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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '31104.mm', 'ambient_counter': 325, 'ambient_order': 31104, 'ambient_tex': 'C_3^4:(D_4\\times \\GL(2,3))', 'central': False, 'central_factor': False, 'centralizer_order': 6, 'characteristic': False, 'core_order': 1, 'counter': 881, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '31104.mm.216.be1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '216.be1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 216, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '144.162', 'subgroup_hash': 162, 'subgroup_order': 144, 'subgroup_tex': 'C_{12}:D_6', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '31104.mm', 'aut_centralizer_order': None, 'aut_label': '216.be1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '5184.m1', 'complements': None, 'conjugacy_class_count': 1, 'contained_in': ['24.x1', '72.z2', '108.v1'], 'contains': ['432.x1', '432.bd1', '432.bx1', '432.el1', '432.ey1', '648.bb1', '648.bd1'], 'core': '31104.a1', 'coset_action_label': None, 'count': 108, 'diagramx': None, 'generators': [45600, 45507, 281173984800, 268241034483, 138240, 368871149043], 'label': '31104.mm.216.be1', 'mobius_quo': None, 'mobius_sub': 0, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '108.v1', 'old_label': '216.be1', 'projective_image': '31104.mm', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '216.be1', 'subgroup_fusion': None, 'weyl_group': '48.51'}
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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': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 2, 12], 'aut_gens': [[1, 2, 12], [1, 2, 60], [73, 2, 12], [1, 38, 84], [1, 10, 12], [121, 38, 12]], 'aut_group': '192.1514', 'aut_hash': 1514, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 192, 'aut_permdeg': 11, 'aut_perms': [368040, 5160, 1, 11520, 3991702], 'aut_phi_ratio': 4.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [2, 3, 2, 1], [2, 6, 2, 1], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 2, 1, 1], [4, 6, 1, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 2, 4, 1], [6, 3, 4, 1], [6, 4, 2, 1], [6, 4, 4, 1], [6, 6, 4, 1], [12, 2, 2, 1], [12, 4, 1, 1], [12, 4, 2, 1], [12, 6, 2, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{12}:C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '16.14', 'autcent_hash': 14, 'autcent_nilpotent': True, 'autcent_order': 16, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 6, 'autcentquo_group': '12.4', 'autcentquo_hash': 4, 'autcentquo_nilpotent': False, 'autcentquo_order': 12, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'D_6', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [2, 3, 2], [2, 6, 2], [3, 1, 2], [3, 2, 3], [4, 2, 1], [4, 6, 1], [6, 1, 2], [6, 2, 7], [6, 3, 4], [6, 4, 6], [6, 6, 4], [12, 2, 2], [12, 4, 3], [12, 6, 2]], 'center_label': '6.2', 'center_order': 6, 'central_product': True, 'central_quotient': '24.14', 'commutator_count': 1, 'commutator_label': '6.