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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '230400.co', 'ambient_counter': 67, 'ambient_order': 230400, 'ambient_tex': '(C_2\\times S_5^2).C_2^3', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': False, 'core_order': 57600, 'counter': 31, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '230400.co.4._.O', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '4.O', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '4.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': None, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^2', 'simple': False, 'solvable': False, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '57600.l', 'subgroup_hash': None, 'subgroup_order': 57600, 'subgroup_tex': 'S_5^2:C_2^2', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '230400.co', 'aut_centralizer_order': None, 'aut_label': None, 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': None, 'complements': None, 'conjugacy_class_count': 1, 'contained_in': None, 'contains': None, 'core': None, 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [126, 1307674368006, 11009533570854], 'label': '230400.co.4._.O', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': None, 'normal_contained_in': None, 'normal_contains': None, 'normalizer': None, 'old_label': '4.O', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '4._.O', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 120, 'aut_gen_orders': [4, 12, 6, 12], 'aut_gens': [[1, 47992326, 87949470], [1, 255247927, 240733687], [1, 200714808, 212373528], [1, 102746569, 62961529], [1, 330233454, 336082471]], 'aut_group': '115200.bv', 'aut_hash': 96265586239778315, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 115200, 'aut_permdeg': 280, 'aut_perms': 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12060020880440239355612892833294588430860074251172825352653765799164536275930441207395343170973481815395968525655446516680587395725371401276161853346076444997944525301639554146592516335817636582774174077990729986870105413428068270619046226711009337923931894522745703462846899977831594102121504237716732310755708462457526803185368607541019014481243889500509534871766520130148903541262088046070810216052765332111972364173486302747468176148947022939816783092791956002963509928483369022931524547722343333112523471183574167241827228997060241253509227375863074536376960085], 'aut_phi_ratio': 7.5, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 20, 2, 1], [2, 30, 1, 2], [2, 100, 1, 2], [2, 120, 2, 1], [2, 225, 1, 2], [2, 300, 2, 1], [3, 40, 1, 1], [3, 400, 1, 1], [4, 60, 2, 1], [4, 600, 1, 2], [4, 900, 1, 2], [4, 900, 2, 1], [4, 1200, 2, 1], [4, 1800, 2, 1], [5, 48, 1, 1], [5, 576, 1, 1], [6, 40, 1, 1], [6, 40, 2, 1], [6, 400, 1, 5], [6, 400, 2, 1], [6, 600, 1, 2], [6, 600, 2, 1], [6, 800, 2, 1], [6, 2400, 2, 1], [8, 3600, 2, 1], [10, 48, 1, 1], [10, 480, 2, 1], [10, 576, 1, 1], [10, 720, 1, 2], [10, 2880, 2, 1], [12, 1200, 1, 2], [12, 1200, 2, 1], [12, 2400, 2, 1], [15, 960, 1, 1], [20, 1440, 2, 1], [30, 960, 1, 1], [30, 960, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^2\\times S_5\\wr C_2', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 120, 'autcentquo_group': '28800.r', 'autcentquo_hash': 1736489480735889307, 'autcentquo_nilpotent': False, 'autcentquo_order': 28800, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_5\\wr C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 20, 2], [2, 30, 2], [2, 100, 2], [2, 120, 2], [2, 225, 2], [2, 300, 2], [3, 40, 1], [3, 400, 1], [4, 60, 2], [4, 600, 2], [4, 900, 4], [4, 1200, 2], [4, 1800, 2], [5, 48, 1], [5, 576, 1], [6, 40, 3], [6, 400, 7], [6, 600, 4], [6, 800, 2], [6, 2400, 2], [8, 3600, 2], [10, 48, 1], [10, 480, 2], [10, 576, 1], [10, 720, 2], [10, 2880, 2], [12, 1200, 4], [12, 2400, 