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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '3840.fe', 'ambient_counter': 135, 'ambient_order': 3840, 'ambient_tex': 'C_2^3:F_5\\times S_4', 'central': False, 'central_factor': False, 'centralizer_order': 40, 'characteristic': False, 'core_order': 480, 'counter': 50, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '3840.fe.8.J', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '8.j1', 'outer_equivalence': True, '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': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '480.1198', 'subgroup_hash': 1198, 'subgroup_order': 480, 'subgroup_tex': 'C_2\\times C_{10}\\times S_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '3840.fe', 'aut_centralizer_order': 80, 'aut_label': '8.J', 'aut_quo_index': 4, 'aut_stab_index': 4, 'aut_weyl_group': '384.20048', 'aut_weyl_index': 320, 'centralizer': '96.A', 'complements': ['480.DZ', '480.IN', '480.EA', '480.IO', '480.IZ', '480.IY', '480.IT', '480.EM', '480.IP', '480.EB'], 'conjugacy_class_count': 4, 'contained_in': ['4.B', '4.J', '4.M'], 'contains': ['16.B', '16.E', '16.G', '16.BL', '24.CW', '32.P', '40.D'], 'core': '8.J', 'coset_action_label': None, 'count': 4, 'diagramx': None, 'generators': [362881, 367873228800, 363000, 274987203241, 274988460112, 5046, 367876471265], 'label': '3840.fe.8.J', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '8.J', 'normal_contained_in': ['4.B', '4.J', '4.M'], 'normal_contains': ['16.B', '16.E', '16.G', '40.D'], 'normalizer': '1.a1', 'old_label': '8.j1', 'projective_image': '1920.240396', 'quotient_action_image': '4.1', 'quotient_action_kernel': '2.1', 'quotient_action_kernel_order': 2, 'quotient_fusion': None, 'short_label': '8.J', 'subgroup_fusion': None, 'weyl_group': '96.186'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '40.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [6, 4, 12], 'aut_gens': [[8111, 114119, 12201, 72009, 88131], [72119, 14116, 92011, 152019, 88131], [94001, 74304, 12201, 88011, 92251], [72319, 78314, 92011, 88011, 88371]], 'aut_group': '2304.jx', 'aut_hash': 7598123446365502787, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2304, 'aut_permdeg': 28, 'aut_perms': [42292377140988702604274396159, 132760386500227288137377290107, 191582298157344838943256642099], 'aut_phi_ratio': 18.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 3, 1, 1], [2, 3, 3, 1], [2, 6, 4, 1], [3, 8, 1, 1], [4, 6, 4, 1], [5, 1, 4, 1], [6, 8, 3, 1], [10, 1, 12, 1], [10, 3, 4, 1], [10, 3, 12, 1], [10, 6, 16, 1], [15, 8, 4, 1], [20, 6, 16, 1], [30, 8, 12, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_4\\times S_4^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '96.186', 'autcent_hash': 186, 'autcent_nilpotent': False, 'autcent_order': 96, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_4\\times S_4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 3, 4], [2, 6, 4], [3, 8, 1], [4, 6, 4], [5, 1, 4], [6, 8, 3], [10, 1, 12], [10, 3, 16], [10, 6, 16], [15, 8, 4], [20, 6, 16], [30, 8, 12]], 'center_label': '20.5', 'center_order': 20, 'central_product': True, 'central_quotient': '24.12', 'commutator_count': 1, 'commutator_label': '12.3', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1198, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['24.12', 1], ['5.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 1, 4], [2, 6, 1, 4], [3, 8, 1, 1], [4, 6, 1, 4], [5, 1, 4, 1], [6, 8, 1, 3], [10, 1, 4, 3], [10, 3, 4, 4], [10, 6, 4, 4], [15, 8, 4, 1], [20, 6, 4, 4], [30, 8, 4, 3]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 13020, 'exponent': 60, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '480.1198', 'hash': 1198, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 3, 2, 1, 2], 'inner_gens': [[8111, 158314, 92011, 72009, 88131], [10011, 114119, 88211, 72009, 92331], [88101, 118319, 12201, 72009, 88131], [8111, 114119, 12201, 72009, 88131], [8111, 34309, 12201, 72009, 88131]], 'inner_hash': 12, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': True, 'inner_tex': 'S_4', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 40], [2, 20], [3, 40]], 'label': '480.1198', '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': False, 'metacyclic': False, 'monomial': True, 'name': 'C2*C10*S4', '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': 16, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 100, 'number_divisions': 40, 'number_normal_subgroups': 52, 'number_subgroup_autclasses': 84, 'number_subgroup_classes': 262, 'number_subgroups': 840, 'old_label': None, 'order': 480, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 