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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, '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': 960, 'characteristic': True, 'core_order': 5, 'counter': 1971, 'cyclic': True, 'direct': False, 'hall': 5, 'label': '3840.fe.768.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': True, 'minimal_normal': True, 'nilpotent': True, 'normal': True, 'old_label': '768.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '768.1088676', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 1088676, 'quotient_metabelian': False, 'quotient_nilpotent': False, 'quotient_order': 768, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': False, 'quotient_tex': 'C_2^6.D_6', 'simple': True, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '5.1', 'subgroup_hash': 1, 'subgroup_order': 5, 'subgroup_tex': 'C_5', 'supersolvable': True, 'sylow': 5}
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gps_subgroup_data • Show schema
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{'ambient': '3840.fe', 'aut_centralizer_order': 30720, 'aut_label': '768.a1', 'aut_quo_index': 16, 'aut_stab_index': 1, 'aut_weyl_group': '4.1', 'aut_weyl_index': 30720, 'centralizer': '4.B', 'complements': ['5.a1'], 'conjugacy_class_count': 1, 'contained_in': ['256.a1', '384.A', '384.B', '384.C', '384.D', '384.E', '384.F', '384.G', '384.H', '384.I', '384.J', '384.K', '384.L', '384.M', '384.N', '384.O', '384.P', '384.Q'], 'contains': ['3840.a1'], 'core': '768.a1', 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [367873228800], 'label': '3840.fe.768.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '768.a1', 'normal_contained_in': ['192.G', '384.A', '384.B', '384.C', '384.D'], 'normal_contains': ['3840.a1'], 'normalizer': '1.a1', 'old_label': '768.a1', 'projective_image': '3840.fe', 'quotient_action_image': '4.1', 'quotient_action_kernel': '192.1537', 'quotient_action_kernel_order': 192, 'quotient_fusion': None, 'short_label': '768.a1', 'subgroup_fusion': None, 'weyl_group': '4.1'}
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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}
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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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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': 24, 'aut_gen_orders': [4, 12, 4, 4, 4, 24], 'aut_gens': [[2176547, 3264371, 3794839, 167041, 1937281, 112145, 2821673], [4165637, 4805029, 3451258, 167041, 2820521, 4535065, 1881985], [1512989, 1445779, 3851311, 2765977, 1881985, 2766377, 2820521], [4829195, 1321349, 797050, 167041, 2820521, 4535065, 1881985], [3244813, 2156989, 2911295, 167041, 1937281, 112145, 2821673], [2176547, 228830, 5219570, 2765977, 4536217, 112145, 2821673], [1511453, 4307786, 2911271, 2820121, 2599687, 3429935, 774551]], 'aut_group': None, 'aut_hash': 682310395033496420, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 98304, 'aut_permdeg': 64, 'aut_perms': [55110440814334738969076067855481968696669171055817734245311353836060435546352983233277608, 118700977202621996963899352035697361684453325604884700575024847079368727030534923453852449, 22014780585350603763932422423397142560056930376663947779681482597695347881141146122643621, 18984299858701996828776199581465716293750222136871005372062322851912330501075141275592250, 15895907316606224007474221313789797023637594549424709827275545554626069507297564239446231, 17039630563487061638446795172603900375748068920931195047239217133909061280381609431719581], 'aut_phi_ratio': 384.