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gps_subgroup_search • Show schema
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{'Agroup': True, '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': 160, 'characteristic': True, 'core_order': 12, 'counter': 1678, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '3840.fe.320.A', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '320.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '320.1607', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 1607, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 320, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^4:F_5', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '12.3', 'subgroup_hash': 3, 'subgroup_order': 12, 'subgroup_tex': 'A_4', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '3840.fe', 'aut_centralizer_order': 5120, 'aut_label': '320.A', 'aut_quo_index': 12, 'aut_stab_index': 1, 'aut_weyl_group': '24.12', 'aut_weyl_index': 5120, 'centralizer': '24.B', 'complements': ['12.BU', '12.O', '12.BI', '12.N'], 'conjugacy_class_count': 1, 'contained_in': ['64.A', '160.A', '160.B', '160.C', '160.BF', '160.BI', '160.BJ', '160.BK', '160.BL', '160.BM', '160.BN'], 'contains': ['960.C', '1280.a1'], 'core': '320.A', 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [362881, 363000, 1624912], 'label': '3840.fe.320.A', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '320.A', 'normal_contained_in': ['64.A', '160.A', '160.B', '160.C'], 'normal_contains': ['960.C'], 'normalizer': '1.a1', 'old_label': '320.a1', 'projective_image': '3840.fe', 'quotient_action_image': '4.2', 'quotient_action_kernel': '80.34', 'quotient_action_kernel_order': 80, 'quotient_fusion': None, 'short_label': '320.A', 'subgroup_fusion': None, 'weyl_group': '24.12'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '3.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 3, 2, 2], 'aut_gens': [[4, 16, 7], [11, 16, 23], [19, 23, 16], [8, 16, 7], [15, 16, 7]], 'aut_group': '24.12', 'aut_hash': 12, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 24, 'aut_permdeg': 4, 'aut_perms': [2, 4, 16, 7], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 4, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_4', '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': 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, 3, 1], [3, 4, 2]], 'center_label': '1.1', 'center_order': 1, 'central_product': False, 'central_quotient': '12.3', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 3, 1, 1], [3, 4, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 4, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[3, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '12.3', 'hash': 3, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [3, 2, 2], 'inner_gens': [[4, 7, 23], [19, 16, 7], [15, 16, 7]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'A_4', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 3, 'irrQ_dim': 3, 'irrR_degree': 3, 'irrep_stats': [[1, 3], [3, 1]], 'label': '12.3', 'linC_count': 1, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 3, 'linQ_degree_count': 1, 'linQ_dim': 3, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 3, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'A4', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 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': 5, 'number_subgroup_classes': 5, 'number_subgroups': 10, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 3], [3, 8]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[3, 7, 16]], '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': 3, 'perfect': False, 'permutation_degree': 4, 'pgroup': 0, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [3, 1]], 'representations': {'PC': {'code': 9572, 'gens': [1, 2, 3], 'pres': [3, -3, -2, 2, 55, 29]}, 'GLZ': {'b': 3, 'd': 3, 'gens': [4828, 4708]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [31, 21], 'family': 'PSL'}, {'d': 2, 'q': 3, 'gens': [362, 766], 'family': 'PSU'}, {'d': 3, 'q': 3, 'gens': [10315, 1638], 'family': 'Omega'}, {'d': 3, 'q': 3, 'gens': [10315, 1638], 'family': 'POmega'}, {'d': 2, 'q': 3, 'gens': [31, 21], 'family': 'PSigmaL'}, {'d': 1, 'q': 4, 'gens': [3, 7], 'family': 'AGL'}], 'GLFp': {'d': 3, 'p': 2, 'gens': [458, 314, 124]}, 'Perm': {'d': 4, 'gens': [4, 16, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'A_4', 'transitive_degree': 4, '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': 4, 'aut_exponent': 120, 'aut_gen_orders': [2, 4, 2, 5, 4, 2, 2, 2, 2, 2, 2], 'aut_gens': [[8201, 72009, 8267, 88211, 12121], [152219, 88011, 72143, 88211, 12281], [8201, 72009, 8267, 88211, 12041], [88011, 152219, 88277, 88211, 12281], [8201, 72009, 8187, 88211, 12121], [8201, 72009, 76153, 88211, 92091], [8201, 152219, 88277, 88211, 12281], [8201, 72009, 8267, 88211, 12081], [8201, 72009, 88277, 88211, 12281], [8201, 72009, 88277, 88211, 92091], [8201, 72209, 88077, 88211, 12081], [8201, 72009, 8067, 88211, 12281]], 'aut_group': '61440.em', 'aut_hash': 5907412864111550192, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 61440, 'aut_permdeg': 80, 'aut_perms': [49025350073374190825637726405524738786552308585670542513475211880934988346775377293147632898065047487493462600875469946, 