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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '10368.qo', 'ambient_counter': 431, 'ambient_order': 10368, 'ambient_tex': '\\SOPlus(4,2)^2.C_2', 'central': False, 'central_factor': False, 'centralizer_order': 9, 'characteristic': True, 'core_order': 1296, 'counter': 61, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '10368.qo.8.o1', 'maximal': False, 'maximal_normal': False, 'metabelian': False, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '8.o1', 'outer_equivalence': True, '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': True, 'standard_generators': False, 'stem': False, 'subgroup': '1296.3510', 'subgroup_hash': 3510, 'subgroup_order': 1296, 'subgroup_tex': 'C_3^4:\\SD_{16}', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '10368.qo', 'aut_centralizer_order': None, 'aut_label': '8.o1', 'aut_quo_index': None, 'aut_stab_index': None, 'aut_weyl_group': None, 'aut_weyl_index': None, 'centralizer': '1152.a1', 'complements': ['1296.bu1', '1296.bt1', '1296.bv1', '1296.k1', '1296.br1', '1296.j1', '1296.bs1', '1296.l1'], 'conjugacy_class_count': 1, 'contained_in': ['4.i1', '4.x1'], 'contains': ['16.h1', '16.j1', '16.l1', '24.bk1', '72.ch1'], 'core': '8.o1', 'coset_action_label': None, 'count': 1, 'diagramx': [7983, 8792, 6937, 8493], 'generators': [716282305200, 8237376000, 480175920, 39330028080, 21675270960, 15, 373, 555770033280], 'label': '10368.qo.8.o1', 'mobius_quo': 0, 'mobius_sub': 0, 'normal_closure': '8.o1', 'normal_contained_in': ['4.i1'], 'normal_contains': ['16.h1', '16.j1', '16.l1', '72.a1'], 'normalizer': '1.a1', 'old_label': '8.o1', 'projective_image': '10368.qo', 'quotient_action_image': '8.3', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '8.o1', 'subgroup_fusion': None, 'weyl_group': '1152.119999'}
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gps_groups • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '36.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 4, 'aut_exponent': 24, 'aut_gen_orders': [6, 6, 8, 8], 'aut_gens': [[1, 6, 144, 432], [437, 682, 288, 864], [267, 646, 864, 288], [1097, 1172, 1152, 720], [1093, 824, 1152, 1008]], 'aut_group': None, 'aut_hash': 8786330501354108419, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 6912, 'aut_permdeg': 38, 'aut_perms': [67193214701799457214848198431288964783874542, 301076424372447345730453301828787478492516754, 393952844133509422815991384448857473292040774, 332945062717043651707083121284606228688881330], 'aut_phi_ratio': 16.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 12, 1, 1], [3, 1, 8, 1], [3, 8, 1, 1], [3, 8, 8, 1], [4, 18, 1, 1], [4, 36, 1, 1], [6, 9, 8, 1], [6, 12, 8, 1], [6, 24, 1, 1], [6, 24, 8, 1], [8, 18, 1, 2], [12, 18, 8, 1], [12, 36, 8, 1], [24, 18, 8, 2]], 'aut_supersolvable': False, 'aut_tex': '(C_3\\times C_6).(D_4\\times S_4).C_2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 24, 'autcent_group': '48.29', 'autcent_hash': 29, 'autcent_nilpotent': False, 'autcent_order': 48, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': False, 'autcent_tex': '\\GL(2,3)', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 24, 'autcentquo_group': '144.182', 'autcentquo_hash': 182, 'autcentquo_nilpotent': False, 'autcentquo_order': 144, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'F_9:C_2', 'cc_stats': [[1, 1, 1], [2, 9, 1], [2, 12, 1], [3, 1, 8], [3, 8, 9], [4, 18, 1], [4, 36, 1], [6, 9, 8], [6, 12, 8], [6, 24, 9], [8, 18, 2], [12, 18, 8], [12, 36, 8], [24, 18, 16]], 'center_label': '9.2', 'center_order': 9, 'central_product': True, 'central_quotient': '144.182', 'commutator_count': 1, 'commutator_label': '36.9', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1', '3.1'], 'composition_length': 8, 'conjugacy_classes_known': True, 'counter': 3510, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['144.182', 1], ['3.