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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '1056.940', 'ambient_counter': 940, 'ambient_order': 1056, 'ambient_tex': 'D_{12}:D_{22}', 'central': False, 'central_factor': True, 'centralizer_order': 8, 'characteristic': True, 'core_order': 264, 'counter': 21, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1056.940.4.l1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': None, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '4.l1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': '4.2', 'quotient_Agroup': True, 'quotient_abelian': True, 'quotient_cyclic': False, 'quotient_hash': 2, 'quotient_metabelian': True, 'quotient_nilpotent': True, 'quotient_order': 4, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'C_2^2', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '264.11', 'subgroup_hash': 11, 'subgroup_order': 264, 'subgroup_tex': 'C_{33}:Q_8', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1056.940', 'aut_centralizer_order': 24, 'aut_label': '4.l1', 'aut_quo_index': 1, 'aut_stab_index': 1, 'aut_weyl_group': None, 'aut_weyl_index': 24, 'centralizer': '132.a1', 'complements': ['264.h1'], 'conjugacy_class_count': 1, 'contained_in': ['2.a1'], 'contains': ['8.e1', '8.f1', '8.g1', '12.l1', '44.l1'], 'core': '4.l1', 'coset_action_label': None, 'count': 1, 'diagramx': [6845, 9620, 1491, 241], 'generators': [265, 528, 532, 18, 96], 'label': '1056.940.4.l1', 'mobius_quo': 0, 'mobius_sub': 2, 'normal_closure': '4.l1', 'normal_contained_in': ['2.a1'], 'normal_contains': ['8.e1', '8.f1', '8.g1'], 'normalizer': '1.a1', 'old_label': '4.l1', 'projective_image': '528.164', 'quotient_action_image': '4.2', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '4.l1', 'subgroup_fusion': None, 'weyl_group': '132.5'}
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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': 330, 'aut_gen_orders': [10, 22, 10, 2, 30], 'aut_gens': [[1, 2, 24], [9, 154, 216], [1, 82, 24], [17, 10, 192], [13, 134, 240], [21, 2, 168]], 'aut_group': None, 'aut_hash': 4845733366138083661, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 2640, 'aut_permdeg': 28, 'aut_perms': [112636314143147892871287292695, 44217293096044286061338725856, 300497921014537021989242925731, 78521236488916391127776035906, 254918034375796383825091923321], 'aut_phi_ratio': 33.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 1], [4, 6, 1, 1], [4, 22, 1, 1], [4, 66, 1, 1], [6, 2, 1, 1], [11, 2, 5, 1], [12, 22, 2, 1], [22, 2, 5, 1], [33, 4, 5, 1], [44, 6, 10, 1], [66, 4, 5, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_{11}:(C_2^2\\times C_{10}\\times S_3)', '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': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 330, 'autcentquo_group': '660.15', 'autcentquo_hash': 15, 'autcentquo_nilpotent': False, 'autcentquo_order': 660, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3\\times F_{11}', 'cc_stats': [[1, 1, 1], [2, 1, 1], [3, 2, 1], [4, 6, 1], [4, 22, 1], [4, 66, 1], [6, 2, 1], [11, 2, 5], [12, 22, 2], [22, 2, 5], [33, 4, 5], [44, 6, 10], [66, 4, 5]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '132.5', 'commutator_count': 1, 'commutator_label': '66.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '3.1', '11.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 11, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [3, 2, 1, 1], [4, 6, 1, 1], [4, 22, 1, 1], [4, 66, 1, 1], [6, 2, 1, 1], [11, 2, 5, 1], [12, 22, 2, 1], [22, 2, 5, 1], [33, 4, 5, 1], [44, 6, 10, 1], [66, 4, 5, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 6, 'exponent': 132, 'exponents_of_order': [3, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 11], 'faithful_reps': [[4, -1, 5]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '132.5', 'hash': 11, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 66, 'inner_gen_orders': [2, 6, 11], 'inner_gens': [[1, 22, 24], [5, 2, 240], [1, 50, 24]], 'inner_hash': 5, 