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 162, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['6.1', 1], ['8.3', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [2, 3, 1, 2], [2, 6, 1, 2], [3, 1, 2, 1], [3, 2, 1, 1], [3, 2, 2, 1], [4, 2, 1, 1], [4, 6, 1, 1], [6, 1, 2, 1], [6, 2, 1, 1], [6, 2, 2, 3], [6, 3, 2, 2], [6, 4, 1, 2], [6, 4, 2, 2], [6, 6, 2, 2], [12, 2, 2, 1], [12, 4, 1, 1], [12, 4, 2, 1], [12, 6, 2, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 4368, 'exponent': 12, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[4, 0, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '72.48', 'hash': 162, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 6], 'inner_gens': [[1, 2, 60], [1, 2, 84], [97, 74, 12]], 'inner_hash': 14, 'inner_nilpotent': False, 'inner_order': 24, 'inner_split': True, 'inner_tex': 'C_2\\times D_6', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 24], [2, 18], [4, 3]], 'label': '144.162', 'linC_count': 66, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 6, 'linQ_degree_count': 88, 'linQ_dim': 6, 'linQ_dim_count': 88, 'linR_count': 88, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C12:D6', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 22, 'number_characteristic_subgroups': 26, 'number_conjugacy_classes': 45, 'number_divisions': 30, 'number_normal_subgroups': 50, 'number_subgroup_autclasses': 74, 'number_subgroup_classes': 116, 'number_subgroups': 260, 'old_label': None, 'order': 144, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 23], [3, 8], [4, 8], [6, 76], [12, 28]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2], 'outer_gen_pows': [36, 0, 0], 'outer_gens': [[1, 38, 12], [1, 10, 12], [73, 2, 12]], 'outer_group': '8.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 8, 'outer_permdeg': 6, 'outer_perms': [1, 6, 120], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^3', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 10, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 3], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 14], [4, 7], [8, 1]], 'representations': {'PC': {'code': 2397610606518029444745, 'gens': [1, 2, 4], 'pres': [6, -2, -2, -3, -2, -2, -3, 31, 1443, 1017, 69, 3604, 88, 3461]}, 'GLZ': {'b': 3, 'd': 6, 'gens': [125101729074820244, 24992885287126483, 108470312588010181]}, 'GLZN': {'d': 2, 'p': 14, 'gens': [31658, 2843, 35685, 24705, 2757, 714]}, 'Perm': {'d': 10, 'gens': [374407, 16, 1, 856920, 7, 1210440]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 6], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}:D_6', 'transitive_degree': 24, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 5, 'aut_exponent': 24, 'aut_gen_orders': [24, 8, 24, 6], 'aut_gens': [[219574, 338677, 368871234956, 85949, 63975, 287404882998, 130855, 45507, 19203664935, 267762211229, 224189, 268240896693], [45507, 94839, 448908205149, 26869, 157572, 482685580, 13599, 184197, 181062729169, 25427100676, 317372, 3669149], [357364, 174329, 456053168073, 19935, 72949, 268244874533, 13599, 276480, 181062677927, 19203793347, 317372, 3957999], [80884, 37817, 25430835860, 24100, 26719, 180587489573, 33383, 184197, 193999606296, 280259739323, 5515, 368871189149], [321987, 98986, 448908002399, 40349, 216254, 180587489573, 188265, 45507, 19203784214, 267762073532, 281995, 180583603893]], 'aut_group': None, 'aut_hash': 6992198563583357107, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 62208, 'aut_permdeg': 48, 'aut_perms': [11014887766334079802049721025038731530413619329645636998016358, 3466187510946102761526114866090088466877265315251262423568960, 4289112569958647093258306098368282720601626128902101700389586, 10315986037247441720429374486163953438941375524515358421304975], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 6, 2, 1], [2, 9, 1, 2], [2, 36, 1, 1], [2, 54, 2, 1], [2, 81, 1, 1], [2, 216, 2, 1], [2, 324, 1, 1], [3, 