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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '4200.o', 'ambient_counter': 15, 'ambient_order': 4200, 'ambient_tex': 'C_{35}:C_{120}', 'central': False, 'central_factor': False, 'centralizer_order': 2100, 'characteristic': True, 'core_order': 420, 'counter': 10, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '4200.o.10.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '10.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '10.1', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 1, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 10, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_5', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '420.12', 'subgroup_hash': 12, 'subgroup_order': 420, 'subgroup_tex': 'C_{420}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '4200.o', 'aut_centralizer_order': 280, 'aut_label': '10.a1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': '96.220', 'aut_weyl_index': 280, 'centralizer': '2.a1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['2.a1.a1', '5.b1.a1'], 'contains': ['20.a1.a1', '30.a1.a1', '50.a1.a1', '70.a1.a1'], 'core': '10.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [6574, 241, 6814, 2710, 4970, 6002, 5458, 3026], 'generators': [30, 24, 600, 80, 60], 'label': '4200.o.10.a1.a1', 'mobius_quo': 0, 'mobius_sub': 5, 'normal_closure': '10.a1.a1', 'normal_contained_in': ['2.a1.a1'], 'normal_contains': ['20.a1.a1', '30.a1.a1', '50.a1.a1', '70.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '10.a1.a1', 'projective_image': '70.3', 'quotient_action_image': '2.1', 'quotient_action_kernel': '5.1', 'quotient_action_kernel_order': 5, 'quotient_fusion': None, 'short_label': '10.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2.1'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '420.12', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 12], 'aut_gens': [[1], [211], [281], [181], [227]], 'aut_group': '96.220', 'aut_hash': 220, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 96, 'aut_permdeg': 14, 'aut_perms': [39916800, 6227020800, 450, 6267345993], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [5, 1, 4, 1], [6, 1, 2, 1], [7, 1, 6, 1], [10, 1, 4, 1], [12, 1, 4, 1], [14, 1, 6, 1], [15, 1, 8, 1], [20, 1, 8, 1], [21, 1, 12, 1], [28, 1, 12, 1], [30, 1, 8, 1], [35, 1, 24, 1], [42, 1, 12, 1], [60, 1, 16, 1], [70, 1, 24, 1], [84, 1, 24, 1], [105, 1, 48, 1], [140, 1, 48, 1], [210, 1, 48, 1], [420, 1, 96, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^3\\times C_{12}', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '96.220', 'autcent_hash': 220, 'autcent_nilpotent': True, 'autcent_order': 96, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\times C_{12}', '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], [3, 1, 2], [4, 1, 2], [5, 1, 4], [6, 1, 2], [7, 1, 6], [10, 1, 4], [12, 1, 4], [14, 1, 6], [15, 1, 8], [20, 1, 8], [21, 1, 12], [28, 1, 12], [30, 1, 8], [35, 1, 24], [42, 1, 12], [60, 1, 16], [70, 1, 24], [84, 1, 24], [105, 1, 48], [140, 1, 48], [210, 1, 48], [420, 1, 96]], 'center_label': '420.12', 'center_order': 420, 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '5.1', '7.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 12, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['4.1', 1], ['5.1', 1], ['7.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [5, 1, 4, 1], [6, 1, 2, 1], [7, 1, 6, 1], [10, 1, 4, 1], [12, 1, 4, 1], [14, 1, 6, 1], [15, 1, 8, 1], [20, 1, 8, 1], [21, 1, 12, 1], [28, 1, 12, 1], [30, 1, 8, 1], [35, 1, 24, 1], [42, 1, 12, 1], [60, 1, 16, 1], [70, 1, 24, 1], [84, 1, 24, 1], [105, 1, 48, 1], [140, 1, 48, 1], [210, 1, 48, 1], [420, 1, 96, 1]], 'element_repr_type': 'PC', 'elementary': 210, 'eulerian_function': 1, 'exponent': 420, 'exponents_of_order': [2, 1, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5, 7], 'faithful_reps': [[1, 0, 96]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '210.12', 'hash': 12, 'hyperelementary': 