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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '11757312.o', 'ambient_counter': 15, 'ambient_order': 11757312, 'ambient_tex': 'C_3^7.C_2^5.\\PSL(2,7)', 'central': False, 'central_factor': False, 'centralizer_order': None, 'characteristic': False, 'core_order': 2, 'counter': 165, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '11757312.o.2187.A', 'maximal': True, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '2187.a1.M', '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': 2187, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': False, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '5376.bx', 'subgroup_hash': 2822916461223089928, 'subgroup_order': 5376, 'subgroup_tex': 'C_2^5:\\GL(3,2)', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '11757312.o', 'aut_centralizer_order': None, 'aut_label': '2187.A', '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': '5878656.A', 'coset_action_label': None, 'count': 2187, 'diagramx': None, 'generators': [1, 96421560501467665, 17427144628534845335887, 96540555984376584, 3934889223163921, 3620423943747102734791, 411382455291201155304], 'label': '11757312.o.2187.A', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '1.a1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '2187.A', 'old_label': '2187.a1.M', 'projective_image': '5878656.d', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '2187.A', 'subgroup_fusion': None, 'weyl_group': '1344.11686'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 168, 'aut_gen_orders': [6, 4, 4], 'aut_gens': [[44005087, 44403136], [135444967, 138314303], [166931407, 99239063], [299502136, 259514647]], 'aut_group': None, 'aut_hash': 5655652934598173233, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 16128, 'aut_permdeg': 87, 'aut_perms': [1696117467777537721514835531342353829993573236458875761830916896152368952642675121933597373534428695177546391462031509412829829673401, 912120696880988689004433103470583860472024339627294667675520478712113368013489734060629464976522246246441004303842779252957804367559, 192476824873964051449787239617661364305864622251138587063375312461722111010398109891403302743861928119651228733732315107912973851331], 'aut_phi_ratio': 10.5, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 7, 1, 1], [2, 7, 3, 1], [2, 42, 2, 1], [2, 42, 6, 1], [3, 224, 1, 1], [4, 84, 1, 1], [4, 84, 3, 1], [4, 168, 2, 1], [4, 168, 6, 1], [6, 224, 1, 1], [6, 224, 3, 2], [7, 192, 1, 2], [14, 192, 3, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4.\\PSL(2,7)\\times 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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 168, 'autcentquo_group': '2688.ch', 'autcentquo_hash': 3418511438992571142, 'autcentquo_nilpotent': False, 'autcentquo_order': 2688, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_2^4:\\GL(3,2)', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 7, 4], [2, 42, 8], [3, 224, 1], [4, 84, 4], [4, 168, 8], [6, 224, 7], [7, 192, 2], [14, 192, 6]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '1344.11686', 'commutator_count': 1, 'commutator_label': '1344.11686', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '168.42'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 50, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['1344.11686', 1], ['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 7, 1, 4], [2, 42, 1, 8], [3, 224, 1, 1], [4, 84, 1, 4], [4, 168, 1, 8], [6, 224, 1, 7], [7, 192, 2, 1], [14, 192, 2, 3]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 342, 'exponent': 84, 'exponents_of_order': [8, 1, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '5376.bx', 'hash': 2822916461223089928, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 84, 'inner_gen_orders': [7, 7], 'inner_gens': [[44005087, 4164496], [4446847, 44403136]], 'inner_hash': 11686, 'inner_nilpotent': False, 'inner_order': 1344, 'inner_split': True, 'inner_tex': 'C_2^3:\\GL(3,2)', 'inner_used': [1, 2], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 4], [3, 8], [6, 4], [7, 12], [8, 4], [14, 4], [21, 8]], 'label': '5376.bx', '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:GL(3,2)', '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, 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': 18, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 44, 'number_divisions': 40, 'number_normal_subgroups': 15, 'number_subgroup_autclasses': 496, 'number_subgroup_classes': 2135, 'number_subgroups': 90793, 'old_label': None, 'order': 5376, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 367], [3, 224], [4, 1680], [6, 1568], [7, 384], [14, 1152]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 6, 'outer_gen_orders': [2, 6], 'outer_gen_pows': [0, 0], 'outer_gens': [[44005103, 44403136], [171451463, 261280207]], 'outer_group': '12.4', 'outer_hash': 4, 'outer_nilpotent': False, 'outer_order': 12, 'outer_permdeg': 5, 'outer_perms': [6, 31], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'D_6', 'pc_rank': None, 'perfect': False, 'permutation_degree': 12, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [6, 8], [7, 12], [8, 4], [14, 4], [21, 8]], 'representations': {'Perm': {'d': 12, 'gens': [44005087, 44403136]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^5:\\GL(3,2)', 'transitive_degree': 28, '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': '4.2', 'all_subgroups_known': False, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 252, 'aut_gen_orders': [6, 42], 'aut_gens': [[2306905122408707975064, 1231427690626232295727], [21434626107824027656320, 15206894711124567711961], [1334180105231328722335, 2915111731300193908207]], 'aut_group': '11757312.o', 'aut_hash': 3646927932059463285, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 11757312, 'aut_permdeg': 756, 'aut_perms': 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'aut_phi_ratio': 3.5, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 189, 2, 1], [2, 378, 1, 2], [2, 567, 1, 2], [2, 1134, 2, 1], [2, 2187, 2, 1], [2, 3402, 1, 2], [2, 10206, 2, 1], [3, 14, 1, 1], [3, 16, 1, 1], [3, 56, 1, 3], [3, 84, 1, 1], [3, 112, 1, 2], [3, 224, 1, 1], [3, 336, 1, 3], [3, 448, 1, 1], [3, 18144, 1, 1], [3, 36288, 1, 1], [4, 13608, 1, 2], [4, 20412, 2, 1], [4, 40824, 2, 2], [4, 61236, 1, 2], [4, 122472, 1, 2], [6, 14, 1, 1], [6, 16, 1, 1], [6, 56, 1, 3], [6, 84, 1, 1], [6, 112, 1, 2], [6, 224, 1, 1], [6, 336, 1, 3], [6, 448, 1, 1], [6, 756, 1, 6], [6, 1512, 1, 10], [6, 1512, 2, 3], [6, 3024, 1, 18], [6, 3402, 1, 2], [6, 4536, 1, 2], [6, 4536, 2, 5], [6, 6048, 1, 14], [6, 6048, 2, 1], [6, 6804, 1, 4], [6, 9072, 2, 2], [6, 12096, 1, 2], [6, 13608, 1, 4], [6, 18144, 1, 1], [6, 18144, 2, 3], [6, 27216, 1, 4], [6, 36288, 1, 1], [6, 40824, 2, 2], [6, 54432, 2, 1], [6, 108864, 2, 1], [6, 163296, 1, 2], [6, 489888, 2, 1], [7, 139968, 1, 2], [9, 36288, 1, 4], [9, 72576, 1, 4], [12, 27216, 1, 10], [12, 40824, 2, 2], [12, 54432, 1, 8], [12, 81648, 2, 5], [12, 122472, 1, 2], [12, 163296, 2, 2], [12, 244944, 1, 2], [14, 139968, 1, 2], [14, 419904, 2, 2], [18, 36288, 1, 4], [18, 72576, 1, 4], [18, 108864, 2, 3], [18, 326592, 1, 2], [21, 279936, 1, 2], [42, 279936, 1, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_3^7.C_2^5.\\PSL(2,7)', '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': 252, 'autcentquo_group': '5878656.d', 'autcentquo_hash': 9080663322717296588, 'autcentquo_nilpotent': False, 'autcentquo_order': 5878656, 'autcentquo_solvable': False, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^7.C_2^4:\\GL(3,2)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 189, 2], [2, 378, 2], [2, 567, 2], [2, 1134, 2], [2, 2187, 2], [2, 3402, 2], [2, 10206, 2], [3, 14, 1], [3, 16, 1], [3, 56, 3], [3, 84, 1], [3, 112, 2], [3, 224, 1], [3, 336, 3], [3, 448, 1], [3, 18144, 1], [3, 36288, 1], [4, 13608, 2], [4, 20412, 2], [4, 40824, 4], [4, 61236, 2], [4, 122472, 2], [6, 14, 1], [6, 16, 1], [6, 56, 3], [6, 84, 1], [6, 112, 2], [6, 224, 1], [6, 336, 3], [6, 448, 1], [6, 756, 6], [6, 1512, 16], [6, 3024, 18], [6, 3402, 2], [6, 4536, 12], [6, 6048, 16], [6, 6804, 4], [6, 9072, 4], [6, 12096, 2], [6, 13608, 4], [6, 18144, 7], [6, 27216, 4], [6, 36288, 1], [6, 40824, 4], [6, 54432, 2], [6, 108864, 2], [6, 163296, 2], [6, 489888, 2], [7, 139968, 2], [9, 36288, 4], [9, 72576, 4], [12, 27216, 10], [12, 40824, 4], [12, 54432, 8], [12, 81648, 10], [12, 122472, 2], [12, 163296, 4], [12, 244944, 2], [14, 139968, 2], [14, 419904, 4], [18, 36288, 4], [18, 72576, 4], [18, 108864, 6], [18, 326592, 2], [21, 279936, 2], [42, 279936, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '5878656.d', 