2], [15, 960, 1], [20, 1440, 2], [30, 960, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '28800.r', 'commutator_count': 1, 'commutator_label': '7200.d', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '60.5', '60.5'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 12, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['28800.r', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 20, 1, 2], [2, 30, 1, 2], [2, 100, 1, 2], [2, 120, 1, 2], [2, 225, 1, 2], [2, 300, 1, 2], [3, 40, 1, 1], [3, 400, 1, 1], [4, 60, 1, 2], [4, 600, 1, 2], [4, 900, 1, 4], [4, 1200, 1, 2], [4, 1800, 1, 2], [5, 48, 1, 1], [5, 576, 1, 1], [6, 40, 1, 3], [6, 400, 1, 7], [6, 600, 1, 4], [6, 800, 1, 2], [6, 2400, 1, 2], [8, 3600, 1, 2], [10, 48, 1, 1], [10, 480, 1, 2], [10, 576, 1, 1], [10, 720, 1, 2], [10, 2880, 1, 2], [12, 1200, 1, 4], [12, 2400, 1, 2], [15, 960, 1, 1], [20, 1440, 1, 2], [30, 960, 1, 3]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 542979360, 'exponent': 120, 'exponents_of_order': [8, 2, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[8, 1, 4], [10, 1, 4], [12, 1, 2], [16, 1, 4], [25, 1, 4], [32, 1, 1], [36, 1, 2], [40, 1, 4], [48, 1, 2], [50, 1, 1], [60, 1, 2]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '57600.l', 'hash': 3328040073188890159, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 120, 'inner_gen_orders': [1, 8, 10], 'inner_gens': [[1, 47992326, 87949470], [1, 47992326, 53479680], [1, 109766454, 87949470]], 'inner_hash': 1736489480735889307, 'inner_nilpotent': False, 'inner_order': 28800, 'inner_split': True, 'inner_tex': 'S_5\\wr C_2', 'inner_used': [2, 3], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 2], [8, 8], [10, 8], [12, 4], [16, 8], [25, 8], [32, 2], [36, 4], [40, 8], [48, 4], [50, 2], [60, 4]], 'label': '57600.l', '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': False, 'name': 'S5^2:C2^2', 'ngens': 3, '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, 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': 51, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 70, 'number_divisions': 70, 'number_normal_subgroups': 21, 'number_subgroup_autclasses': 1829, 'number_subgroup_classes': 2688, 'number_subgroups': 846009, 'old_label': None, 'order': 57600, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 1591], [3, 440], [4, 10920], [5, 624], [6, 11720], [8, 7200], [10, 8784], [12, 9600], [15, 960], [20, 2880], [30, 2880]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[1, 47992326, 87949471], [1, 47992327, 87949470]], 'outer_group': '4.2', 'outer_hash': 2, 'outer_nilpotent': True, 'outer_order': 4, 'outer_permdeg': 4, 'outer_perms': [1, 6], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^2', 'pc_rank': None, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 2], [8, 8], [10, 8], [12, 4], [16, 8], [25, 8], [32, 2], [36, 4], [40, 8], [48, 4], [50, 2], [60, 4]], 'representations': {'Perm': {'d': 12, 'gens': [1, 47992326, 87949470]}}, 'schur_multiplier': [2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'S_5^2:C_2^2', 'transitive_degree': 20, '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': '16.14', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 120, 'aut_gen_orders': [4, 12, 4, 20, 12, 8, 60], 'aut_gens': [[8213651349126, 8133447536334, 8220160972926, 4552432340401, 1439044594248], [8145941897928, 7176486562560, 10828829819089, 10918416943134, 704256820248], [8302549651609, 4377795972607, 4384991157535, 8233294003488, 11775982539174], [4210558370215, 6995063658240, 4398878575135, 5704589727054, 9161033334055], [4212420710695, 11003306549761, 10755063935569, 8307852094368, 9241429120854], [4211386865400, 10916609765328, 5607359694847, 8307818265726, 1394903503128], [4311588153894, 8214405741361, 8308014624289, 8128219196448, 1838971330855], [10922684376649, 8302589245566, 8307852094494, 10755063935695, 1426558210855]], 'aut_group': None, 'aut_hash': 6281824195318411014, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 7372800, 'aut_permdeg': 1040, 'aut_perms': 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'aut_phi_ratio': 120.