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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '12960.bu', 'ambient_counter': 47, 'ambient_order': 12960, 'ambient_tex': 'C_3^3:(F_5\\times S_4)', 'central': False, 'central_factor': False, 'centralizer_order': 5, 'characteristic': True, 'core_order': 1620, 'counter': 18, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '12960.bu.8.a1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '8.a1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '8.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2\\times C_4', 'simple': False, 'solvable': True, 'special_labels': ['D', 'L1', 'D1', 'C3'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '1620.423', 'subgroup_hash': 423, 'subgroup_order': 1620, 'subgroup_tex': 'C_5\\times C_3^3:A_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '12960.bu', 'aut_centralizer_order': None, 'aut_label': '8.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '2592.a1.a1', 'complements': ['1620.i1.a1', '1620.h1.a1'], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '4.b1.a1', '4.c1.a1'], 'contains': ['24.b1.a1', '32.a1.a1', '40.a1.a1', '216.bc1.a1'], 'core': '8.a1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [9127, 6728, 9515, 3908, 9280, 6515, 9502, 3486], 'generators': [32692635423, 1959955200, 77, 239909760, 44111738160, 18593280, 1174320], 'label': '12960.bu.8.a1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '8.a1.a1', 'normal_contained_in': ['4.a1.a1', '4.b1.a1', '4.c1.a1'], 'normal_contains': ['24.b1.a1', '40.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '8.a1.a1', 'projective_image': '12960.bu', 'quotient_action_image': '8.2', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '8.a1.a1', 'subgroup_fusion': None, 'weyl_group': '2592.ko'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '15.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 36, 'aut_gen_orders': [4, 12, 6], 'aut_gens': [[7265664720, 33, 123752880, 124472880, 11300400, 1481684400, 13056785280], [19814462640, 64, 31672897920, 19686960, 31618097280, 2917600560, 18547920], [28509344640, 90, 1480561200, 31658024880, 1481684400, 13056785280, 18547920], [9834860160, 96, 2917968480, 12933054000, 1481684400, 31618097280, 11300400]], 'aut_group': '5184.og', 'aut_hash': 2215668191011695968, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 5184, 'aut_permdeg': 76, 'aut_perms': [1026556821513419140336280521023689745718308630968542806015879577874289629258352542221502806493090832624560761947, 537960755947752782949279735788529542609866772938259016153712387664458286402684272750970257175865546567327740689, 1414607185222458365636390665046957328860530135430560358216175995562091162716545937694085454433662172121963706538], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 27, 1, 1], [3, 4, 2, 1], [3, 6, 1, 1], [3, 12, 1, 1], [3, 36, 2, 1], [5, 1, 4, 1], [6, 54, 1, 1], [9, 36, 4, 1], [10, 27, 4, 1], [15, 4, 8, 1], [15, 6, 4, 1], [15, 12, 4, 1], [15, 36, 8, 1], [30, 54, 4, 1], [45, 36, 16, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times S_3\\wr S_3', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 36, 'autcentquo_group': '1296.3490', 'autcentquo_hash': 3490, 'autcentquo_nilpotent': False, 'autcentquo_order': 1296, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\wr S_3', 'cc_stats': [[1, 1, 1], [2, 27, 1], [3, 4, 2], [3, 6, 1], [3, 12, 1], [3, 36, 2], [5, 1, 4], [6, 54, 1], [9, 36, 4], [10, 27, 4], [15, 4, 8], [15, 6, 4], [15, 12, 4], [15, 36, 8], [30, 54, 4], [45, 36, 16]], 'center_label': '5.1', 'center_order': 5, 'central_product': True, 'central_quotient': '324.160', 'commutator_count': 1, 'commutator_label': '108.40', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 423, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['324.160', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 27, 1, 1], [3, 4, 2, 1], [3, 6, 1, 1], [3, 12, 1, 1], [3, 36, 2, 1], [5, 1, 4, 1], [6, 54, 1, 1], [9, 36, 2, 2], [10, 27, 4, 1], [15, 4, 8, 1], [15, 6, 4, 1], [15, 12, 4, 1], [15, 36, 8, 1], [30, 54, 4, 1], [45, 36, 8, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 312, 'exponent': 90, 'exponents_of_order': [4, 2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[4, 0, 24], [6, 0, 8], [12, 0, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1620.423', 'hash': 423, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 18, 'inner_gen_orders': [3, 1, 2, 2, 3, 3, 3], 'inner_gens': [[7265664720, 33, 124472880, 1139040, 13056785280, 11300400, 1481684400], [7265664720, 33, 123752880, 124472880, 11300400, 1481684400, 13056785280], [7386100560, 33, 123752880, 124472880, 18547920, 1481684400, 31618097280], [7266435840, 33, 123752880, 124472880, 11300400, 2917600560, 31618097280], [8754238800, 33, 138625920, 124472880, 11300400, 1481684400, 13056785280], [28501321680, 33, 123752880, 3041705520, 11300400, 1481684400, 13056785280], [45070830720, 33, 31658024880, 31658744880, 11300400, 