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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'ambient': '384.16399', 'ambient_counter': 16399, 'ambient_order': 384, 'ambient_tex': '(C_6\\times D_8):C_4', 'central': False, 'central_factor': False, 'centralizer_order': 96, 'characteristic': True, 'core_order': 12, 'counter': 112, 'cyclic': True, 'direct': False, 'hall': 0, 'label': '384.16399.32.e1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': True, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '32.e1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '32.45', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 45, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 32, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^3\\times C_4', 'simple': False, 'solvable': True, 'special_labels': ['D', 'L1', 'D1'], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '12.2', 'subgroup_hash': 2, 'subgroup_order': 12, 'subgroup_tex': 'C_{12}', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '384.16399', 'aut_centralizer_order': None, 'aut_label': '32.e1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '4.d1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['16.b1', '16.f1', '16.g1', '16.h1', '16.i1'], 'contains': ['64.b1', '96.e1'], 'core': '32.e1', 'coset_action_label': None, 'count': 1, 'diagramx': None, 'generators': [288, 16, 192], 'label': '384.16399.32.e1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '32.e1', 'normal_contained_in': ['16.b1', '16.f1', '16.g1', '16.h1', '16.i1'], 'normal_contains': ['64.b1', '96.e1'], 'normalizer': '1.a1', 'old_label': '32.e1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '32.e1', 'subgroup_fusion': None, 'weyl_group': None}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '12.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': True, 'aut_cyclic': False, 'aut_derived_length': 1, 'aut_exponent': 2, 'aut_gen_orders': [2, 2], 'aut_gens': [[1], [5], [7]], 'aut_group': '4.2', 'aut_hash': 2, 'aut_nilpotency_class': 1, 'aut_nilpotent': True, 'aut_order': 4, 'aut_permdeg': 4, 'aut_perms': [1, 6], 'aut_phi_ratio': 1.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [6, 1, 2, 1], [12, 1, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^2', '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': 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, 1], [3, 1, 2], [4, 1, 2], [6, 1, 2], [12, 1, 4]], 'center_label': '12.2', 'center_order': 12, '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', '3.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': True, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['3.1', 1], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 2, 1], [4, 1, 2, 1], [6, 1, 2, 1], [12, 1, 4, 1]], 'element_repr_type': 'PC', 'elementary': 6, 'eulerian_function': 1, 'exponent': 12, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2], 'factors_of_order': [2, 3], 'faithful_reps': [[1, 0, 4]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '6.2', 'hash': 2, 'hyperelementary': 6, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1], 'inner_gens': [[1]], 'inner_hash': 1, 'inner_nilpotent': True, 'inner_order': 1, 'inner_split': True, 'inner_tex': 'C_1', 'inner_used': [], 'irrC_degree': 1, 'irrQ_degree': 4, 'irrQ_dim': 4, 'irrR_degree': 2, 'irrep_stats': [[1, 12]], 'label': '12.2', 'linC_count': 4, 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 3, 'linQ_dim': 4, 'linQ_dim_count': 3, 'linR_count': 2, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C12', 'ngens': 3, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 6, 'number_characteristic_subgroups': 6, 'number_conjugacy_classes': 12, 'number_divisions': 6, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 6, 'number_subgroups': 6, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 2], [6, 2], [12, 4]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2, 2], 'outer_gen_pows': [0, 0], 'outer_gens': [[5], [7]], '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': 1, 'perfect': False, 'permutation_degree': 7, 'pgroup': 0, 'primary_abelian_invariants': [4, 3], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 3], [4, 1]], 'representations': {'PC': {'code': 3865, 'gens': [1], 'pres': [3, -2, -2, -3, 6, 16]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [20970031]}, 'GLFp': {'d': 2, 'p': 5, 'gens': [619]}, 'Perm': {'d': 7, 'gens': [2400, 4, 744]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [12], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{12}', 'transitive_degree': 