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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1024.dih', 'ambient_counter': 2244, 'ambient_order': 1024, 'ambient_tex': 'C_4^2.C_2\\wr C_4', 'central': False, 'central_factor': False, 'centralizer_order': 8, 'characteristic': True, 'core_order': 128, 'counter': 14, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1024.dih.8.c1.a1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': True, 'normal': True, 'old_label': '8.c1.a1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '8.3', 'quotient_Agroup': False, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 3, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 8, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_4', 'simple': False, 'solvable': True, 'special_labels': [], 'split': False, 'standard_generators': False, 'stem': False, 'subgroup': '128.580', 'subgroup_hash': 580, 'subgroup_order': 128, 'subgroup_tex': 'C_4^2.D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1024.dih', 'aut_centralizer_order': None, 'aut_label': '8.c1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '128.b1.a1', 'complements': [], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '4.f1.a1', '4.g1.a1'], 'contains': ['16.a1.a1', '16.k1.a1', '16.l1.a1'], 'core': '8.c1.a1', 'coset_action_label': None, 'count': 1, 'diagramx': [6284, 5479, 5029, 5465, 6263, 5688, 5104, 5690], 'generators': [2, 40, 800], 'label': '1024.dih.8.c1.a1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '8.c1.a1', 'normal_contained_in': ['4.a1.a1'], 'normal_contains': ['16.a1.a1'], 'normalizer': '1.a1.a1', 'old_label': '8.c1.a1', 'projective_image': None, 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '8.c1.a1', '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': '16.10', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 8, 'aut_gen_orders': [2, 2, 4, 4, 2, 2, 2, 2, 2, 4, 4, 8], 'aut_gens': [[1, 4, 16], [9, 76, 112], [65, 68, 80], [107, 108, 56], [67, 76, 118], [11, 76, 24], [1, 4, 24], [9, 4, 16], [1, 4, 80], [3, 68, 16], [105, 4, 16], [111, 4, 16], [29, 68, 16]], 'aut_group': '8192.xr', 'aut_hash': 321439569190445723, 'aut_nilpotency_class': 5, 'aut_nilpotent': True, 'aut_order': 8192, 'aut_permdeg': 64, 'aut_perms': [6382826291138653554064403237684880946559214595854113308470577290474783806205814997461915, 1951654802941483242724066409337910538833069058931548738248399773203795026350478892933207, 23481750491273717399967886340474360160184858778971989973299472997152560844471852722042890, 53736698121981710667986001048751537040432128177979731903438980638858641152213367297136505, 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'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '128.2328', 'autcent_hash': 2328, 'autcent_nilpotent': True, 'autcent_order': 128, 'autcent_solvable': True, 'autcent_split': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^7', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 4, 'autcentquo_group': '64.138', 'autcentquo_hash': 138, 'autcentquo_nilpotent': True, 'autcentquo_order': 64, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2\\wr C_2^2', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 2, 2], [4, 1, 4], [4, 2, 10], [8, 2, 16], [8, 8, 8]], 'center_label': '8.2', 'center_order': 8, 'central_product': False, 'central_quotient': '16.11', 'commutator_count': 1, 'commutator_label': '8.2', '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'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 580, '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], [4, 1, 2, 2], [4, 2, 1, 6], [4, 2, 2, 2], [8, 2, 2, 8], [8, 8, 2, 4]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 84, 'exponent': 8, 'exponents_of_order': [7], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [], 'familial': False, 'frattini_label': '16.10', 'frattini_quotient': '8.5', 'hash': 580, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 4, 'inner_gen_orders': [2, 2, 4], 'inner_gens': [[1, 76, 112], [73, 4, 16], [33, 4, 16]], 'inner_hash': 11, 'inner_nilpotent': True, 'inner_order': 16, 'inner_split': False, 'inner_tex': 'C_2\\times D_4', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 16], [2, 28]], 'label': '128.580', 'linC_count': 192, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 10, 'linQ_degree_count': 32, 'linQ_dim': 10, 'linQ_dim_count': 16, 'linR_count': 32, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4^2.D4', 'ngens': 3, 'nilpotency_class': 3, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 12, 