1481684400, 13056785280]], 'inner_hash': 160, 'inner_nilpotent': False, 'inner_order': 324, 'inner_split': True, 'inner_tex': 'C_3^3:A_4', 'inner_used': [1, 3, 5], 'irrC_degree': 4, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 8, 'irrep_stats': [[1, 15], [3, 5], [4, 30], [6, 10], [12, 5]], 'label': '1620.423', 'linC_count': 24, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 2, 'linQ_dim': 10, 'linQ_dim_count': 2, 'linR_count': 24, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C5*C3^3:A4', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 16, 'number_characteristic_subgroups': 8, 'number_conjugacy_classes': 65, 'number_divisions': 18, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 56, 'number_subgroup_classes': 64, 'number_subgroups': 680, 'old_label': None, 'order': 1620, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 27], [3, 98], [5, 4], [6, 54], [9, 144], [10, 108], [15, 392], [30, 216], [45, 576]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[18840850560, 33, 123752880, 1139040, 13056785280, 1481684400, 11300400], [7265664720, 33, 123752880, 124472880, 18547920, 2917600560, 31618097280], [7265664720, 64, 123752880, 124472880, 18547920, 2917600560, 31618097280]], 'outer_group': '16.10', 'outer_hash': 10, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [120, 5040, 9], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2\\times C_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [3, 5], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [3, 1], [4, 1], [6, 2], [8, 4], [12, 2], [24, 2], [32, 3], [48, 1]], 'representations': {'PC': {'code': 3552120128390912526718033353088286088139476897879, 'gens': [1, 2, 3, 5, 6], 'pres': [7, -3, -2, 2, -3, 3, 3, -5, 295, 11405, 219, 58, 30243, 234, 844, 851, 438, 30245, 8335, 166]}, 'Perm': {'d': 14, 'gens': [7265664720, 33, 123752880, 124472880, 11300400, 1481684400, 13056785280]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [15], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_5\\times C_3^3:A_4', 'transitive_degree': 45, '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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 180, 'aut_gen_orders': [12, 20, 6], 'aut_gens': [[6786702755, 1089855376], [51783822838, 9160624187], [21758648518, 650770627], [53427558995, 639767536]], 'aut_group': '25920.bd', 'aut_hash': 6704104401456500954, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 25920, 'aut_permdeg': 99, 'aut_perms': [69136137250280612481572958485813197444312980330054196134537437936790772265862786946454241334978912393207347723492985328558664641845989337302672633007667115, 834484672098030940939833005286140257551159019105742219667204013240578110282186700508376914914310125915418337869234919861132376413674318824075142198333235208, 481839283351971465731886625543464632026973670077711098839541402515251951151992390418314915174333486587782164740791703766190543683821628956997829304681752014], 'aut_phi_ratio': 7.5, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [2, 27, 1, 1], [2, 54, 1, 1], [2, 135, 1, 1], [2, 270, 1, 1], [3, 6, 1, 1], [3, 8, 1, 1], [3, 12, 1, 1], [3, 72, 1, 1], [4, 45, 1, 2], [4, 54, 1, 1], [4, 90, 1, 2], [4, 135, 1, 2], [4, 270, 1, 1], [4, 810, 1, 2], [5, 4, 1, 1], [6, 30, 1, 1], [6, 40, 1, 1], [6, 54, 1, 1], [6, 60, 1, 1], [6, 108, 1, 1], [6, 270, 1, 1], [6, 360, 1, 1], [6, 540, 1, 1], [9, 144, 1, 1], [10, 108, 1, 1], [10, 216, 1, 1], [12, 108, 1, 1], [12, 180, 1, 8], [12, 360, 1, 2], [12, 540, 1, 1], [12, 1080, 1, 2], [15, 24, 1, 1], [15, 32, 1, 1], [15, 48, 1, 1], [15, 288, 1, 1], [18, 720, 1, 1], [20, 216, 1, 1], [30, 216, 1, 1], [30, 432, 1, 1], [45, 288, 2, 1], [60, 216, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_5\\times S_3\\wr S_3', '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': 180, 'autcentquo_group': '25920.bd', 'autcentquo_hash': 6704104401456500954, 'autcentquo_nilpotent': False, 'autcentquo_order': 25920, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_5\\times S_3\\wr S_3', 'cc_stats': [[1, 1, 1], [2, 5, 1], [2, 27, 1], [2, 54, 1], [2, 135, 1], [2, 270, 1], [3, 6, 1], [3, 8, 1], [3, 12, 1], [3, 72, 1], [4, 45, 2], [4, 54, 1], [4, 90, 2], [4, 135, 2], [4, 270, 1], [4, 810, 2], [5, 4, 1], [6, 30, 1], [6, 40, 1], [6, 54, 1], [6, 60, 1], [6, 108, 1], [6, 270, 1], [6, 360, 1], [6, 540, 1], [9, 144, 1], [10, 108, 1], [10, 216, 1], [12, 108, 1], [12, 180, 8], [12, 360, 2], [12, 540, 1], [12, 1080, 2], [15, 24, 1], [15, 32, 1], [15, 48, 1], [15, 288, 1], [18, 720, 1], [20, 216, 1], [30, 216, 1], [30, 432, 1], [45, 288, 2], [60, 216, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '12960.bu', 'commutator_count': 1, 'commutator_label': '1620.423', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '5.