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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '11520.ec', 'ambient_counter': 107, 'ambient_order': 11520, 'ambient_tex': 'C_2^5:C_6\\times A_5', 'central': False, 'central_factor': False, 'centralizer_order': 960, 'characteristic': True, 'core_order': 64, 'counter': 756, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '11520.ec.180.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '180.a1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '180.19', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 19, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 180, 'quotient_simple': False, 'quotient_solvable': False, 'quotient_supersolvable': False, 'quotient_tex': '\\GL(2,4)', 'simple': False, 'solvable': True, 'special_labels': ['F', 'F'], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '64.261', 'subgroup_hash': 261, 'subgroup_order': 64, 'subgroup_tex': 'D_4\\times C_2^3', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '11520.ec', 'aut_centralizer_order': None, 'aut_label': '180.a1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '12.b1', 'complements': ['64.a1'], 'conjugacy_class_count': 1, 'contained_in': ['36.g1', '60.a1', '60.l1', '60.m1', '90.g1'], 'contains': ['360.a1', '360.b1', '360.c1', '360.h1', '360.s1', '360.z1', '360.ev1'], 'core': '180.a1', 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [593, 5040, 121, 1, 367920, 5046], 'label': '11520.ec.180.a1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '180.a1', 'normal_contained_in': ['3.a1', '60.a1'], 'normal_contains': ['360.a1', '360.b1', '360.c1', '720.a1'], 'normalizer': '1.a1', 'old_label': '180.a1', 'projective_image': '2880.gn', 'quotient_action_image': '3.1', 'quotient_action_kernel': '60.5', 'quotient_action_kernel_order': 60, 'quotient_fusion': None, 'short_label': '180.a1', 'subgroup_fusion': None, 'weyl_group': '12.5'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '32.51', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 168, 'aut_gen_orders': [14, 4], 'aut_gens': [[2561, 2565, 1543, 1541, 1587], [1571, 2561, 517, 1589, 2581], [1539, 517, 3623, 1573, 3635]], 'aut_group': '688128.be', 'aut_hash': 4970985371203673109, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 688128, 'aut_permdeg': 32, 'aut_perms': [255721047101209868832962344144395246, 45877573149809264889240548981106065], 'aut_phi_ratio': 21504.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 14, 1], [2, 2, 16, 1], [4, 2, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^5.C_2^7:\\GL(3,2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 168, 'autcent_group': None, 'autcent_hash': 6173500181054804164, 'autcent_nilpotent': False, 'autcent_order': 344064, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^5.C_2^6.\\PSL(2,7)', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 15], [2, 2, 16], [4, 2, 8]], 'center_label': '16.14', 'center_order': 16, 'central_product': True, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 261, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15], [2, 2, 1, 16], [4, 2, 1, 8]], 'element_repr_type': 'GLZq', 'elementary': 2, 'eulerian_function': 465, 'exponent': 4, 'exponents_of_order': [6], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '32.51', 'hash': 261, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [1, 1, 1, 2, 2], 'inner_gens': [[2561, 2565, 1543, 1541, 1587], [2561, 2565, 1543, 1541, 1587], [2561, 2565, 1543, 1541, 1587], [2561, 2565, 1543, 1541, 1555], [2561, 2565, 1543, 1573, 1587]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': False, 'inner_tex': 'C_2^2', 'inner_used': [4, 5], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 8]], 'label': '64.261', 'linC_count': None, 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, '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': 'D4*C2^3', 'ngens': 5, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 40, 'number_divisions': 40, 'number_normal_subgroups': 425, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 681, 'number_subgroups': 937, 'old_label': None, 'order': 64, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 47], [4, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 168, 'outer_gen_orders': [8, 4], 'outer_gen_pows': [513, 513], 'outer_gens': [[3619, 3591, 1571, 567, 3607], [3587, 517, 2565, 1541, 533]], 'outer_group': None, 'outer_hash': 7746879706912876876, 'outer_nilpotent': False, 'outer_order': 172032, 'outer_permdeg': 24, 'outer_perms': [293898027136311178599264, 162476987823691988414844], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^9.C_2.