60119581156028890999581315414004249444827387058370037492189737494596582400038799275715707027045981928158509088062026201, 18123927055978385163063809491932357058416511671447136907776662850404004780590239419292927236130227217638445601561667591, 67345060822156377175898025373876979689528025287788974054346393673347593106710029978837203027173761825957728107354060313, 36882893708014204317770185246418867376989588852428523381580234880407469869021505020821924512279709675571547158650089265, 58880233772761448599323437482682060313122578461128561271066647670117146434371160747044816388730763820297134578723954497, 56586992218535762344068502313280633982018660403982890442720423052057808113398275761780704532682378148920457326782915897, 18610942809485206206483866411288087231142932412090157893951092716973154915322893535018256346356593846898117109856444952, 58393141507497903767986708670244997705040297750655949393584958733813788686623438231619873530927563821238767732021137865, 16821851566420700669441589025079341304268486135700155355531620029888210640808446913820803825140542420671528256129638598, 56587137401636791244841022468849926715725592744305928270457974203278043670559215025380910084678440820234978881766837743], 'aut_phi_ratio': 480.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 6, 1], [2, 2, 4, 1], [2, 5, 2, 1], [2, 5, 6, 1], [2, 10, 4, 1], [4, 10, 8, 2], [5, 4, 1, 1], [10, 4, 1, 1], [10, 4, 6, 1], [10, 4, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'F_5\\times C_2^6:(C_2\\times S_4)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 12, 'autcent_group': '1536.408640869', 'autcent_hash': 4599629986281706675, 'autcent_nilpotent': False, 'autcent_order': 1536, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^6:S_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, 7], [2, 2, 4], [2, 5, 8], [2, 10, 4], [4, 10, 16], [5, 4, 1], [10, 4, 15]], 'center_label': '8.5', 'center_order': 8, 'central_product': True, 'central_quotient': '40.12', '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', '2.1', '5.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 1607, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 2], ['80.34', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 7], [2, 2, 1, 4], [2, 5, 1, 8], [2, 10, 1, 4], [4, 10, 2, 8], [5, 4, 1, 1], [10, 4, 1, 7], [10, 4, 2, 4]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 52080, 'exponent': 20, 'exponents_of_order': [6, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '160.236', 'hash': 1607, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 20, 'inner_gen_orders': [1, 1, 4, 1, 10], 'inner_gens': [[8201, 72009, 8267, 88211, 12121], [8201, 72009, 8267, 88211, 12121], [8201, 72009, 8267, 88211, 92171], [8201, 72009, 8267, 88211, 12121], [8201, 72009, 88197, 88211, 12121]], 'inner_hash': 12, 'inner_nilpotent': False, 'inner_order': 40, 'inner_split': False, 'inner_tex': 'C_2\\times F_5', 'inner_used': [3, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 8], [4, 16]], 'label': '320.1607', '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': False, 'monomial': True, 'name': 'C2^4:F5', 'ngens': 7, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 13, 'number_characteristic_subgroups': 13, 'number_conjugacy_classes': 56, 'number_divisions': 44, 'number_normal_subgroups': 196, 'number_subgroup_autclasses': 94, 'number_subgroup_classes': 674, 'number_subgroups': 2762, 'old_label': None, 'order': 320, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 95], [4, 160], [5, 4], [10, 60]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 12, 'outer_gen_orders': [2, 3, 2, 2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [8001, 8001, 8001, 8001, 8001, 92011, 92011, 8001, 8001, 8001], 'outer_gens': [[72209, 72009, 76153, 88211, 12121], [72009, 72209, 8067, 88211, 12321], [8201, 72009, 76153, 88211, 12121], [8201, 152219, 8267, 88211, 92331], [88011, 72009, 8267, 88211, 12121], [88011, 152219, 72343, 88211, 76089], [88011, 72009, 8067, 88211, 156099], [8201, 72009, 8267, 88211, 92331], [8201, 72009, 8067, 88211, 12121], [8201, 72009, 72343, 88211, 12121]], 'outer_group': '1536.408632661', 'outer_hash': 6321312703651895429, 'outer_nilpotent': False, 'outer_order': 1536, 'outer_permdeg': 16, 'outer_perms': [19200362299, 349272367736, 4215700702080, 2966098055470, 1327353718609, 6719474385938, 1326874376551, 18498, 2966098020480, 1327353713280], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': 'C_2^6:S_4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 13, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 4], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 16], [4, 8], [8, 4]], 'representations': {'PC': {'code': 22228473201735780442636291, 'gens': [1, 2, 3, 5, 6], 'pres': [7, -2, -2, -2, -2, -2, 2, -5, 58, 1195, 1538, 124, 2372, 1595]}, 'GLZN': {'d': 2, 'p': 20, 'gens': [8009, 8201, 72009, 8023, 8081, 152019, 12201]}, 'Perm': {'d': 13, 'gens': [522588249, 90720, 131760, 83553120, 1037836816, 1037836800, 34]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 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_2^4:F_5', 'transitive_degree': 80, 'wreath_data': None, 'wreath_product': False}