'inner_nilpotent': False, 'inner_order': 132, 'inner_split': True, 'inner_tex': 'S_3\\times D_{11}', 'inner_used': [1, 2, 3], 'irrC_degree': 4, 'irrQ_degree': 20, 'irrQ_dim': 40, 'irrR_degree': 8, 'irrep_stats': [[1, 4], [2, 25], [4, 10]], 'label': '264.11', 'linC_count': 65, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 14, 'linQ_degree_count': 6, 'linQ_dim': 16, 'linQ_dim_count': 4, 'linR_count': 40, 'linR_degree': 6, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C33:Q8', 'ngens': 5, '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': 13, 'number_characteristic_subgroups': 14, 'number_conjugacy_classes': 39, 'number_divisions': 13, 'number_normal_subgroups': 14, 'number_subgroup_autclasses': 24, 'number_subgroup_classes': 24, 'number_subgroups': 136, 'old_label': None, 'order': 264, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 1], [3, 2], [4, 94], [6, 2], [11, 10], [12, 44], [22, 10], [33, 20], [44, 60], [66, 20]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 10, 'outer_gen_orders': [2, 10], 'outer_gen_pows': [0, 4], 'outer_gens': [[1, 10, 240], [9, 2, 168]], 'outer_group': '20.5', 'outer_hash': 5, 'outer_nilpotent': True, 'outer_order': 20, 'outer_permdeg': 9, 'outer_perms': [40320, 810], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times C_{10}', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 3], [4, 1], [10, 2], [20, 3]], 'representations': {'PC': {'code': 5015419889255009599100019, 'gens': [1, 2, 5], 'pres': [5, -2, -2, -2, -3, -11, 60, 221, 26, 302, 42, 323, 3009]}, 'GLZN': {'d': 2, 'p': 66, 'gens': [6612431, 433456, 12986313, 287893, 12413213]}, 'Perm': {'d': 22, 'gens': [56462890705918064047, 109750244592178713600, 163613728419837696000, 6706022400, 4364893]}}, 'schur_multiplier': [], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_{33}:Q_8', 'transitive_degree': 264, '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': '16.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 660, 'aut_gen_orders': [30, 60, 20, 10, 30, 20], 'aut_gens': [[1, 2, 12, 24], [7, 272, 714, 654], [3, 2, 780, 414], [529, 274, 756, 984], [529, 10, 636, 360], [5, 530, 828, 408], [533, 802, 900, 216]], 'aut_group': None, 'aut_hash': 6709908377699679727, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 63360, 'aut_permdeg': 34, 'aut_perms': [113936647647098396960056320805686379326, 258741739291806077402258517165206241274, 82115929450324270997848270034861892367, 60126761486738799796909464324319688430, 216362907859637179592504478392042652751, 209468832212250367007725386982198796893], 'aut_phi_ratio': 198.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 3, 1], [2, 22, 3, 1], [2, 66, 3, 1], [3, 2, 1, 1], [4, 2, 3, 1], [4, 6, 1, 1], [4, 22, 1, 1], [4, 66, 1, 1], [6, 2, 1, 1], [6, 44, 3, 1], [11, 2, 5, 1], [12, 4, 3, 1], [12, 22, 2, 1], [22, 2, 5, 1], [22, 12, 15, 1], [33, 4, 5, 1], [44, 4, 15, 1], [44, 6, 10, 1], [66, 4, 5, 1], [132, 8, 15, 1]], 'aut_supersolvable': False, 'aut_tex': '(C_{33}\\times A_4).C_5.C_2^5', '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': 330, 'autcentquo_group': None, 'autcentquo_hash': 6135319449156764412, 'autcentquo_nilpotent': False, 'autcentquo_order': 3960, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_{11}:(C_{10}\\times S_3^2)', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 6, 3], [2, 22, 3], [2, 66, 3], [3, 2, 1], [4, 2, 3], [4, 6, 1], [4, 22, 1], [4, 66, 1], [6, 2, 1], [6, 44, 3], [11, 2, 5], [12, 4, 3], [12, 22, 2], [22, 2, 5], [22, 12, 15], [33, 4, 5], [44, 4, 15], [44, 6, 10], [66, 4, 5], [132, 8, 15]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '528.164', 'commutator_count': 1, 'commutator_label': '66.4', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '11.1'], 'composition_length': 7, 'conjugacy_classes_known': True, 'counter': 940, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 6, 1, 3], [2, 22, 1, 3], [2, 66, 1, 3], [3, 2, 1, 1], [4, 2, 1, 3], [4, 6, 1, 1], [4, 22, 1, 1], [4, 66, 1, 1], [6, 2, 1, 1], [6, 44, 1, 3], [11, 