39], [3, 8], [4, 24], [5, 4], [6, 24], [10, 156], [15, 32], [20, 96], [30, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 2, 4, 4, 2, 6], 'outer_gen_pows': [8001, 8001, 8001, 8001, 8001, 8001], 'outer_gens': [[8111, 114119, 12201, 72009, 88291], [72119, 114119, 12201, 72009, 88291], [88301, 114119, 12201, 72009, 88051], [8111, 114119, 12201, 72009, 88371], [8111, 118309, 12201, 152019, 88291], [8111, 118309, 12201, 88011, 88291]], 'outer_group': '96.186', 'outer_hash': 186, 'outer_nilpotent': False, 'outer_order': 96, 'outer_permdeg': 8, 'outer_perms': [16, 16696, 11529, 18, 10800, 5776], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_4\\times S_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 5], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 4], [3, 8], [4, 8], [8, 4], [12, 8]], 'representations': {'PC': {'code': 167401518874537311523096749856555267, 'gens': [1, 2, 4, 5, 6], 'pres': [7, -2, -2, -3, -2, 2, 2, -5, 141, 36, 170, 7059, 3370, 1781, 1272, 3169, 124]}, 'GLZN': {'d': 2, 'p': 20, 'gens': [8201, 92011, 72009, 8081, 130311, 130106, 12201]}, 'Perm': {'d': 13, 'gens': [39916800, 7, 16, 46200, 43545600, 482630400, 1037836800]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 10], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2\\times C_{10}\\times S_4', 'transitive_degree': 60, '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': '32.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [2, 4, 12, 4, 4, 4, 4], 'aut_gens': [[1, 355938468607, 362886, 19639451809, 5040, 5167, 5047, 281177544377, 274987198207, 274988822521], [1, 355938473641, 362886, 261534902088, 5040, 5167, 5047, 88179448343, 367873591807, 367874853121], [5166, 355938468607, 367920, 361207509534, 5040, 5167, 120, 6268930553, 174875868841, 174878743270], [362880, 6267305641, 5160, 261534901975, 6, 367927, 367926, 281177176343, 174875868841, 174875501392], [5047, 281173939206, 362887, 19639456854, 363007, 367927, 1, 355939735673, 367873228806, 367874853120], [5047, 6266942640, 5161, 261535259809, 368041, 5167, 1, 281177539223, 174875505840, 174877493167], [120, 6266942640, 127, 361207146535, 368041, 5167, 5166, 355939730639, 105859353606, 105862953905], [5166, 6266937606, 127, 361207151575, 363007, 5167, 120, 355939735552, 105859353606, 105862596070]], 'aut_group': None, 'aut_hash': 6217586336571907406, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 122880, 'aut_permdeg': 64, 'aut_perms': [44391485488040302680661689665926819842244810558103219467980264652084765340760102209707485, 47327110434022500045578688771510961093294273863991824978495792264984498376394756623113072, 94765130617761165152456020907633026175284793391318681449674968508276886126758368254390656, 3328990630340957902295040764340901341339460562767598521943068367155029838188150988906234, 85315007539601126376202475874172034014856133863059332223440490062275284644105866506108575, 84941871540829582757462117064721940898460206615184614527664149294237035830539702919831288, 47310058857907721996006917288001530206552294734858393630059066450632039134410689695407114], 'aut_phi_ratio': 120.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 3, 1, 2], [2, 3, 2, 1], [2, 5, 2, 2], [2, 6, 2, 1], [2, 6, 4, 1], [2, 10, 2, 1], [2, 12, 2, 1], [2, 15, 2, 2], [2, 30, 2, 1], [2, 30, 4, 1], [2, 60, 2, 1], [3, 8, 1, 1], [4, 6, 4, 1], [4, 10, 4, 2], [4, 12, 2, 1], [4, 30, 4, 3], [4, 60, 2, 1], [4, 60, 4, 4], [5, 4, 1, 1], [6, 8, 1, 1], [6, 8, 2, 1], [6, 16, 2, 1], [6, 40, 2, 2], [6, 80, 2, 1], [10, 4, 1, 1], [10, 4, 2, 1], [10, 4, 4, 1], [10, 12, 1, 2], [10, 12, 2, 1], [10, 12, 4, 1], [10, 24, 4, 2], [12, 80, 4, 2], [15, 32, 1, 1], [20, 24, 4, 2], [30, 32, 1, 1], [30, 32, 2, 1], [30, 32, 4, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_5\\times A_4).C_2^6.C_2^5', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '128.1578', 'autcent_hash': 1578, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '960.11361', 'autcentquo_hash': 11361, 'autcentquo_nilpotent': False, 'autcentquo_order': 960, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2\\times F_5\\times S_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 3, 4], [2, 5, 4], [2, 6, 6], [2, 10, 2], [2, 12, 2], [2, 15, 4], [2, 30, 6], [2, 60, 2], [3, 8, 1], [4, 6, 4], [4, 10, 8], [4, 12, 2], [4, 30, 12], [4, 60, 18], [5, 4, 1], [6, 8, 3], [6, 16, 2], [6, 40, 4], [6, 80, 2], [10, 4, 7], [10, 12, 8], [10, 24, 8], [12, 80, 8], [15, 32, 1], [20, 24, 8], [30, 32, 7]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '960.11361', 'commutator_count': 1, 'commutator_label': '120.43', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '5.