]], 'center_label': '40.9', 'center_order': 40, 'central_product': True, 'central_quotient': '20.4', 'commutator_count': 1, 'commutator_label': '10.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '5.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 814, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['5.1', 1], ['80.10', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [4, 1, 2, 2], [4, 2, 1, 2], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 4, 2], [8, 10, 2, 4], [10, 1, 4, 3], [10, 2, 2, 3], [10, 2, 4, 18], [20, 1, 8, 2], [20, 2, 4, 6], [20, 2, 8, 8], [40, 10, 8, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 15624, 'exponent': 40, 'exponents_of_order': [5, 2], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '200.50', 'hash': 814, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 2, 5], 'inner_gens': [[1, 42, 80], [41, 2, 720], [1, 162, 80]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 20, 'inner_split': False, 'inner_tex': 'D_{10}', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 80], [2, 180]], 'label': '800.814', 'linC_count': 3072, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 12, 'linQ_degree_count': 112, 'linQ_dim': 12, 'linQ_dim_count': 64, 'linR_count': 128, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2*C20.C20', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 30, 'number_characteristic_subgroups': 38, 'number_conjugacy_classes': 260, 'number_divisions': 62, 'number_normal_subgroups': 98, 'number_subgroup_autclasses': 96, 'number_subgroup_classes': 190, 'number_subgroups': 348, 'old_label': None, 'order': 800, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 7], [4, 8], [5, 24], [8, 80], [10, 168], [20, 192], [40, 320]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4, 4, 2, 2, 4, 4], 'outer_gen_pows': [160, 0, 0, 161, 640, 0, 160], 'outer_gens': [[1, 619, 80], [441, 418, 120], [1, 426, 600], [41, 618, 120], [441, 262, 80], [1, 794, 600], [401, 95, 120]], 'outer_group': '512.7532447', 'outer_hash': 4915265279642952395, 'outer_nilpotent': True, 'outer_order': 512, 'outer_permdeg': 256, 'outer_perms': [3351420997441564218358268902411913479159657493826267810330236048856557932648040541199429033638573679534385236761579286752622453834659209196469497364528281837615773381332991441720763706312010036751654037016737086978980444447626923416257000087726962301453049482980116191644636239804228716604033535434076817434242209394676317903812967464033320814929868736349640377509901275329690304114034327425191721099287585925245445057969147086206755273600519270248147559408472448399934603890627864045948933400820910794816, 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16886489560170907139104339757139346843833659381527036342568411691253074512659287659199805163421461676905535890472705704012745085661521545526254813787978549282726740490912796641080840609373565258757943493440444568750125879184500355517387747882952796669024761392729555811984356170015543823925267628836932156707229236681194509303836128507807185048470734384847728200821207666192117541315748818888086748111350308978360894715769214987546597750576543117384778753346737599474190420740284700642790754786034671726720, 20250534411837292006293845272057749475546931713119513737653539723662764634817085820016875225879751520896521637963411317049813230063496608179262399218777219657262997525461931438821775376600410963138862424763416288596563866137482860682147199428596967445697711807658332518229613029073337318554676850901399153664005565662244818559395097524979078262005995604212442288732687003387570803360876081428069221455346307993793752367042862637176032947312480107060399330479208240719131173111528517662174819298184983374096], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3.C_2^6', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 20, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 4, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [4, 18], [8, 20], [16, 12]], 'representations': {'PC': {'code': 1732327612206632604896316850227, 'gens': [1, 2, 6], 'pres': [7, -2, -2, -2, -2, -5, -2, -5, 589, 36, 58, 80, 15132, 124, 15693]}, 'GLZN': {'d': 2, 'p': 50, 'gens': [125031, 3875031, 125049, 126251, 6125049, 3156825, 5375043]}, 'Perm': {'d': 20, 'gens': [122419506127984830, 1, 126774, 263569876491398400, 899688, 1229808, 384231431418912000]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 20], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_{20}.C_{20}', 'transitive_degree': 80, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '5.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': True, 'aut_derived_length': 1, 'aut_exponent': 4, 'aut_gen_orders': [4], 'aut_gens': [[1], [2]], 'aut_group': '4.1', 'aut_hash': 1, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [9], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [5, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 4, 'autcent_group': '4.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_4', '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], [5, 1, 4]], 'center_label': '5.1', 'center_order': 5, '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': ['5.1'], 'composition_length': 1, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [5, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 5, 'eulerian_function': 1, 'exponent': 5, 'exponents_of_order': [1], 'factors_of_aut_order': [2], 'factors_of_order': [5], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '5.1', 'hash': 1, 'hyperelementary': 5, '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': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 5]], 'label': '5.1', 'linC_count': 4, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 1, 'linQ_dim': 4, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C5', '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': 5, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 5, 'order_factorization_type': 1, 'order_stats': [[1, 1], [5, 4]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [4], 'outer_gen_pows': [0], 'outer_gens': [[2]], 'outer_group': '4.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_4', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 5, 'pgroup': 5, 'primary_abelian_invariants': [5], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [4, 1]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -5]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [26329372]}, 'Lie': [{'d': 1, 'q': 5, 'gens': [33], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [131]}, 'Perm': {'d': 5, 'gens': [96]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [5], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_5', 'transitive_degree': 5, 'wreath_data': None, 'wreath_product': False}