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 9, 1, 1], [2, 12, 1, 1], [3, 1, 2, 4], [3, 8, 1, 1], [3, 8, 2, 4], [4, 18, 1, 1], [4, 36, 1, 1], [6, 9, 2, 4], [6, 12, 2, 4], [6, 24, 1, 1], [6, 24, 2, 4], [8, 18, 2, 1], [12, 18, 2, 4], [12, 36, 2, 4], [24, 18, 4, 4]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 48, 'exponent': 24, 'exponents_of_order': [4, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '1296.3510', 'hash': 3510, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [2, 8, 3, 3], 'inner_gens': [[1, 114, 144, 864], [37, 6, 1152, 1008], [1, 726, 144, 432], [865, 1158, 144, 432]], 'inner_hash': 182, 'inner_nilpotent': False, 'inner_order': 144, 'inner_split': True, 'inner_tex': 'F_9:C_2', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 36], [2, 27], [8, 18]], 'label': '1296.3510', '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': True, 'name': 'C3^4:SD16', '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, 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': 16, 'number_conjugacy_classes': 81, 'number_divisions': 40, 'number_normal_subgroups': 48, 'number_subgroup_autclasses': 84, 'number_subgroup_classes': 185, 'number_subgroups': 1404, 'old_label': None, 'order': 1296, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 21], [3, 80], [4, 54], [6, 384], [8, 36], [12, 432], [24, 288]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 24, 'outer_gen_orders': [2, 3], 'outer_gen_pows': [0, 0], 'outer_gens': [[5, 8, 288, 864], [53, 58, 144, 432]], 'outer_group': '48.29', 'outer_hash': 29, 'outer_nilpotent': False, 'outer_order': 48, 'outer_permdeg': 8, 'outer_perms': [36887, 21251], 'outer_solvable': True, 'outer_supersolvable': False, 'outer_tex': '\\GL(2,3)', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 15, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 17], [4, 5], [8, 6], [16, 8]], 'representations': {'PC': {'code': 197949893735698911509618857050639637444915248700743, 'gens': [1, 3, 7, 8], 'pres': [8, -2, -3, -2, -2, -2, -3, -3, 3, 16, 2738, 66, 2691, 91, 116, 10774, 4062, 710, 55303, 10775, 799, 2343]}, 'Perm': {'d': 15, 'gens': [13986206640, 19320900480, 147, 3, 7355945520, 20797746480, 106978152960, 188857635120]}}, 'schur_multiplier': [3], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6, 6], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_3^4:\\SD_{16}', 'transitive_degree': 72, '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': 24, 'aut_gen_orders': [8, 8, 12, 8, 4], 'aut_gens': [[55, 107935072560, 208417633200, 186, 20, 25032193516, 8237376000, 14936145840, 13579781094], [403, 208417633200, 350350479360, 15, 378, 387519597381, 289397203200, 262660970880, 697625994556], [67, 208417633200, 555770033280, 181, 15, 619350203338, 268931537280, 270726986880, 182111695974], [60, 655840649520, 450751835520, 373, 366, 264017336339, 120233433600, 94933036800, 623823419534], [60, 208417633200, 555770033280, 174, 378, 94455209574, 697625994240, 374151461040, 289397203859], [403, 299510305920, 655840649520, 15, 186, 46463558421, 13579781040, 14936145840, 8237376719]], 'aut_group': None, 'aut_hash': 2723700116397070647, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20736, 'aut_permdeg': 42, 'aut_perms': [130464360219826856029674814068861990777124882991480, 344278660541932616747885775268561816191527414885975, 347622154334153664000197527635309835759639941002759, 310139888288768074471352824155073463450625365151793, 495964547811237692664051156303754274909148729688058], 'aut_phi_ratio': 6.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 6, 2, 1], [2, 9, 1, 2], [2, 12, 1, 1], [2, 54, 2, 1], [2, 72, 2, 1], [2, 81, 1, 1], [2, 108, 1, 1], [3, 4, 2, 1], [3, 8, 1, 1], [3, 32, 2, 1], [4, 18, 1, 2], [4, 36, 1, 1], [4, 108, 2, 1], [4, 162, 1, 2], [4, 216, 1, 1], [4, 216, 2, 1], [4, 324, 1, 2], [4, 648, 1, 1], [6, 12, 2, 1], [6, 24, 1, 1], [6, 36, 2, 1], [6, 48, 2, 2], [6, 72, 1, 1], [6, 96, 2, 2], [6, 108, 2, 1], [6, 144, 2, 2], [6, 216, 1, 1], [6, 288, 2, 1], [8, 18, 1, 2], [8, 108, 2, 2], [8, 162, 1, 2], [8, 324, 1, 2], [12, 72, 2, 1], [12, 144, 1, 1], [12, 144, 2, 1], [12, 216, 2, 1], [12, 432, 1, 1], [12, 432, 2, 1], [24, 72, 2, 2], [24, 216, 2, 2]], 'aut_supersolvable': False, 'aut_tex': 'C_3^4.C_4^2.C_2^4', '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': 2723700116397070647, 'autcentquo_nilpotent': False, 'autcentquo_order': 20736, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'C_3^4.C_4^2.C_2^4', 'cc_stats': [[1, 1, 1], [2, 6, 2], [2, 9, 2], [2, 12, 1], [2, 54, 2], [2, 72, 2], [2, 81, 1], [2, 108, 1], [3, 4, 2], [3, 8, 1], [3, 32, 2], [4, 18, 2], [4, 36, 1], [4, 108, 2], [4, 162, 2], [4, 216, 3], [4, 324, 2], [4, 648, 1], [6, 12, 2], [6, 24, 1], [6, 36, 2], [6, 48, 4], [6, 72, 1], [6, 96, 4], [6, 108, 2], [6, 144, 4], [6, 216, 1], [6, 288, 2], [8, 18, 2], [8, 108, 4], [8, 162, 2], [8, 324, 2], [12, 72, 2], [12, 144, 3], [12, 216, 2], [12, 432, 3], [24, 72, 4], [24, 216, 