12, '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': 2, 'aut_exponent': 24, 'aut_gen_orders': [12, 24, 24, 4, 8, 2, 12, 8], 'aut_gens': [[1, 4, 8, 48], [131, 172, 202, 338], [307, 172, 200, 242], [105, 268, 8, 48], [323, 124, 232, 338], [217, 148, 200, 240], [227, 196, 42, 336], [137, 220, 200, 240], [105, 270, 40, 242]], 'aut_group': '2304.mr', 'aut_hash': 77725528710035942, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 49152, 'aut_permdeg': 80, 'aut_perms': [25325922187374790716897865279284023695810030426007519325222971678555700316930721134460753545964608974784322500886514175, 37738626250667243400090034281670302434288001311919733487764149809536011992626237628091932226789357233979609410184885136, 69201123915436103136685402496115726502476091362841838177009866561322118211449485752704232421747363380632236078932926153, 65092656478760072406797268818370091637391959207735942188907293197990986439723416806497938234433594667258549541953421831, 55609795187984690490074703915446140444513489572210156366423699090412675830197678766438105196690343615093060235617978806, 13525425714007736389855335379242956520978702301713199107206008846295433020931767170236117245022759157278274245779001689, 43011895489514534270605743046254027081996897996973206223479146561517218986519987946784745898561222412093200297843472785, 28313394593590848119473308930739187508386619108344388554406437748965424403325681781744041913047982964307353731868885456], 'aut_phi_ratio': 384.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 4, 8, 1], [3, 2, 1, 1], [4, 2, 1, 2], [4, 2, 2, 1], [4, 6, 8, 1], [4, 12, 8, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 2, 4, 1], [6, 8, 8, 1], [8, 2, 4, 1], [8, 4, 2, 1], [8, 12, 4, 1], [12, 4, 1, 2], [12, 4, 2, 1], [24, 4, 4, 2]], 'aut_supersolvable': True, 'aut_tex': 'C_6\\times C_2^4:S_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '256.56092', 'autcent_hash': 56092, 'autcent_nilpotent': True, 'autcent_order': 256, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^8', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '192.1514', 'autcentquo_hash': 1514, 'autcentquo_nilpotent': False, 'autcentquo_order': 192, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{12}:C_2^4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [2, 4, 8], [3, 2, 1], [4, 2, 4], [4, 6, 8], [4, 12, 8], [6, 2, 7], [6, 8, 8], [8, 2, 4], [8, 4, 2], [8, 12, 4], [12, 4, 4], [24, 4, 8]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '96.209', 'commutator_count': 1, 'commutator_label': '12.2', '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'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 16399, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 4, 1, 8], [3, 2, 1, 1], [4, 2, 1, 4], [4, 6, 2, 4], [4, 12, 2, 4], [6, 2, 1, 3], [6, 2, 2, 2], [6, 8, 1, 8], [8, 2, 2, 2], [8, 4, 1, 2], [8, 12, 2, 2], [12, 4, 1, 4], [24, 4, 2, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 131040, 'exponent': 24, 'exponents_of_order': [7, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '8.2', 'frattini_quotient': '48.51', 'hash': 16399, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 6, 4], 'inner_gens': [[1, 4, 232, 48], [1, 4, 8, 336], [209, 4, 8, 48], [1, 100, 8, 48]], 'inner_hash': 209, 'inner_nilpotent': False, 'inner_order': 96, 'inner_split': False, 'inner_tex': 'D_4\\times D_6', 'inner_used': [1, 2, 3, 4], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 32], [2, 24], [4, 16]], 'label': '384.16399', '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': '(C6*D8):C4', 'ngens': 8, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 23, 'number_characteristic_subgroups': 31, 'number_conjugacy_classes': 72, 'number_divisions': 54, 'number_normal_subgroups': 221, 'number_subgroup_autclasses': 156, 'number_subgroup_classes': 528, 'number_subgroups': 1534, 'old_label': None, 'order': 384, 'order_factorization_type': 51, 'order_stats': [[1, 1], [2, 39], [3, 2], [4, 152], [6, 78], [8, 64], [12, 16], [24, 32]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 4, 'outer_gen_orders': [4, 2, 4, 4, 4, 4], 'outer_gen_pows': [0, 48, 32, 0, 0, 0], 'outer_gens': [[209, 246, 234, 338], [123, 76, 232, 240], [305, 126, 10, 50], [217, 196, 40, 50], [305, 126, 40, 242], [131, 30, 234, 50]], 'outer_group': '512.554852', 'outer_hash': 342028529733335859, 'outer_nilpotent': True, 'outer_order': 512, 'outer_permdeg': 64, 'outer_perms': [49116753021834969415457974817642303822341629679936597269500253675262018115131217854324804, 38491069700616731613885928757911513239420268479790818095632778324429188133322251085452739, 