\\PSL(2,7)', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 10, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 2, 2], 'quasisimple': False, 'rank': 5, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 32], [2, 8]], 'representations': {'PC': {'code': 68288528, 'gens': [1, 2, 3, 4, 5], 'pres': [6, -2, 2, 2, 2, 2, -2, 202, 88]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [141602720900, 705686952884, 706461792886, 706460730980, 706461792404]}, 'GLFp': {'d': 5, 'p': 2, 'gens': [4562006, 7292993, 5128257, 20225089, 19896385, 6210625]}, 'GLZq': {'d': 2, 'q': 8, 'gens': [529, 2565, 545, 515, 2611, 1539]}, 'Perm': {'d': 10, 'gens': [127, 368045, 806416, 136, 368056, 16]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2, 2], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4\\times C_2^3', 'transitive_degree': 32, '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': '24.15', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 60, 'aut_gen_orders': [20, 4, 6, 30, 12, 20], 'aut_gens': [[362886, 363599, 2424528, 5167, 5047, 13413242449, 5040, 349234870728, 1], [367920, 368512, 1197654, 5167, 362881, 281176358695, 6, 354981295128, 367927], [362886, 362887, 2424528, 5167, 5638, 268721095536, 5040, 93408099817, 592], [367920, 367921, 1192729, 5167, 5759, 281176364143, 127, 519748416, 713], [5046, 5638, 2782368, 368041, 7, 88177045969, 362880, 355421611848, 367927], [5046, 5047, 2424649, 368041, 598, 180624717942, 363001, 349234865977, 368518], [362886, 363599, 2424649, 5167, 362881, 88177040935, 5161, 355421606814, 367927]], 'aut_group': None, 'aut_hash': 6293584833498415370, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 184320, 'aut_permdeg': 31, 'aut_perms': [4478945397383180209950366877038068, 198719575044961869040821523165454, 5180759045212816889729792409698710, 8142179124918957848175911260254388, 2671505500455493811344944802997601, 175432144929647036742395551854506], 'aut_phi_ratio': 60.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 4, 1], [2, 3, 1, 2], [2, 3, 2, 1], [2, 6, 4, 1], [2, 15, 1, 2], [2, 15, 2, 1], [2, 30, 4, 1], [2, 45, 1, 2], [2, 45, 2, 1], [2, 90, 4, 1], [3, 4, 2, 1], [3, 20, 1, 1], [3, 80, 2, 1], [4, 2, 2, 1], [4, 6, 2, 1], [4, 30, 2, 1], [4, 90, 2, 1], [5, 12, 2, 1], [6, 4, 2, 1], [6, 4, 4, 1], [6, 8, 8, 1], [6, 20, 1, 1], [6, 20, 2, 1], [6, 40, 4, 1], [6, 60, 1, 2], [6, 60, 2, 3], [6, 60, 4, 1], [6, 80, 2, 1], [6, 80, 4, 1], [6, 120, 4, 1], [6, 120, 8, 1], [6, 160, 8, 1], [10, 12, 2, 1], [10, 12, 4, 1], [10, 24, 8, 1], [10, 36, 2, 2], [10, 36, 4, 1], [10, 72, 8, 1], [12, 8, 4, 1], [12, 40, 2, 1], [12, 120, 2, 1], [12, 120, 4, 1], [12, 160, 4, 1], [15, 48, 4, 1], [20, 24, 4, 1], [20, 72, 4, 1], [30, 48, 4, 1], [30, 48, 8, 1], [30, 96, 16, 1], [60, 96, 8, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_5\\times C_2\\wr C_2^2\\times S_4', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '5760.co', 'autcentquo_hash': 6537139331229567448, 'autcentquo_nilpotent': False, 'autcentquo_order': 5760, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2\\times S_4\\times S_5', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 4], [2, 3, 4], [2, 6, 4], [2, 15, 4], [2, 30, 4], [2, 45, 4], [2, 90, 4], [3, 4, 2], [3, 20, 1], [3, 80, 2], [4, 2, 2], [4, 6, 2], [4, 30, 2], [4, 90, 2], [5, 12, 2], [6, 4, 6], [6, 8, 8], [6, 20, 3], [6, 40, 4], [6, 60, 12], [6, 80, 6], [6, 120, 12], [6, 160, 8], [10, 12, 6], [10, 24, 