'commutator_count': 1, 'commutator_label': '2939328.a', 'complements_known': False, 'complete': False, 'complex_characters_known': False, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '3.1', '168.42'], 'composition_length': 13, 'conjugacy_classes_known': True, 'counter': 15, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['5878656.d', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 189, 1, 2], [2, 378, 1, 2], [2, 567, 1, 2], [2, 1134, 1, 2], [2, 2187, 1, 2], [2, 3402, 1, 2], [2, 10206, 1, 2], [3, 14, 1, 1], [3, 16, 1, 1], [3, 56, 1, 3], [3, 84, 1, 1], [3, 112, 1, 2], [3, 224, 1, 1], [3, 336, 1, 3], [3, 448, 1, 1], [3, 18144, 1, 1], [3, 36288, 1, 1], [4, 13608, 1, 2], [4, 20412, 1, 2], [4, 40824, 1, 4], [4, 61236, 1, 2], [4, 122472, 1, 2], [6, 14, 1, 1], [6, 16, 1, 1], [6, 56, 1, 3], [6, 84, 1, 1], [6, 112, 1, 2], [6, 224, 1, 1], [6, 336, 1, 3], [6, 448, 1, 1], [6, 756, 1, 6], [6, 1512, 1, 16], [6, 3024, 1, 18], [6, 3402, 1, 2], [6, 4536, 1, 12], [6, 6048, 1, 16], [6, 6804, 1, 4], [6, 9072, 1, 4], [6, 12096, 1, 2], [6, 13608, 1, 4], [6, 18144, 1, 7], [6, 27216, 1, 4], [6, 36288, 1, 1], [6, 40824, 1, 4], [6, 54432, 1, 2], [6, 108864, 1, 2], [6, 163296, 1, 2], [6, 489888, 1, 2], [7, 139968, 2, 1], [9, 36288, 1, 4], [9, 72576, 1, 4], [12, 27216, 1, 10], [12, 40824, 1, 4], [12, 54432, 1, 8], [12, 81648, 1, 10], [12, 122472, 1, 2], [12, 163296, 1, 4], [12, 244944, 1, 2], [14, 139968, 2, 1], [14, 419904, 2, 2], [18, 36288, 1, 4], [18, 72576, 1, 4], [18, 108864, 1, 6], [18, 326592, 1, 2], [21, 279936, 2, 1], [42, 279936, 2, 1]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': None, 'exponent': 252, 'exponents_of_order': [8, 8, 1], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2, 3, 7], 'faithful_reps': [[14, 1, 4], [16, 1, 1], [28, 1, 2], [42, 1, 4], [48, 0, 2], [56, 1, 12], [84, 1, 12], [96, 1, 1], [112, 1, 11], [128, 1, 1], [168, 1, 18], [224, 1, 6], [336, 1, 16], [448, 1, 4], [672, 1, 3], [896, 1, 1]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '11757312.o', 'hash': 3646927932059463285, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 252, 'inner_gen_orders': [6, 21], 'inner_gens': [[2306905122408707975064, 7054515102012122874631], [20475717160538015174304, 1231427690626232295727]], 'inner_hash': 9080663322717296588, 'inner_nilpotent': False, 'inner_order': 5878656, 'inner_split': True, 'inner_tex': 'C_3^7.C_2^4:\\GL(3,2)', 'inner_used': [1, 2], 'irrC_degree': 14, 'irrQ_degree': 14, 'irrQ_dim': None, 'irrR_degree': None, 'irrep_stats': [[1, 4], [3, 8], [6, 4], [7, 12], [8, 4], [14, 12], [16, 2], [21, 8], [28, 4], [42, 8], [48, 4], [56, 24], [84, 24], [96, 2], [112, 22], [128, 2], [168, 36], [224, 12], [336, 32], [448, 8], [672, 6], [896, 2]], 'label': '11757312.o', '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': 'C3^7.C2^5.PSL(2,7)', '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, 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, 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': 200, 'number_characteristic_subgroups': 11, 'number_conjugacy_classes': 240, 'number_divisions': 234, 'number_normal_subgroups': 17, 'number_subgroup_autclasses': None, 'number_subgroup_classes': None, 'number_subgroups': None, 'old_label': None, 'order': 11757312, 'order_factorization_type': 321, 'order_stats': [[1, 1], [2, 36127], [3, 56618], [4, 598752], [6, 2453894], [7, 279936], [9, 435456], [12, 3075408], [14, 1959552], [18, 1741824], [21, 559872], [42, 559872]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[2306905122408707975065, 1231427690626232295727]], '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': None, 'perfect': False, 'permutation_degree': 23, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': False, 'ratrep_stats': [[1, 4], [6, 8], [7, 12], [8, 4], [14, 12], [16, 2], [21, 8], [28, 4], [42, 8], [56, 24], [84, 24], [96, 4], [112, 22], [128, 2], [168, 36], [224, 12], [336, 32], [448, 8], [672, 6], [896, 2]], 'representations': {'Perm': {'d': 23, 'gens': [2306905122408707975064, 1231427690626232295727]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 13, 'solvable': False, 'subgroup_inclusions_known': False, 'subgroup_index_bound': 64, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^7.C_2^5.\\PSL(2,7)', 'transitive_degree': 42, 'wreath_data': None, 'wreath_product': False}