210, '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': 96, 'irrQ_dim': 96, 'irrR_degree': 2, 'irrep_stats': [[1, 420]], 'label': '420.12', 'linC_count': 96, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 12, 'linQ_dim': 14, 'linQ_dim_count': 12, 'linR_count': 48, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C420', 'ngens': 5, 'nilpotency_class': 1, 'nilpotent': True, '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': 24, 'number_characteristic_subgroups': 24, 'number_conjugacy_classes': 420, 'number_divisions': 24, 'number_normal_subgroups': 24, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 24, 'number_subgroups': 24, 'old_label': None, 'order': 420, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 2], [5, 4], [6, 2], [7, 6], [10, 4], [12, 4], [14, 6], [15, 8], [20, 8], [21, 12], [28, 12], [30, 8], [35, 24], [42, 12], [60, 16], [70, 24], [84, 24], [105, 48], [140, 48], [210, 48], [420, 96]], 'outer_abelian': True, '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': [[211], [281], [181], [227]], 'outer_group': '96.220', 'outer_hash': 220, 'outer_nilpotent': True, 'outer_order': 96, 'outer_permdeg': 14, 'outer_perms': [39916800, 6227020800, 450, 6267345993], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_{12}', 'pc_rank': 1, 'perfect': False, 'permutation_degree': 19, 'pgroup': 0, 'primary_abelian_invariants': [4, 3, 5, 7], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 3], [6, 2], [8, 3], [12, 3], [16, 1], [24, 3], [48, 3], [96, 1]], 'representations': {'PC': {'code': 9598160346925019, 'gens': [1], 'pres': [5, -2, -2, -3, -5, -7, 10, 26, 57, 118]}, 'GLFp': {'d': 2, 'p': 29, 'gens': [669318]}, 'Perm': {'d': 19, 'gens': [19918495973376000, 174356582400, 159667200, 4320, 6423296495616000]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [420], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{420}', 'transitive_degree': 420, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '120.4', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 420, 'aut_gen_orders': [28, 30, 6, 12, 12, 28, 14], 'aut_gens': [[1, 120], [1259, 2640], [539, 3120], [3659, 2880], [373, 2760], [2563, 2040], [2269, 960], [4069, 3480]], 'aut_group': None, 'aut_hash': 1713921373283272042, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 26880, 'aut_permdeg': 146, 'aut_perms': [27327507077808166340904620113717602822339456339813189497539763384193513649709320055198267630915093057224169381677773982664811166285990292182887871570546232673704948711073691290249953419160092662327160307363329559856959333700481399948967198581585032491563, 16442182010430093492814323628300080919206004951580524308385344313346506683228830355613282543023494537987351091766623934847012176518995580599454097496539380134243428285614460420833917050943076636424287597673783492133160702927901168930276372932360599862607, 103039676664946325414521225739773319789457210035602882775403731070026534627847153084215957513127751697421375677129348393164879889540642399961212240318123703938167583639423753587916035492757558625373927959392344285069193454722172702784344198429017893735579, 59134287927417643549489644962664872422950940531585488103846795044614314139683761763652752163262084438730247736028820478001969241180569262744691290012714146637551866151118674327472484392596601286880285138333682307816442045960531843994326656273145243775969, 82580188842327126339322766337092108373436530378279404457449261876427887158926620605585130756628989597701907870480131909595930500221538546293953161386317807007442514885910428774925236108989292702461340866303971243191125839643048294837415053728349160791568, 94298499635882168900445588036840031572804194744043023404746148550139990329212738988426449852953046279826934224906591602878167159909648700385139180027503484720712036917648847086873570429715282638255303794490880198842505842964743503978223099118541933331770, 31704895294623023097880882468589924573347173949509848334512613208874437384560494458658421534251994723203046984609479123319145071048627425981526588381980104750738129307550875414030591702281701373832421226585142248503511607689548425314904446408433742502221], 'aut_phi_ratio': 28.