8], [10, 36, 8], [10, 72, 8], [12, 8, 4], [12, 40, 2], [12, 120, 6], [12, 160, 4], [15, 48, 4], [20, 24, 4], [20, 72, 4], [30, 48, 12], [30, 96, 16], [60, 96, 8]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '2880.gn', 'commutator_count': 1, 'commutator_label': '480.1187', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '60.5'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 107, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['2.1', 1], ['60.5', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 4], [2, 3, 1, 4], [2, 6, 1, 4], [2, 15, 1, 4], [2, 30, 1, 4], [2, 45, 1, 4], [2, 90, 1, 4], [3, 4, 2, 1], [3, 20, 1, 1], [3, 80, 2, 1], [4, 2, 1, 2], [4, 6, 1, 2], [4, 30, 1, 2], [4, 90, 1, 2], [5, 12, 2, 1], [6, 4, 2, 3], [6, 8, 2, 4], [6, 20, 1, 3], [6, 40, 1, 4], [6, 60, 1, 4], [6, 60, 2, 4], [6, 80, 2, 3], [6, 120, 1, 4], [6, 120, 2, 4], [6, 160, 2, 4], [10, 12, 2, 3], [10, 24, 2, 4], [10, 36, 2, 4], [10, 72, 2, 4], [12, 8, 2, 2], [12, 40, 1, 2], [12, 120, 1, 2], [12, 120, 2, 2], [12, 160, 2, 2], [15, 48, 4, 1], [20, 24, 2, 2], [20, 72, 2, 2], [30, 48, 4, 3], [30, 96, 4, 4], [60, 96, 4, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 2276820, 'exponent': 60, 'exponents_of_order': [8, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '5760.gx', 'hash': 3725610209488732842, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [2, 2, 3, 2, 2, 6, 2, 6, 2], 'inner_gens': [[362886, 363599, 2419494, 5167, 5047, 13412874529, 5040, 349234865694, 1], [362886, 363599, 2419494, 5167, 5166, 13412874648, 5040, 349234865575, 120], [5046, 5759, 2424528, 368041, 367921, 13412874529, 362880, 349234507854, 1], [362886, 363599, 2782368, 5167, 5047, 13412879575, 5040, 349234502808, 1], [362886, 363478, 2782368, 5167, 5047, 13412879575, 5040, 349234502808, 1], [367920, 368512, 2787414, 363007, 362887, 13413242449, 6, 181065387294, 1], [362886, 363599, 2419494, 5167, 5047, 13412874529, 5040, 349234865694, 1], [5046, 5638, 2787414, 368041, 367921, 274468751575, 362880, 349234870728, 1], [362886, 363478, 2424528, 5167, 5047, 13413242449, 5040, 349234870728, 1]], 'inner_hash': 5094570882729245637, 'inner_nilpotent': False, 'inner_order': 2880, 'inner_split': False, 'inner_tex': 'C_2^4:\\GL(2,4)', 'inner_used': [1, 2, 3, 5, 6, 8], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 24], [2, 6], [3, 56], [4, 24], [5, 24], [6, 14], [8, 6], [9, 16], [10, 6], [12, 8], [15, 8], [18, 4], [24, 2], [30, 2]], 'label': '11520.ec', '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': 'C2^5:C6*A5', '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, 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': 60, 'number_characteristic_subgroups': 30, 'number_conjugacy_classes': 200, 'number_divisions': 120, 'number_normal_subgroups': 114, 'number_subgroup_autclasses': 2815, 'number_subgroup_classes': 12547, 'number_subgroups': 564134, 'old_label': None, 'order': 11520, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 767], [3, 188], [4, 256], [5, 24], [6, 4228], [10, 1128], [12, 1472], [15, 192], [20, 384], [30, 2112], [60, 768]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4], 'outer_gen_pows': [0, 0, 712, 0, 0], 'outer_gens': [[367920, 368633, 1192608, 5167, 5047, 13414469335, 6, 349232913048, 1], [367920, 719, 1192608, 5167, 362881, 19681764769, 6, 354981300174, 367927], [362886, 5753, 2424649, 5167, 362881, 19679812135, 5161, 354982522014, 367927], [367920, 126, 1192608, 5167, 363472, 19681765056, 6, 354981300463, 368518], [362886, 5160, 2424528, 5167, 5638, 13413242736, 5040, 349234871017, 592]], 'outer_group': '64.261', 'outer_hash': 261, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 10, 'outer_perms': [806400, 806520, 368040, 368056, 368062], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_4\\times C_2^3', 'pc_rank': None, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 10], [3, 8], [4, 10], [5, 8], [6, 10], [8, 10], [10, 10], [12, 18], [15, 8], [16, 2], [18, 8], [20, 2], [24, 4], [30, 2], [36, 2]], 'representations': {'Perm': {'d': 15, 'gens': [362886, 363599, 2424528, 5167, 5047, 13413242449, 5040, 349234870728, 1]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^5:C_6\\times A_5', 'transitive_degree': 120, 'wreath_data': None, 'wreath_product': False}