19752278409905974174035987017447314104840518641781757671279472299919274609770297635184712, 13763188445504584964349036056280053182419974803545057271961848545223128504163645018112819, 104596468713609471969534590775735163157434369760806567425191686522287613487506992933186420, 3931178583723290297293269908147281016371592272511188427261208687650522143313480472366217], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4.C_2^5', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 23, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 4], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 28], [4, 6], [8, 2], [16, 2]], 'representations': {'PC': {'code': 10914695718026757047911251863816, 'gens': [1, 3, 4, 6], 'pres': [8, -2, -2, -2, -2, -3, -2, -2, -2, 16, 7427, 91, 1284, 4053, 141, 4054, 166]}, 'Perm': {'d': 23, 'gens': [1139598738262160507649, 2453505025166418101776, 1131805242336905297418, 258018868696477109760, 3696002616839408910720, 16, 4772140498719079315200, 840]}}, 'schur_multiplier': [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_6\\times D_8):C_4', 'transitive_degree': 96, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '32.45', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 0, 'aut_exponent': 168, 'aut_gen_orders': [7, 3], 'aut_gens': [[1, 2, 4, 8], [20, 19, 2, 14], [1, 22, 19, 28]], 'aut_group': '21504.y', 'aut_hash': 2095861522221326572, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 21504, 'aut_permdeg': 16, 'aut_perms': [19981329134312, 7769561399125], 'aut_phi_ratio': 1344.0, 'aut_solvable': False, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 14, 1], [4, 1, 16, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_2^4:C_2^3:\\GL(3,2)', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 168, 'autcent_group': '21504.y', 'autcent_hash': 2095861522221326572, 'autcent_nilpotent': False, 'autcent_order': 21504, 'autcent_solvable': False, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': 'C_2^4:C_2^3:\\GL(3,2)', '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, 15], [4, 1, 16]], 'center_label': '32.45', 'center_order': 32, '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', '2.1', '2.1', '2.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 45, 'cyclic': False, 'derived_length': 1, 'dihedral': False, 'direct_factorization': [['2.1', 3], ['4.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 15], [4, 1, 2, 8]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 15, 'exponent': 4, 'exponents_of_order': [5], 'factors_of_aut_order': [2, 3, 7], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '16.14', 'hash': 45, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1, 1, 1], 'inner_gens': [[1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 8]], '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, 32]], 'label': '32.45', 'linC_count': 13440, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 1792, 'linQ_dim': 5, 'linQ_dim_count': 1792, 'linR_count': 1792, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^3*C4', 'ngens': 4, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 32, 'number_divisions': 24, 'number_normal_subgroups': 118, 'number_subgroup_autclasses': 12, 'number_subgroup_classes': 118, 'number_subgroups': 118, 'old_label': None, 'order': 32, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 15], [4, 16]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 168, 'outer_gen_orders': [7, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[20, 19, 2, 14], [1, 22, 19, 28]], 'outer_group': '21504.y', 'outer_hash': 2095861522221326572, 'outer_nilpotent': False, 'outer_order': 21504, 'outer_permdeg': 16, 'outer_perms': [19981329134312, 7769561399125], 'outer_solvable': False, 'outer_supersolvable': False, 'outer_tex': 'C_2^4:C_2^3:\\GL(3,2)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 10, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 2, 4], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 8]], 'representations': {'PC': {'code': 16392, 'gens': [1, 2, 3, 4], 'pres': [5, -2, 2, 2, 2, -2, 58]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [141602720900, 705686952884, 706461792892, 706460730980]}, 'GLFp': {'d': 4, 'p': 5, 'gens': [122109387504, 122297461254, 30594431879, 30704310001, 30590220001]}, 'Perm': {'d': 10, 'gens': [22, 362880, 5040, 120, 7]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 4], 'solvability_type': 2, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^3\\times C_4', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