2, 5, 1], [12, 4, 1, 3], [12, 22, 2, 1], [22, 2, 5, 1], [22, 12, 5, 3], [33, 4, 5, 1], [44, 4, 5, 3], [44, 6, 10, 1], [66, 4, 5, 1], [132, 8, 5, 3]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 5809440, 'exponent': 132, 'exponents_of_order': [5, 1, 1], 'factors_of_aut_order': [2, 3, 5, 11], 'factors_of_order': [2, 3, 11], 'faithful_reps': [[8, 1, 5]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '528.164', 'hash': 940, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 66, 'inner_gen_orders': [2, 6, 2, 22], 'inner_gens': [[1, 538, 12, 24], [5, 2, 12, 552], [1, 2, 12, 1032], [1, 530, 60, 24]], 'inner_hash': 164, 'inner_nilpotent': False, 'inner_order': 528, 'inner_split': True, 'inner_tex': 'D_6\\times D_{22}', 'inner_used': [1, 2, 3, 4], 'irrC_degree': 8, 'irrQ_degree': 40, 'irrQ_dim': 80, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 48], [4, 33], [8, 5]], 'label': '1056.940', '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': 'D12:D22', 'ngens': 7, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 22, 'number_characteristic_subgroups': 24, 'number_conjugacy_classes': 102, 'number_divisions': 40, 'number_normal_subgroups': 108, 'number_subgroup_autclasses': 132, 'number_subgroup_classes': 332, 'number_subgroups': 3928, 'old_label': None, 'order': 1056, 'order_factorization_type': 311, 'order_stats': [[1, 1], [2, 283], [3, 2], [4, 100], [6, 134], [11, 10], [12, 56], [22, 190], [33, 20], [44, 120], [66, 20], [132, 120]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 30, 'outer_gen_orders': [2, 2, 30], 'outer_gen_pows': [4, 264, 0], 'outer_gens': [[533, 2, 12, 504], [1, 266, 276, 504], [531, 268, 810, 750]], 'outer_group': '120.45', 'outer_hash': 45, 'outer_nilpotent': False, 'outer_order': 120, 'outer_permdeg': 12, 'outer_perms': [40320, 3628800, 43547329], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_{10}\\times D_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 22, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 8], [4, 1], [8, 1], [10, 8], [20, 4], [40, 2]], 'representations': {'PC': {'code': 19746279684243059227381153025889569180010009, 'gens': [1, 2, 4, 5], 'pres': [7, -2, -2, -3, -2, -2, -2, -11, 7533, 36, 11258, 2788, 9671, 3035, 102, 3554, 124, 3947]}, 'Perm': {'d': 22, 'gens': [51469039215450224047, 493380308827929600, 112305146080038912000, 153650588794230528000, 212155499332288512000, 6706022400, 4364893]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_{12}:D_{22}', 'transitive_degree': 264, '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': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 6, 'aut_gen_orders': [2, 3], 'aut_gens': [[1, 2], [3, 2], [2, 3]], 'aut_group': '6.1', 'aut_hash': 1, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6, 'aut_permdeg': 3, 'aut_perms': [1, 4], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'aut_supersolvable': True, 'aut_tex': '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': 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, 3]], 'center_label': '4.2', 'center_order': 4, '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'], 'composition_length': 2, 'conjugacy_classes_known': True, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'dihedral': True, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': True, 'inner_exponent': 1, 'inner_gen_orders': [1, 1], 'inner_gens': [[1, 2], [1, 2]], '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, 4]], 'label': '4.2', 'linC_count': 3, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 3, 'linQ_dim': 2, 'linQ_dim_count': 3, 'linR_count': 3, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': [0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[3, 2], [2, 3]], 'outer_group': '6.1', 'outer_hash': 1, 'outer_nilpotent': False, 'outer_order': 6, 'outer_permdeg': 3, 'outer_perms': [1, 4], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3', '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]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