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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': 1, 'aut_exponent': 60, 'aut_gen_orders': [3, 4, 2], 'aut_gens': [[66, 21], [66, 45], [82, 139], [67, 84]], 'aut_group': '240.189', 'aut_hash': 189, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 240, 'aut_permdeg': 7, 'aut_perms': [1584, 408, 415], 'aut_phi_ratio': 5.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 15, 1, 1], [3, 1, 2, 1], [3, 20, 1, 1], [3, 20, 2, 1], [5, 12, 2, 1], [6, 15, 2, 1], [15, 12, 4, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2\\times S_5', 'autcent_abelian': True, 'autcent_cyclic': True, 'autcent_exponent': 2, 'autcent_group': '2.1', 'autcent_hash': 1, 'autcent_nilpotent': True, 'autcent_order': 2, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 60, 'autcentquo_group': '120.34', 'autcentquo_hash': 34, 'autcentquo_nilpotent': False, 'autcentquo_order': 120, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_5', 'cc_stats': [[1, 1, 1], [2, 15, 1], [3, 1, 2], [3, 20, 3], [5, 12, 2], [6, 15, 2], [15, 12, 4]], 'center_label': '3.1', 'center_order': 3, 'central_product': True, 'central_quotient': '60.5', 'commutator_count': 1, 'commutator_label': '60.5', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['3.1', '60.5'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 19, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['60.5', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 15, 1, 1], [3, 1, 2, 1], [3, 20, 1, 1], [3, 20, 2, 1], [5, 12, 2, 1], [6, 15, 2, 1], [15, 12, 4, 1]], 'element_repr_type': 'Lie', 'elementary': 1, 'eulerian_function': 76, 'exponent': 30, 'exponents_of_order': [2, 2, 1], 'factors_of_aut_order': [2, 3, 5], 'factors_of_order': [2, 3, 5], 'faithful_reps': [[3, 0, 4], [4, 0, 2], [5, 0, 2]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '180.19', 'hash': 19, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 30, 'inner_gen_orders': [3, 3], 'inner_gens': [[66, 45], [177, 21]], 'inner_hash': 5, 'inner_nilpotent': False, 'inner_order': 60, 'inner_split': True, 'inner_tex': 'A_5', 'inner_used': [1, 2], 'irrC_degree': 3, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 6, 'irrep_stats': [[1, 3], [3, 6], [4, 3], [5, 3]], 'label': '180.19', 'linC_count': 4, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 1, 'linQ_dim': 6, 'linQ_dim_count': 1, 'linR_count': 2, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': False, 'name': 'GL(2,4)', 'ngens': 2, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 8, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 15, 'number_divisions': 8, 'number_normal_subgroups': 4, 'number_subgroup_autclasses': 20, 'number_subgroup_classes': 21, 'number_subgroups': 148, 'old_label': None, 'order': 180, 'order_factorization_type': 222, 'order_stats': [[1, 1], [2, 15], [3, 62], [5, 24], [6, 30], [15, 48]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [205, 65], 'outer_gens': [[216, 131], [193, 21]], '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': 8, 'pgroup': 0, 'primary_abelian_invariants': [3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1], [2, 1], [4, 1], [5, 1], [6, 1], [8, 1], [10, 1], [12, 1]], 'representations': {'GLZ': {'b': 3, 'd': 6, 'gens': [125101743028352636, 41624327359088224]}, 'Lie': [{'d': 2, 'q': 4, 'gens': [66, 21], 'family': 'GL'}, {'d': 3, 'q': 4, 'gens': [131586, 4373, 131331, 4368], 'family': 'CSO'}, {'d': 3, 'q': 4, 'gens': [131586, 4373, 131331, 4368], 'family': 'CO'}], 'GLFp': {'d': 4, 'p': 2, 'gens': [27484, 38957, 59379]}, 'Perm': {'d': 8, 'gens': [17283, 22348]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [3], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\GL(2,4)', 'transitive_degree': 15, 'wreath_data': None, 'wreath_product': False}