4]], 'center_label': '1.1', 'center_order': 1, 'central_product': True, 'central_quotient': '10368.qo', 'commutator_count': 1, 'commutator_label': '648.715', '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'], 'composition_length': 11, 'conjugacy_classes_known': True, 'counter': 431, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['144.182', 1], ['72.40', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 6, 1, 2], [2, 9, 1, 2], [2, 12, 1, 1], [2, 54, 1, 2], [2, 72, 1, 2], [2, 81, 1, 1], [2, 108, 1, 1], [3, 4, 1, 2], [3, 8, 1, 1], [3, 32, 1, 2], [4, 18, 1, 2], [4, 36, 1, 1], [4, 108, 1, 2], [4, 162, 1, 2], [4, 216, 1, 3], [4, 324, 1, 2], [4, 648, 1, 1], [6, 12, 1, 2], [6, 24, 1, 1], [6, 36, 1, 2], [6, 48, 1, 4], [6, 72, 1, 1], [6, 96, 1, 4], [6, 108, 1, 2], [6, 144, 1, 4], [6, 216, 1, 1], [6, 288, 1, 2], [8, 18, 2, 1], [8, 108, 2, 2], [8, 162, 2, 1], [8, 324, 2, 1], [12, 72, 1, 2], [12, 144, 1, 3], [12, 216, 1, 2], [12, 432, 1, 3], [24, 72, 2, 2], [24, 216, 2, 2]], 'element_repr_type': 'Perm', 'elementary': 1, 'eulerian_function': 170951639040, 'exponent': 24, 'exponents_of_order': [7, 4], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[32, 1, 8]], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '10368.qo', 'hash': 4072227654749320783, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 24, 'inner_gen_orders': [2, 3, 3, 3, 3, 8, 8, 4, 8], 'inner_gens': [[55, 107935072560, 208417633200, 373, 15, 25032193516, 8237376000, 14936145840, 13579781094], [55, 107935072560, 208417633200, 186, 20, 619350203116, 268931537280, 181633132800, 182111695974], [55, 107935072560, 208417633200, 186, 20, 455730186556, 374630140080, 287601637680, 548951437494], [396, 107935072560, 208417633200, 186, 20, 25032193798, 8237376000, 14936145840, 13579781430], [60, 107935072560, 208417633200, 186, 20, 25032193859, 8237376000, 14936145840, 13579781094], [55, 450751835520, 107935072560, 373, 366, 25032193516, 8237376000, 43789823280, 13579781041], [55, 208417633200, 555770033280, 186, 20, 25032193516, 8237376000, 7320463920, 13579781094], [55, 299510305920, 655840649520, 186, 20, 46463558716, 13579781040, 14936145840, 8237376054], [55, 299510305920, 350350479360, 378, 20, 25032193541, 8237376000, 43789823280, 13579781094]], 'inner_hash': 4072227654749320783, 'inner_nilpotent': False, 'inner_order': 10368, 'inner_split': False, 'inner_tex': '\\SOPlus(4,2)^2.C_2', 'inner_used': [1, 2, 3, 4, 5, 6, 7, 8, 9], 'irrC_degree': 32, 'irrQ_degree': 32, 'irrQ_dim': 32, 'irrR_degree': 32, 'irrep_stats': [[1, 16], [2, 16], [4, 19], [8, 20], [16, 2], [32, 8]], 'label': '10368.qo', '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': 'SO+(4,2)^2.C2', 'ngens': 9, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 54, 'number_characteristic_subgroups': 64, 'number_conjugacy_classes': 81, 'number_divisions': 72, 'number_normal_subgroups': 118, 'number_subgroup_autclasses': 1318, 'number_subgroup_classes': 2197, 'number_subgroups': 105240, 'old_label': None, 'order': 10368, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 483], [3, 80], [4, 2556], [6, 2352], [8, 1440], [12, 2304], [24, 1152]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [55], 'outer_gens': [[55, 107935072560, 208417633200, 366, 181, 25032193254, 8237376000, 14936145840, 13579781356]], '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': 6, 'perfect': False, 'permutation_degree': 15, '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, 21], [8, 13], [16, 6], [32, 8]], 'representations': {'PC': {'code': '2582467937227575674226118623295126587000088774182148228047009580157038234771536636310276251827157607484859552583325213669433648171277254898719591430665565288487', 'gens': [1, 2, 5, 7, 10, 11], 'pres': [11, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 88, 133, 56, 398, 90, 22004, 11015, 6406, 587, 158, 52805, 26416, 13755, 566, 41894, 13569, 3108, 4659, 3284, 226, 87303, 26770, 4957, 10600, 260, 82376, 3187, 11118, 9545, 6675, 1406, 557589, 69750, 5884, 4443]}, 'Perm': {'d': 15, 'gens': [55, 107935072560, 208417633200, 186, 20, 25032193516, 8237376000, 14936145840, 13579781094]}}, 'schur_multiplier': [2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 12, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': '\\SOPlus(4,2)^2.C_2', 'transitive_degree': 48, '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': 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}