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 8, 1], [6, 1, 2, 1], [7, 2, 3, 1], [8, 35, 4, 1], [10, 1, 4, 1], [10, 2, 2, 1], [10, 2, 8, 1], [12, 1, 4, 1], [14, 2, 3, 1], [15, 1, 8, 1], [15, 2, 4, 1], [15, 2, 16, 1], [20, 1, 8, 1], [20, 2, 4, 1], [20, 2, 16, 1], [21, 2, 6, 1], [24, 35, 8, 1], [28, 2, 6, 1], [30, 1, 8, 1], [30, 2, 4, 1], [30, 2, 16, 1], [35, 2, 12, 2], [35, 2, 48, 1], [40, 35, 16, 1], [42, 2, 6, 1], [60, 1, 16, 1], [60, 2, 8, 1], [60, 2, 32, 1], [70, 2, 12, 2], [70, 2, 48, 1], [84, 2, 12, 1], [105, 2, 24, 2], [105, 2, 96, 1], [120, 35, 32, 1], [140, 2, 24, 2], [140, 2, 96, 1], [210, 2, 24, 2], [210, 2, 96, 1], [420, 2, 48, 2], [420, 2, 192, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^3\\times C_4\\times F_5\\times F_7', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.45', 'autcent_hash': 45, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\times C_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 420, 'autcentquo_group': '840.139', 'autcentquo_hash': 139, 'autcentquo_nilpotent': False, 'autcentquo_order': 840, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5\\times F_7', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 1, 2], [4, 1, 2], [5, 1, 4], [5, 2, 10], [6, 1, 2], [7, 2, 3], [8, 35, 4], [10, 1, 4], [10, 2, 10], [12, 1, 4], [14, 2, 3], [15, 1, 8], [15, 2, 20], [20, 1, 8], [20, 2, 20], [21, 2, 6], [24, 35, 8], [28, 2, 6], [30, 1, 8], [30, 2, 20], [35, 2, 72], [40, 35, 16], [42, 2, 6], [60, 1, 16], [60, 2, 40], [70, 2, 72], [84, 2, 12], [105, 2, 144], [120, 35, 32], [140, 2, 144], [210, 2, 144], [420, 2, 288]], 'center_label': '60.4', 'center_order': 60, 'central_product': True, 'central_quotient': '70.3', 'commutator_count': 1, 'commutator_label': '35.1', 'complements_known': True, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '5.1', '5.1', '7.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 15, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['280.3', 1], ['3.1', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [5, 1, 4, 1], [5, 2, 2, 1], [5, 2, 4, 2], [6, 1, 2, 1], [7, 2, 3, 1], [8, 35, 4, 1], [10, 1, 4, 1], [10, 2, 2, 1], [10, 2, 4, 2], [12, 1, 4, 1], [14, 2, 3, 1], [15, 1, 8, 1], [15, 2, 4, 1], [15, 2, 8, 2], [20, 1, 8, 1], [20, 2, 4, 1], [20, 2, 8, 2], [21, 2, 6, 1], [24, 35, 8, 1], [28, 2, 6, 1], [30, 1, 8, 1], [30, 2, 4, 1], [30, 2, 8, 2], [35, 2, 12, 2], [35, 2, 24, 2], [40, 35, 16, 1], [42, 2, 6, 1], [60, 1, 16, 1], [60, 2, 8, 1], [60, 2, 16, 2], [70, 2, 12, 2], [70, 2, 24, 2], [84, 2, 12, 1], [105, 2, 24, 2], [105, 2, 48, 2], [120, 35, 32, 1], [140, 2, 24, 2], [140, 2, 48, 2], [210, 2, 24, 2], [210, 2, 48, 2], [420, 2, 48, 2], [420, 2, 96, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 288, 'exponent': 840, 'exponents_of_order': [3, 2, 1, 1], 'factors_of_aut_order': [2, 3, 5, 7], 'factors_of_order': [2, 3, 5, 7], 'faithful_reps': [[2, 0, 192]], 'familial': False, 'frattini_label': '4.1', 'frattini_quotient': '1050.31', 'hash': 3605652675087346187, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 70, 'inner_gen_orders': [2, 35], 'inner_gens': [[1, 4080], [241, 120]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 70, 'inner_split': True, 'inner_tex': 'D_{35}', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 192, 'irrQ_dim': 192, 'irrR_degree': None, 'irrep_stats': [[1, 120], [2, 1020]], 'label': '4200.o', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'linQ_degree_count': None, 'linQ_dim': None, 'linQ_dim_count': None, 'linR_count': None, 'linR_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C35:C120', '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, 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': 52, 'number_characteristic_subgroups': 52, 'number_conjugacy_classes': 1140, 'number_divisions': 64, 'number_normal_subgroups': 52, 'number_subgroup_autclasses': 76, 'number_subgroup_classes': 88, 'number_subgroups': 288, 