'number_characteristic_subgroups': 16, 'number_conjugacy_classes': 44, 'number_divisions': 28, 'number_normal_subgroups': 72, 'number_subgroup_autclasses': 46, 'number_subgroup_classes': 110, 'number_subgroups': 156, 'old_label': None, 'order': 128, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 7], [4, 24], [8, 96]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 2, 2, 2, 2, 2], 'outer_gen_pows': [0, 1, 96, 116, 4, 0, 0, 4, 0], 'outer_gens': [[3, 12, 16], [3, 36, 54], [63, 108, 16], [85, 68, 80], [5, 68, 80], [1, 68, 24], [1, 68, 16], [71, 4, 16], [1, 4, 80]], 'outer_group': '512.6344809', 'outer_hash': 3030424736949194728, 'outer_nilpotent': True, 'outer_order': 512, 'outer_permdeg': 14, 'outer_perms': [7620480, 46678429135, 32736493495, 83467704, 6314117329, 87091200, 6234641280, 6314117328, 6314117040], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^6:D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 20, 'pgroup': 2, 'primary_abelian_invariants': [2, 2, 4], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 16], [8, 4]], 'representations': {'PC': {'code': 77370292940516920133303, 'gens': [1, 3, 5], 'pres': [7, 2, 2, 2, 2, 2, 2, 2, 14, 456, 1598, 58, 3924, 102, 4037, 124]}, 'GLZN': {'d': 2, 'p': 48, 'gens': [110675, 111745, 1880081, 2802473, 110617, 2545103, 111169]}, 'GLZq': {'d': 2, 'q': 16, 'gens': [15033, 37993, 36873, 12483, 14467, 22669, 4225]}, 'Perm': {'d': 20, 'gens': [149961924304955401, 276169695258241567, 406261760842430777, 406261760842430760, 534682123137203047, 650199434828076120, 534682123137203040]}}, 'schur_multiplier': [2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4^2.D_4', 'transitive_degree': 64, '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': '8.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 16, 'aut_gen_orders': [4, 4, 8, 8, 4, 8, 8], 'aut_gens': [[1, 4, 32, 128], [97, 876, 32, 944], [529, 332, 32, 944], [843, 964, 96, 976], [57, 756, 32, 440], [817, 516, 32, 192], [243, 124, 32, 200], [619, 276, 96, 456]], 'aut_group': None, 'aut_hash': 6035287362225783817, 'aut_nilpotency_class': 6, 'aut_nilpotent': True, 'aut_order': 32768, 'aut_permdeg': 384, 'aut_perms': 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121116202515857131693175270394903060287782326590893427296884840546681502782847368983377637768094054727219491726515702673805784498837396501777188262554444986790555631449357518333798713007489182546289972581580364585743475825813619222495595712493318454287667253198833844530375034317958509908908435206370373258256078415499847206318884780246526904798544360876566717527029131620482740436583019525421386916797435100502808850086696990085101693881497931301906515858124252546846274535451060367964363641602104203482897877334686246562581383486586635876468814826593127296784223513731306053954367316877363549980508930334251293702663360564612238381269994632698656074816324761957768297739449627310921295252255686834588472794674837360993842814552903794442529919042471615114499477268268397042357835615768844163484675079505229156758381951695335316], 'aut_phi_ratio': 64.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 1], [2, 32, 1, 1], [2, 64, 2, 1], [4, 2, 1, 2], [4, 4, 1, 1], [4, 4, 2, 1], [4, 8, 1, 1], [4, 32, 1, 1], [4, 64, 4, 1], [8, 4, 4, 1], [8, 8, 2, 1], [8, 8, 8, 1], [8, 32, 2, 1], [16, 16, 8, 1], [16, 64, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_4^3.C_2^4.C_2^5', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 8, 'autcentquo_group': None, 'autcentquo_hash': 6208140336951712643, 'autcentquo_nilpotent': True, 'autcentquo_order': 8192, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2^3.C_2^6.C_2^4', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 4, 1], [2, 32, 1], [2, 64, 2], [4, 2, 2], [4, 4, 3], [4, 8, 1], [4, 32, 1], [4, 64, 4], [8, 4, 4], [8, 8, 10], [8, 32, 2], [16, 16, 8], [16, 64, 4]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '512.1838', 'commutator_count': 2, 'commutator_label': '128.179', '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', '2.1', '2.1', '2.1'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 2244, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 1], [2, 32, 1, 1], [2, 64, 1, 2], [4, 2, 1, 2], [4, 4, 1, 3], [4, 8, 1, 1], [4, 32, 1, 1], [4, 64, 2, 2], [8, 4, 2, 2], [8, 8, 1, 2], [8, 8, 2, 4], [8, 32, 2, 1], [16, 16, 2, 4], [16, 64, 4, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 12, 'exponent': 16, 'exponents_of_order': [10], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[8, 1, 8]], 'familial': False, 'frattini_label': '256.2844', 