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 30, 1, 2], [2, 30, 2, 1], [2, 40, 2, 1], [2, 60, 2, 1], [2, 100, 2, 2], [2, 120, 8, 1], [2, 200, 2, 1], [2, 225, 1, 2], [2, 225, 2, 1], [2, 450, 2, 1], [2, 600, 2, 1], [3, 40, 1, 1], [3, 400, 1, 1], [4, 40, 2, 1], [4, 120, 2, 2], [4, 600, 2, 3], [4, 900, 2, 2], [4, 1200, 2, 1], [4, 1800, 2, 3], [4, 1800, 8, 1], [4, 2400, 4, 1], [5, 48, 1, 1], [5, 576, 1, 1], [6, 40, 1, 1], [6, 40, 2, 1], [6, 80, 2, 2], [6, 400, 1, 1], [6, 400, 2, 5], [6, 600, 1, 2], [6, 600, 2, 1], [6, 800, 2, 4], [6, 1200, 2, 2], [6, 1600, 2, 1], [6, 2400, 8, 1], [8, 7200, 4, 1], [10, 48, 1, 1], [10, 48, 2, 1], [10, 96, 2, 1], [10, 576, 1, 1], [10, 576, 2, 1], [10, 576, 4, 1], [10, 720, 1, 2], [10, 720, 2, 1], [10, 960, 2, 1], [10, 1440, 2, 1], [10, 2880, 8, 1], [12, 80, 2, 1], [12, 800, 2, 1], [12, 1200, 2, 3], [12, 1600, 2, 1], [12, 2400, 2, 3], [12, 4800, 4, 1], [15, 960, 1, 1], [20, 960, 2, 1], [20, 2880, 2, 2], [30, 960, 1, 1], [30, 960, 2, 1], [30, 1920, 2, 2], [60, 1920, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^6.C_2^2.A_5^2.D_4', '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': 120, 'autcentquo_group': '57600.l', 'autcentquo_hash': 3328040073188890159, 'autcentquo_nilpotent': False, 'autcentquo_order': 57600, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_5^2:C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 30, 4], [2, 40, 2], [2, 60, 2], [2, 100, 4], [2, 120, 8], [2, 200, 2], [2, 225, 4], [2, 450, 2], [2, 600, 2], [3, 40, 1], [3, 400, 1], [4, 40, 2], [4, 120, 4], [4, 600, 6], [4, 900, 4], [4, 1200, 2], [4, 1800, 14], [4, 2400, 4], [5, 48, 1], [5, 576, 1], [6, 40, 3], [6, 80, 4], [6, 400, 11], [6, 600, 4], [6, 800, 8], [6, 1200, 4], [6, 1600, 2], [6, 2400, 8], [8, 7200, 4], [10, 48, 3], [10, 96, 2], [10, 576, 7], [10, 720, 4], [10, 960, 2], [10, 1440, 2], [10, 2880, 8], [12, 80, 2], [12, 800, 2], [12, 1200, 6], [12, 1600, 2], [12, 2400, 6], [12, 4800, 4], [15, 960, 1], [20, 960, 2], [20, 2880, 4], [30, 960, 3], [30, 1920, 4], [60, 1920, 2]], 'center_label': '4.2', 'center_order': 4, 'central_product': None, 'central_quotient': '57600.l', 'commutator_count': 1, 'commutator_label': '14400.ca', 'complements_known': False, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '60.5', '60.5'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 67, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': None, 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 30, 1, 4], [2, 40, 1, 2], [2, 60, 1, 2], [2, 100, 1, 4], [2, 120, 1, 8], [2, 200, 1, 2], [2, 225, 1, 4], [2, 450, 1, 2], [2, 600, 1, 2], [3, 40, 1, 1], [3, 400, 1, 1], [4, 40, 1, 2], [4, 120, 1, 4], [4, 600, 1, 6], [4, 900, 1, 4], [4, 1200, 1, 2], [4, 1800, 1, 14], [4, 2400, 1, 4], [5, 48, 1, 1], [5, 576, 1, 1], [6, 40, 1, 3], [6, 80, 1, 4], [6, 400, 1, 11], [6, 600, 1, 4], [6, 800, 1, 8], [6, 1200, 1, 4], [6, 1600, 1, 2], [6, 2400, 1, 8], [8, 7200, 1, 4], [10, 48, 1, 3], [10, 96, 1, 2], [10, 576, 1, 7], [10, 720, 1, 4], [10, 960, 1, 2], [10, 1440, 1, 2], [10, 2880, 1, 8], [12, 80, 1, 2], [12, 800, 1, 2], [12, 1200, 1, 6], [12, 1600, 1, 2], [12, 2400, 1, 6], [12, 4800, 1, 4], [15, 960, 1, 1], [20, 960, 1, 2], [20, 2880, 1, 4], [30, 960, 1, 3], [30, 1920, 1, 4], [60, 1920, 1, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 120, 'exponents_of_order': [10, 2, 2], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '115200.bv', 'hash': 2885533212428351551, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 120, 'inner_gen_orders': [12, 4, 10, 10, 12], 'inner_gens': [[8213651349126, 