'old_label': None, 'order': 4200, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 2], [5, 24], [6, 2], [7, 6], [8, 140], [10, 24], [12, 4], [14, 6], [15, 48], [20, 48], [21, 12], [24, 280], [28, 12], [30, 48], [35, 144], [40, 560], [42, 12], [60, 96], [70, 144], [84, 24], [105, 288], [120, 1120], [140, 288], [210, 288], [420, 576]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 2, 4, 12], 'outer_gen_pows': [0, 0, 0, 0, 0], 'outer_gens': [[91, 120], [61, 120], [41, 120], [1, 2640], [47, 360]], 'outer_group': '384.19878', 'outer_hash': 19878, 'outer_nilpotent': True, 'outer_order': 384, 'outer_permdeg': 17, 'outer_perms': [20922789888000, 479001600, 87178291200, 864, 21010451213667], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3\\times C_4\\times C_{12}', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 28, 'pgroup': 0, 'primary_abelian_invariants': [8, 3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 6], [6, 2], [8, 11], [12, 3], [16, 9], [24, 5], [32, 3], [48, 10], [96, 8], [192, 2]], 'representations': {'PC': {'code': '1032505579665446479890739142834439131', 'gens': [1, 6], 'pres': [7, -2, -2, -2, -3, -5, -5, -7, 14, 36, 58, 108, 171365, 250, 176406]}, 'GLFp': {'d': 2, 'p': 421, 'gens': [14401362975, 9327307689, 231129]}, 'Perm': {'d': 28, 'gens': [11727096416380429380663575797, 262560264741663892, 23003126742653342006645589847]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [120], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{35}:C_{120}', 'transitive_degree': 840, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': False, 'abelian_quotient': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 20, 'aut_gen_orders': [4, 5], 'aut_gens': [[1, 2], [3, 6], [5, 2]], 'aut_group': '20.3', 'aut_hash': 3, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20, 'aut_permdeg': 5, 'aut_perms': [17, 89], 'aut_phi_ratio': 5.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [5, 2, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'F_5', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 20, 'autcentquo_group': '20.3', 'autcentquo_hash': 3, 'autcentquo_nilpotent': False, 'autcentquo_order': 20, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'F_5', 'cc_stats': [[1, 1, 1], [2, 5, 1], [5, 2, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '10.1', 'commutator_count': 1, 'commutator_label': '5.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '5.1'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 1, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [5, 2, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 10, 'exponents_of_order': [1, 1], 'factors_of_aut_order': [2, 5], 'factors_of_order': [2, 5], 'faithful_reps': [[2, 1, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '10.1', 'hash': 1, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 10, 'inner_gen_orders': [2, 5], 'inner_gens': [[1, 8], [5, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 10, 'inner_split': False, 'inner_tex': 'D_5', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 2], [2, 2]], 'label': '10.1', 'linC_count': 2, 'linC_degree': 2, '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': 'D5', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 4, 'number_subgroup_classes': 4, 'number_subgroups': 8, 'old_label': None, 'order': 10, 'order_factorization_type': 11, 'order_stats': [[1, 1], [2, 5], [5, 4]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [1], 'outer_gens': [[1, 6]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 5, 'pgroup': 0, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [4, 1]], 'representations': {'PC': {'code': 227, 'gens': [1, 2], 'pres': [2, -2, -5, 33]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [23707240, 26329372]}, 'Lie': [{'d': 2, 'q': 4, 'gens': [81, 94], 'family': 'CSOMinus'}], 'GLFp': {'d': 2, 'p': 5, 'gens': [131, 129]}, 'Perm': {'d': 5, 'gens': [23, 96]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_5', 'transitive_degree': 5, 'wreath_data': None, 'wreath_product': False}