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 135, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['24.12', 1], ['80.34', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 3, 1, 4], [2, 5, 1, 4], [2, 6, 1, 6], [2, 10, 1, 2], [2, 12, 1, 2], [2, 15, 1, 4], [2, 30, 1, 6], [2, 60, 1, 2], [3, 8, 1, 1], [4, 6, 1, 4], [4, 10, 2, 4], [4, 12, 1, 2], [4, 30, 1, 4], [4, 30, 2, 4], [4, 60, 1, 2], [4, 60, 2, 8], [5, 4, 1, 1], [6, 8, 1, 3], [6, 16, 1, 2], [6, 40, 1, 4], [6, 80, 1, 2], [10, 4, 1, 3], [10, 4, 2, 2], [10, 12, 1, 4], [10, 12, 2, 2], [10, 24, 1, 4], [10, 24, 2, 2], [12, 80, 2, 4], [15, 32, 1, 1], [20, 24, 1, 4], [20, 24, 2, 2], [30, 32, 1, 3], [30, 32, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 511842240, 'exponent': 60, 'exponents_of_order': [8, 1, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '1920.240396', 'hash': 4547106905351228588, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 60, 'inner_gen_orders': [2, 2, 2, 4, 2, 2, 2, 6, 10, 10], 'inner_gens': [[1, 355938468607, 362886, 19639451809, 5040, 5167, 5047, 281177181496, 274987198207, 274988822521], [1, 355938468607, 362886, 261535264855, 5040, 5167, 5047, 88179816377, 367873591807, 367875216121], [1, 355938468607, 362886, 19639456855, 5040, 5167, 5047, 281177181377, 274987198207, 274988459521], [1, 186810992041, 367920, 19639451809, 6, 5167, 5047, 88179811222, 105859721641, 105860977921], [1, 355938468607, 362886, 19639456855, 5040, 5167, 5047, 281177544377, 274987198207, 274988822521], [1, 355938468607, 362886, 19639451809, 5040, 5167, 5047, 281177181377, 274987198207, 274988459521], [1, 355938468607, 362886, 19639451809, 5040, 5167, 5047, 281177181496, 274987198207, 274988822521], [120, 186810987007, 7, 361207151695, 5040, 368046, 5166, 281177544377, 367873591807, 367876829225], [1, 281174302207, 362886, 361207514455, 5040, 5167, 5047, 186814229177, 274987198207, 274988822521], [1, 281174302207, 126, 361207146649, 5040, 367927, 5047, 186812616953, 274987198207, 274988822521]], 'inner_hash': 11361, 'inner_nilpotent': False, 'inner_order': 960, 'inner_split': False, 'inner_tex': 'C_2\\times F_5\\times S_4', 'inner_used': [1, 2, 3, 4, 8, 10], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 24], [3, 32], [4, 20], [6, 8], [8, 8], [12, 16]], 'label': '3840.fe', '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': False, 'metacyclic': False, 'monomial': None, 'name': 'C2^3:F5*S4', 'ngens': 10, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 55, 'number_characteristic_subgroups': 58, 'number_conjugacy_classes': 140, 'number_divisions': 110, 'number_normal_subgroups': 288, 'number_subgroup_autclasses': 2090, 'number_subgroup_classes': 8666, 'number_subgroups': 136628, 'old_label': None, 'order': 3840, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 479], [3, 8], [4, 1568], [5, 4], [6, 376], [10, 316], [12, 640], [15, 32], [20, 192], [30, 224]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 4, 2, 4], 'outer_gen_pows': [0, 6266937600, 0, 0, 0], 'outer_gens': [[1, 355938105606, 127, 19639089048, 368041, 5167, 5047, 281177181376, 274986835206, 274988459520], [1, 88176574207, 362886, 19639089048, 5040, 5167, 5047, 186814229177, 174875863807, 174877125120], [1, 355938110640, 127, 19639456849, 368041, 5167, 5047, 281177176342, 274986835206, 274988822521], [1, 355938473641, 367920, 19639089048, 6, 5167, 5047, 281177539343, 274987203241, 274988827567], [5047, 355938468607, 367920, 19639456849, 5040, 5167, 1, 281177544377, 274987203241, 274988464566]], 'outer_group': '128.1135', 'outer_hash': 1135, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 12, 'outer_perms': [127463, 40764976, 40682898, 90842416, 262417805], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:D_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 4], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 24], [3, 16], [4, 16], [6, 16], [8, 8], [12, 8], [16, 2], [24, 4]], 'representations': {'PC': {'code': '541636586301990200893064802099770244214994057639815912320292674596918750264115766767858569610469060864', 'gens': [1, 3, 5, 7, 8, 9, 10], 'pres': [10, 2, 2, 2, 5, 2, 3, 2, 2, 2, 2, 20, 58442, 552, 82, 643, 653, 164004, 8524, 144, 148805, 38425, 75606, 37826, 3826, 2366, 48027, 4847, 1977, 64828, 6528, 3838, 1249, 3059]}, 'Perm': {'d': 15, 'gens': [1, 355938468607, 362886, 19639451809, 5040, 5167, 5047, 281177544377, 274987198207, 274988822521]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 4], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3:F_5\\times S_4', 'transitive_degree': 120, '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}