4311493406928, 5862225196926, 5868782075521, 32619732054], [10841447595000, 8133447536334, 8475015996846, 8303652846721, 9161475684774], [11017994848566, 8300633327694, 8220160972926, 8233239173041, 2746689972168], [6827733158406, 4224888501534, 4561148027646, 4552432340401, 11859245803848], [5604932067960, 10918936264608, 10929394027566, 5530752062641, 1439044594248]], 'inner_hash': 3328040073188890159, 'inner_nilpotent': False, 'inner_order': 57600, 'inner_split': None, 'inner_tex': 'S_5^2:C_2^2', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 12], [8, 16], [10, 16], [12, 8], [16, 20], [20, 4], [24, 2], [25, 16], [32, 12], [36, 16], [40, 16], [48, 8], [50, 12], [60, 8], [80, 4], [96, 2], [120, 2]], 'label': '230400.co', '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': False, 'name': '(C2*S5^2).C2^3', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [1, 3, 0, 3, 0, 0, 1, 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, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 19, 0, 0, 0, 0, 27, 0, 0, 35, 0, 15, 1], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 89, 'number_characteristic_subgroups': 19, 'number_conjugacy_classes': 190, 'number_divisions': 190, 'number_normal_subgroups': 113, 'number_subgroup_autclasses': None, 'number_subgroup_classes': None, 'number_subgroups': None, 'old_label': None, 'order': 230400, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 5087], [3, 440], [4, 44960], [5, 624], [6, 40840], [8, 28800], [10, 35088], [12, 45760], [15, 960], [20, 13440], [30, 10560], [60, 3840]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [1428466596480, 1426590870174, 2044594021680, 11982356754480, 131331075120, 11982872084400, 11769603108480], 'outer_gens': [[11018514850680, 8146062006175, 10831863208320, 5862065171887, 2014325918935], [4311588153894, 8214405741361, 8308014624289, 8128219196448, 1838971330855], [6820508540568, 8222552432767, 6927336058015, 4224398578128, 9161069622055], [6926926769287, 5512844373534, 10744005933486, 10757018769841, 524554289689], [5517155710201, 5692100561569, 5694422227327, 4567375008000, 11944033078728], [6840350133127, 10924633042129, 8321698470961, 8303573335806, 132292707414], [7012634272680, 5619731362848, 4311573240240, 11005621764607, 118959770568]], 'outer_group': '128.1755', 'outer_hash': 1755, 'outer_nilpotent': True, 'outer_order': 128, 'outer_permdeg': 16, 'outer_perms': [14570092279818, 1336367651876, 8027649593058, 8031681278130, 18711782990350, 10555353003872, 11505577670619], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:D_4', 'pc_rank': None, 'perfect': False, 'permutation_degree': 16, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 12], [8, 16], [10, 16], [12, 8], [16, 20], [20, 4], [24, 2], [25, 16], [32, 12], [36, 16], [40, 16], [48, 8], [50, 12], [60, 8], [80, 4], [96, 2], [120, 2]], 'representations': {'Perm': {'d': 16, 'gens': [8213651349126, 8133447536334, 8220160972926, 4552432340401, 1439044594248]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 100, 'supersolvable': False, 'sylow_subgroups_known': False, 'tex_name': '(C_2\\times S_5^2).C_2^3', 'transitive_degree': 80, 'wreath_data': None, 'wreath_product': False}
-
gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': 'S_3', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '6.1', 'autcent_hash': 1, 'autcent_nilpotent': False, 'autcent_order': 6, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3', '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]], 'center_label': '4.2', 'center_order': 4, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', '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, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 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': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