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 1, 4, 1], [2, 2, 4, 1], [2, 3, 1, 2], [2, 3, 2, 1], [2, 3, 4, 1], [2, 6, 4, 1], [2, 6, 8, 1], [2, 12, 4, 1], [3, 8, 1, 1], [4, 2, 8, 1], [4, 6, 8, 2], [4, 12, 4, 1], [4, 12, 8, 2], [6, 8, 1, 1], [6, 8, 2, 1], [6, 8, 4, 1], [6, 16, 4, 1], [12, 16, 8, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_2^4\\times A_4).C_2^6.C_2^3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': None, 'autcent_hash': 8092891664598236742, 'autcent_nilpotent': True, 'autcent_order': 2048, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^7.C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '48.48', 'autcentquo_hash': 48, 'autcentquo_nilpotent': False, 'autcentquo_order': 48, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2\\times S_4', 'cc_stats': [[1, 1, 1], [2, 1, 7], [2, 2, 4], [2, 3, 8], [2, 6, 12], [2, 12, 4], [3, 8, 1], [4, 2, 8], [4, 6, 16], [4, 12, 20], [6, 8, 7], [6, 16, 4], [12, 16, 8]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '96.226', 'commutator_count': 1, 'commutator_label': '24.13', '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'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 1088676, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['16.3', 1], ['2.1', 1], ['24.12', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 2, 1, 4], [2, 3, 1, 8], [2, 6, 1, 12], [2, 12, 1, 4], [3, 8, 1, 1], [4, 2, 2, 4], [4, 6, 1, 8], [4, 6, 2, 4], [4, 12, 1, 4], [4, 12, 2, 8], [6, 8, 1, 7], [6, 16, 1, 4], [12, 16, 2, 4]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 1031940, 'exponent': 12, 'exponents_of_order': [8, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '192.1537', 'hash': 1088676, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 3, 2, 2, 2, 2], 'inner_gens': [[2176547, 3264371, 3794839, 167041, 166289, 1880833, 4590361], [2176547, 3264371, 852370, 2820121, 4590361, 112145, 166289], [2176547, 616331, 3794839, 2765977, 4536217, 112145, 166289], [2176547, 610139, 3851287, 167041, 1937281, 112145, 2821673], [3908371, 610139, 3851287, 167041, 1937281, 112145, 2821673], [3908371, 3264371, 3794839, 167041, 1937281, 112145, 2821673], [3908371, 610139, 1195951, 167041, 1937281, 112145, 2821673]], 'inner_hash': 226, 'inner_nilpotent': False, 'inner_order': 96, 'inner_split': False, 'inner_tex': 'C_2^2\\times S_4', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 24], [3, 32], [4, 4], [6, 8]], 'label': '768.1088676', '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': 'C2^6.D6', 'ngens': 9, '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': 24, 'number_characteristic_subgroups': 23, 'number_conjugacy_classes': 100, 'number_divisions': 80, 'number_normal_subgroups': 245, 'number_subgroup_autclasses': 625, 'number_subgroup_classes': 4333, 'number_subgroups': 24370, 'old_label': None, 'order': 768, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 159], [3, 8], [4, 352], [6, 120], [12, 128]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 4, 2, 4, 2, 4, 2], 'outer_gen_pows': [110593, 110593, 110593, 110593, 110593, 110593, 110593, 110593, 110593], 'outer_gens': [[1254187, 1494883, 3794839, 167041, 1937281, 112145, 2821673], [1254187, 4147963, 3794839, 167041, 3484079, 5198623, 2599687], [1254187, 608987, 3794839, 167041, 2820521, 4535065, 1936129], [1254187, 3264371, 312383, 167041, 2600839, 775703, 3485231], [2176547, 4147963, 3794839, 167041, 1937281, 112145, 2821673], [3907603, 4147963, 255911, 167041, 3484079, 5198623, 2599687], [4167173, 3264371, 3794839, 167041, 1937281, 112145, 2821673], [2176547, 1273397, 255911, 167041, 1937281, 112145, 2821673], [2176547, 3926477, 3794839, 167041, 1937281, 112145, 2821673]], 'outer_group': '1024.djz', 'outer_hash': 3485275617416775616, 'outer_nilpotent': True, 'outer_order': 1024, 'outer_permdeg': 16, 'outer_perms': [1321086781009, 5381, 998282935, 5411760077117, 1320607416529, 8388882069037, 1321086797501, 2884555267996, 1307674752701], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^6.C_2^4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 14, '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, 8], [6, 16]], 'representations': {'PC': {'code': '1376816049273714397685297681486812746537707285327052095361552', 'gens': [1, 3, 4, 6, 7, 8, 9], 'pres': [9, -2, -2, -2, -2, -3, -2, 2, 2, 2, 18, 381, 102, 382, 7799, 4244, 1985, 15126, 11364, 5325, 2877, 10375, 58328, 4886, 2465, 2231]}, 'GLZN': {'d': 2, 'p': 48, 'gens': [2764825, 2800678, 553715, 111745, 774151, 2821273, 2176547, 4534313, 1169270]}, 'Perm': {'d': 14, 'gens': [6267300485, 362880, 6266939160, 13452330360, 6266949120, 11520, 3, 7, 16]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 4], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^6.D_6', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