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 47, 'cyclic': False, 'derived_length': 4, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 5, 1, 1], [2, 27, 1, 1], [2, 54, 1, 1], [2, 135, 1, 1], [2, 270, 1, 1], [3, 6, 1, 1], [3, 8, 1, 1], [3, 12, 1, 1], [3, 72, 1, 1], [4, 45, 2, 1], [4, 54, 1, 1], [4, 90, 2, 1], [4, 135, 2, 1], [4, 270, 1, 1], [4, 810, 2, 1], [5, 4, 1, 1], [6, 30, 1, 1], [6, 40, 1, 1], [6, 54, 1, 1], [6, 60, 1, 1], [6, 108, 1, 1], [6, 270, 1, 1], [6, 360, 1, 1], [6, 540, 1, 1], [9, 144, 1, 1], [10, 108, 1, 1], [10, 216, 1, 1], [12, 108, 1, 1], [12, 180, 2, 4], [12, 360, 2, 1], [12, 540, 1, 1], [12, 1080, 2, 1], [15, 24, 1, 1], [15, 32, 1, 1], [15, 48, 1, 1], [15, 288, 1, 1], [18, 720, 1, 1], [20, 216, 1, 1], [30, 216, 1, 1], [30, 432, 1, 1], [45, 288, 2, 1], [60, 216, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 936, 'exponent': 180, 'exponents_of_order': [5, 4, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[24, 1, 4], [32, 1, 3], [48, 1, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '12960.bu', 'hash': 8789333695697293874, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 180, 'inner_gen_orders': [12, 6], 'inner_gens': [[6786702755, 19165230107], [20290194079, 1089855376]], 'inner_hash': 8789333695697293874, 'inner_nilpotent': False, 'inner_order': 12960, 'inner_split': True, 'inner_tex': 'C_3^3:(F_5\\times S_4)', 'inner_used': [1, 2], 'irrC_degree': 24, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 8], [2, 4], [3, 8], [4, 2], [6, 8], [8, 5], [12, 12], [16, 2], [24, 4], [32, 3], [48, 2]], 'label': '12960.bu', 'linC_count': 16, 'linC_degree': 10, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 8, 'linQ_dim': 10, 'linQ_dim_count': 8, 'linR_count': 8, 'linR_degree': 10, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C3^3:(F5*S4)', 'ngens': 2, '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, 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': 56, 'number_characteristic_subgroups': 21, 'number_conjugacy_classes': 58, 'number_divisions': 46, 'number_normal_subgroups': 21, 'number_subgroup_autclasses': 604, 'number_subgroup_classes': 604, 'number_subgroups': 39576, 'old_label': None, 'order': 12960, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 491], [3, 98], [4, 2484], [5, 4], [6, 1462], [9, 144], [10, 324], [12, 4968], [15, 392], [18, 720], [20, 216], [30, 648], [45, 576], [60, 432]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[6914079395, 3554046736]], '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': 6, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [3, 4], [4, 3], [6, 6], [8, 3], [12, 10], [16, 3], [24, 4], [32, 1], [48, 3], [64, 1]], 'representations': {'PC': {'code': '16500429729639889220832731541831533987278594701809027114276306356327820339233178969965035922901911054525563653951813149773448576349263095068389063894892861597890384661', 'gens': [1, 2, 5, 7, 9, 10], 'pres': [10, 2, 2, 2, 3, 2, 5, 2, 3, 3, 3, 52800, 63081, 51, 301562, 82, 91843, 274804, 274214, 82224, 55684, 144, 11525, 2895, 2905, 92406, 121816, 65126, 93486, 16146, 206, 691207, 57617, 172827, 28837, 3247, 259208, 388818, 21628, 48638, 1148, 96009, 24019, 36029, 12039, 36049, 3669]}, 'Perm': {'d': 14, 'gens': [6786702755, 1089855376]}}, 'schur_multiplier': [2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^3:(F_5\\times S_4)', 'transitive_degree': 45, 'wreath_data': None, 'wreath_product': False}
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 3], [5, 3]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [1, 22], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [4, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '8.3', 'autcent_hash': 3, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'D_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], [2, 1, 3], [4, 1, 4]], 'center_label': '8.2', 'center_order': 8, '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', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [4, 1, 2, 2]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 8]], 'label': '8.2', 'linC_count': 12, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 4, 'linQ_dim': 3, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2*C4', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 8, 'number_divisions': 6, 'number_normal_subgroups': 8, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 8, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 3], [4, 4]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 4], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 3], [5, 3]], 'outer_group': '8.3', 'outer_hash': 3, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 4, 'outer_perms': [1, 22], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 6, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2]], 'representations': {'PC': {'code': 34, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 16]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [3362, 16426]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [251, 504]}, 'Perm': {'d': 6, 'gens': [22, 120, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_4', 'transitive_degree': 8, 'wreath_data': None, 'wreath_product': False}