4, 2, 1], [3, 8, 1, 1], [3, 24, 1, 1], [3, 32, 2, 1], [3, 48, 1, 1], [3, 96, 2, 1], [3, 192, 2, 1], [4, 18, 1, 1], [4, 54, 1, 1], [4, 162, 1, 1], [4, 324, 2, 1], [4, 486, 1, 1], [4, 648, 1, 1], [4, 972, 1, 1], [6, 12, 2, 1], [6, 36, 2, 1], [6, 48, 2, 1], [6, 72, 1, 3], [6, 96, 2, 1], [6, 108, 2, 1], [6, 144, 2, 2], [6, 216, 1, 1], [6, 288, 2, 4], [6, 432, 1, 1], [6, 432, 2, 3], [6, 576, 2, 1], [6, 648, 1, 2], [6, 864, 2, 2], [8, 54, 1, 2], [8, 324, 2, 2], [8, 486, 1, 2], [8, 972, 1, 2], [12, 144, 1, 1], [12, 216, 2, 1], [12, 432, 1, 1], [12, 648, 2, 1], [12, 864, 1, 1], [12, 1296, 1, 2], [24, 216, 2, 2], [24, 648, 2, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_3:S_3.C_6^2.C_6.D_4.C_2', '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': 24, 'autcentquo_group': None, 'autcentquo_hash': 6992198563583357107, 'autcentquo_nilpotent': False, 'autcentquo_order': 62208, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3:S_3.C_6^2.C_6.D_4.C_2', 'cc_stats': [[1, 1, 1], [2, 6, 2], [2, 9, 2], [2, 36, 1], [2, 54, 2], [2, 81, 1], [2, 216, 2], [2, 324, 1], [3, 4, 2], [3, 8, 1], [3, 24, 1], [3, 32, 2], [3, 48, 1], [3, 96, 2], [3, 192, 2], [4, 18, 1], [4, 54, 1], [4, 162, 1], [4, 324, 2], [4, 486, 1], [4, 648, 1], [4, 972, 1], [6, 12, 2], [6, 36, 2], [6, 48, 2], [6, 72, 3], [6, 96, 2], [6, 108, 2], [6, 144, 4], [6, 216, 1], [6, 288, 8], [6, 432, 7], [6, 576, 2], [6, 648, 2], [6, 864, 4], [8, 54, 2], [8, 324, 4], [8, 486, 2], [8, 972, 2], [12, 144, 1], [12, 216, 2], [12, 432, 1], [12, 648, 2], [12, 864, 1], [12, 1296, 2], [24, 216, 4], [24, 648, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '31104.mm', 'commutator_count': 1, 'commutator_label': '3888.kp', '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', '3.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 12, 'conjugacy_classes_known': True, 'counter': 325, 'cyclic': False, 'derived_length': 5, 'dihedral': False, 'direct_factorization': [['432.734', 1], ['72.40', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 6, 1, 2], [2, 9, 1, 2], [2, 36, 1, 1], [2, 54, 1, 2], [2, 81, 1, 1], [2, 216, 1, 2], [2, 324, 1, 1], [3, 4, 1, 2], [3, 8, 1, 1], [3, 24, 1, 1], [3, 32, 1, 2], [3, 48, 1, 1], [3, 96, 1, 2], [3, 192, 1, 2], [4, 18, 1, 1], [4, 54, 1, 1], [4, 162, 1, 1], [4, 324, 1, 2], [4, 486, 1, 1], [4, 648, 1, 1], [4, 972, 1, 1], [6, 12, 1, 2], [6, 36, 1, 2], [6, 48, 1, 2], [6, 72, 1, 3], [6, 96, 1, 2], [6, 108, 1, 2], [6, 144, 1, 4], [6, 216, 1, 1], [6, 288, 1, 8], [6, 432, 1, 7], [6, 576, 1, 2], [6, 648, 1, 2], [6, 864, 1, 4], [8, 54, 2, 1], [8, 324, 2, 2], [8, 486, 2, 1], [8, 972, 2, 1], [12, 144, 1, 1], [12, 216, 1, 2], [12, 432, 1, 1], [12, 648, 1, 2], [12, 864, 1, 1], [12, 1296, 1, 2], [24, 216, 2, 2], [24, 648, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 129024000, 'exponent': 24, 'exponents_of_order': [7, 5], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[32, 1, 8], [64, 1, 4]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '31104.mm', 'hash': 8544079931638136565, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [3, 4, 6, 2, 4, 4, 3, 3, 6, 6, 2, 2], 'inner_gens': [[219574, 33770, 368871511436, 5065, 20073, 287404839096, 33383, 45507, 19203708873, 267762176186, 5515, 268241218294], [45507, 338677, 368871267857, 238635, 166153, 287404931606, 349453, 321987, 19203715186, 267762162350, 224189, 268240981706], [357364, 68681, 368871234956, 85706, 192555, 19679226400, 209798, 80884, 181062832779, 25427226239, 