'frattini_quotient': '4.2', 'hash': 505083576705387077, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 8, 'inner_gen_orders': [8, 8, 2, 8], 'inner_gens': [[1, 148, 32, 440], [977, 4, 96, 160], [1, 68, 32, 128], [809, 100, 32, 128]], 'inner_hash': 1034948244345798672, 'inner_nilpotent': True, 'inner_order': 512, 'inner_split': None, 'inner_tex': 'C_4^2.C_4\\wr C_2', 'inner_used': [1, 2], 'irrC_degree': 8, 'irrQ_degree': 16, 'irrQ_dim': 16, 'irrR_degree': 8, 'irrep_stats': [[1, 8], [2, 10], [4, 17], [8, 11]], 'label': '1024.dih', 'linC_count': 8, 'linC_degree': 8, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 16, 'linQ_degree_count': 4, 'linQ_dim': 16, 'linQ_dim_count': 4, 'linR_count': 8, 'linR_degree': 8, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C4^2.C2wrC4', 'ngens': 2, 'nilpotency_class': 7, 'nilpotent': True, 'normal_counts': [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': 19, 'number_characteristic_subgroups': 27, 'number_conjugacy_classes': 46, 'number_divisions': 30, 'number_normal_subgroups': 27, 'number_subgroup_autclasses': 213, 'number_subgroup_classes': 363, 'number_subgroups': 3813, 'old_label': None, 'order': 1024, 'order_factorization_type': 7, 'order_stats': [[1, 1], [2, 167], [4, 312], [8, 160], [16, 384]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2, 4], 'outer_gen_pows': [96, 312, 362, 112, 608], 'outer_gens': [[625, 36, 32, 192], [643, 28, 32, 136], [73, 660, 32, 504], [353, 532, 32, 640], [859, 468, 96, 400]], 'outer_group': '64.193', 'outer_hash': 193, 'outer_nilpotent': True, 'outer_order': 64, 'outer_permdeg': 32, 'outer_perms': [61359211852608767692984943193077998, 65480704795541004387875626219965596, 113324326726187840421577805918431882, 225069911148678500642314332909604560, 52387529969665291155189941225975086], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4:C_4', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 32, 'pgroup': 2, 'primary_abelian_invariants': [2, 4], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 4], [4, 9], [8, 9], [16, 4]], 'representations': {'PC': {'code': '237605593520588123342671235491883536806422556954066425666572975069626498067439039', 'gens': [1, 3, 6, 8], 'pres': [10, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 20, 971, 4442, 912, 82, 14403, 1773, 113, 29604, 2014, 244, 1465, 175, 35207, 39697, 3227, 237, 70568, 37458, 7228, 268, 57609]}, 'Perm': {'d': 32, 'gens': [76209879730140522703248896331659647, 93185859240955738096818815502524759]}}, 'schur_multiplier': [2, 4], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 4], 'solvability_type': 4, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4^2.C_2\\wr C_4', 'transitive_degree': 32, 'wreath_data': None, 'wreath_product': False}
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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': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 6], [3, 2]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [5, 9], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [4, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', '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': 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, 1], [2, 2, 2], [4, 2, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, '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'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [4, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 6], [5, 2]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 1]], 'label': '8.3', 'linC_count': 1, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D4', 'ngens': 2, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [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': 5, 'number_divisions': 5, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 10, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 5], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [2], 'outer_gens': [[3, 2]], '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': 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], [2, 1]], 'representations': {'PC': {'code': 294, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 37, 16]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [29, 56, 24], 'family': 'COPlus'}, {'d': 1, 'q': 4, 'gens': [7, 16, 1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [12, 55, 56]}, 'Perm': {'d': 4, 'gens': [6, 16, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4', 'transitive_degree': 4, 'wreath_data': ['C_2', 'C_2', '2T1'], 'wreath_product': True}