281995, 19199981398], [184197, 63975, 368871235051, 85949, 338677, 287404794107, 249909, 80884, 19203877717, 267762176045, 224189, 268240896598], [138690, 254325, 368871256885, 26869, 63975, 287404741624, 37525, 184197, 19203673026, 267762171478, 224189, 268241026433], [138690, 285303, 448908225677, 349883, 227590, 287404882998, 299453, 184197, 25427252056, 181062650850, 362879, 268241159803], [357364, 146625, 368871319399, 210193, 254325, 287404746372, 130855, 321987, 19203797686, 267761975634, 224189, 268241066174], [219574, 108554, 368871189835, 40442, 294198, 287404785526, 263606, 45507, 19203864102, 267762176186, 178682, 268240896693], [184197, 336160, 448908080657, 37727, 55180, 281174127750, 33383, 219574, 19203664935, 267762252114, 143305, 268241066174], [138690, 105306, 448908185641, 40585, 135289, 25909700681, 238510, 80884, 19203780199, 267762211229, 362879, 479139840], [321987, 338677, 368871466315, 85949, 63975, 287404785526, 130855, 80884, 19203633879, 267762073439, 224189, 268241218294], [321987, 157572, 88698844651, 85801, 72949, 287404805054, 329656, 45507, 19203686839, 181062829439, 317372, 268240896693]], 'inner_hash': 8544079931638136565, 'inner_nilpotent': False, 'inner_order': 31104, 'inner_split': False, 'inner_tex': 'C_3^4:(D_4\\times \\GL(2,3))', 'inner_used': [1, 2, 3, 4, 6, 9, 12], 'irrC_degree': 32, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 32, 'irrep_stats': [[1, 8], [2, 14], [3, 8], [4, 15], [6, 2], [8, 21], [12, 8], [16, 10], [32, 9], [64, 4]], 'label': '31104.mm', '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': 'C3^4:(D4*GL(2,3))', 'ngens': 12, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 66, 'number_characteristic_subgroups': 36, 'number_conjugacy_classes': 99, 'number_divisions': 90, 'number_normal_subgroups': 56, 'number_subgroup_autclasses': 2304, 'number_subgroup_classes': 3900, 'number_subgroups': 455324, 'old_label': None, 'order': 31104, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 1011], [3, 728], [4, 2988], [6, 12840], [8, 4320], [12, 5760], [24, 3456]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [281173939200], 'outer_gens': [[321987, 130523, 268241039405, 85949, 250296, 25909652073, 130855, 80884, 193999531302, 200222893291, 86399, 368871367824]], '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': 7, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [3, 8], [4, 17], [6, 2], [8, 14], [12, 8], [16, 14], [32, 9], [64, 4]], 'representations': {'PC': {'code': '289240225309904697145300414803216816498756940631423779280166841968956687538306808054858849802593097209095332256794318724414138242288663906687019254134928771531784250423364171744759658183547036411589649429950511965082798836901904032295', 'gens': [1, 2, 4, 6, 8, 11, 12], 'pres': [12, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 3, 3, 241, 61, 290, 879, 135, 324869, 971153, 203069, 8393, 773, 209, 1020114, 437502, 150570, 79350, 19734, 2446855, 827155, 570271, 3883, 5815, 52483, 14191, 283, 1181960, 352532, 575456, 114956, 13016, 9140, 12392, 320, 1284501, 13485, 11577, 1824790, 798370, 47590, 9586, 6430, 1690, 2985995, 1493015, 41543, 31187, 12191, 5291]}, 'Perm': {'d': 15, 'gens': [219574, 338677, 368871234956, 85949, 63975, 287404882998, 130855, 45507, 19203664935, 267762211229, 224189, 268240896693]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:(D_4\\times \\GL(2,3))', 'transitive_degree': 48, 'wreath_data': None, 'wreath_product': False}
