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gps_subgroup_search • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'ambient': '1728.46903', 'ambient_counter': 46903, 'ambient_order': 1728, 'ambient_tex': 'C_6^2:(C_2\\times S_4)', 'central': False, 'central_factor': False, 'centralizer_order': 36, 'characteristic': False, 'core_order': 144, 'counter': 94, 'cyclic': False, 'direct': False, 'hall': 0, 'label': '1728.46903.12.b1.b1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': True, 'old_label': '12.b1.b1', 'outer_equivalence': False, 'perfect': False, 'proper': True, 'quotient': '12.4', 'quotient_Agroup': True, 'quotient_abelian': False, 'quotient_cyclic': False, 'quotient_hash': 4, 'quotient_metabelian': True, 'quotient_nilpotent': False, 'quotient_order': 12, 'quotient_simple': False, 'quotient_solvable': True, 'quotient_supersolvable': True, 'quotient_tex': 'D_6', 'simple': False, 'solvable': True, 'special_labels': [], 'split': True, 'standard_generators': False, 'stem': False, 'subgroup': '144.193', 'subgroup_hash': 193, 'subgroup_order': 144, 'subgroup_tex': 'C_2^2:C_6^2', 'supersolvable': False, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '1728.46903', 'aut_centralizer_order': 72, 'aut_label': '12.b1', 'aut_quo_index': 2, 'aut_stab_index': 3, 'aut_weyl_group': '96.226', 'aut_weyl_index': 216, 'centralizer': '48.a1.a1', 'complements': ['144.dr1.a1', '144.dq1.a1', '144.du1.a1', '144.dr1.c1', '144.dr1.d1', '144.dq1.b1', '144.du1.b1', '144.dr1.f1'], 'conjugacy_class_count': 1, 'contained_in': ['4.a1.a1', '6.b1.b1', '6.j1.b1', '6.l1.b1'], 'contains': ['24.c1.b1', '24.cf1.b1', '36.a1.a1', '36.c1.b1', '36.u1.b1', '48.g1.b1'], 'core': '12.b1.b1', 'coset_action_label': None, 'count': 1, 'diagramx': [1503, 4948, 5348, 5675, 622, 4361, 3257, 3796], 'generators': [31671377, 892585, 25489843, 7705165, 4555816, 8046913], 'label': '1728.46903.12.b1.b1', 'mobius_quo': 0, 'mobius_sub': -6, 'normal_closure': '12.b1.b1', 'normal_contained_in': ['4.a1.a1', '6.b1.b1'], 'normal_contains': ['24.c1.b1', '36.a1.a1', '36.c1.b1'], 'normalizer': '1.a1.a1', 'old_label': '12.b1.b1', 'projective_image': '864.4692', 'quotient_action_image': '12.4', 'quotient_action_kernel': '1.1', 'quotient_action_kernel_order': 1, 'quotient_fusion': None, 'short_label': '12.b1.b1', 'subgroup_fusion': None, 'weyl_group': '48.48'}
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gps_groups • Show schema
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{'Agroup': True, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '36.14', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 3, 3, 3, 2, 2], 'aut_gens': [[1, 2, 12, 24], [1, 2, 12, 120], [1, 3, 12, 24], [1, 10, 72, 60], [1, 2, 84, 108], [7, 9, 12, 24], [1, 50, 12, 24], [1, 86, 12, 24], [1, 74, 12, 24]], 'aut_group': '864.4673', 'aut_hash': 4673, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 864, 'aut_permdeg': 10, 'aut_perms': [379560, 374400, 5, 3, 1905840, 1941120, 23, 7], 'aut_phi_ratio': 18.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1], [2, 3, 1, 1], [2, 3, 3, 1], [3, 1, 2, 1], [3, 4, 6, 1], [6, 1, 6, 1], [6, 3, 2, 1], [6, 3, 6, 1], [6, 4, 18, 1]], 'aut_supersolvable': False, 'aut_tex': 'S_4\\times S_3^2', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 6, 'autcent_group': '36.10', 'autcent_hash': 10, 'autcent_nilpotent': False, 'autcent_order': 36, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'S_3^2', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '24.12', 'autcentquo_hash': 12, 'autcentquo_nilpotent': False, 'autcentquo_order': 24, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_4', 'cc_stats': [[1, 1, 1], [2, 1, 3], [2, 3, 4], [3, 1, 2], [3, 4, 6], [6, 1, 6], [6, 3, 8], [6, 4, 18]], 'center_label': '12.5', 'center_order': 12, 'central_product': True, 'central_quotient': '12.3', 'commutator_count': 1, 'commutator_label': '4.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1', '3.1'], 'composition_length': 6, 'conjugacy_classes_known': True, 'counter': 193, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['12.3', 1], ['2.1', 2], ['3.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 3, 1, 4], [3, 1, 2, 1], [3, 4, 2, 3], [6, 1, 2, 3], [6, 3, 2, 4], [6, 4, 2, 9]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 4, 'exponent': 6, 'exponents_of_order': [4, 2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '1.1', 'frattini_quotient': '144.193', 'hash': 193, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [1, 3, 2, 2], 'inner_gens': [[1, 2, 12, 24], [1, 2, 84, 108], [1, 74, 12, 24], [1, 86, 12, 24]], 'inner_hash': 3, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': True, 'inner_tex': 'A_4', 'inner_used': [2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 36], [3, 12]], 'label': '144.193', 'linC_count': 144, 'linC_degree': 4, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 5, 'linQ_degree_count': 18, 'linQ_dim': 5, 'linQ_dim_count': 18, 'linR_count': 18, 'linR_degree': 5, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C2^2:C6^2', 'ngens': 6, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 10, 'number_characteristic_subgroups': 10, 'number_conjugacy_classes': 48, 'number_divisions': 28, 'number_normal_subgroups': 40, 'number_subgroup_autclasses': 40, 'number_subgroup_classes': 98, 'number_subgroups': 234, 'old_label': None, 'order': 144, 'order_factorization_type': 32, 'order_stats': [[1, 1], [2, 15], [3, 26], [6, 102]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 6, 6], 'outer_gen_pows': [0, 0, 0], 'outer_gens': [[1, 10, 72, 108], [7, 58, 72, 60], [7, 5, 72, 60]], 'outer_group': '72.46', 'outer_hash': 46, 'outer_nilpotent': False, 'outer_order': 72, 'outer_permdeg': 8, 'outer_perms': [744, 5761, 27], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'S_3\\times D_6', 'pc_rank': 4, 'perfect': False, 'permutation_degree': 11, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 3, 3], 'quasisimple': False, 'rank': 2, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 16], [3, 4], [6, 4]], 'representations': {'PC': {'code': 25221789117966582921, 'gens': [1, 2, 4, 5], 'pres': [6, -2, -2, -3, -2, 2, -3, 31, 1017, 447, 1630, 286, 88]}, 'GLZ': {'b': 3, 'd': 5, 'gens': [140827878942, 48233790121]}, 'GLFq': {'d': 3, 'q': 4, 'gens': [164358, 184618, 127769, 56040, 41268, 197379]}, 'Perm': {'d': 11, 'gens': [3669120, 7983360, 843, 840, 7, 16]}}, 'schur_multiplier': [2, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [6, 6], 'solvability_type': 8, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2:C_6^2', 'transitive_degree': 36, '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': '8.5', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 3, 'aut_exponent': 12, 'aut_gen_orders': [2, 6, 6, 12, 2, 6, 6], 'aut_gens': [[594539, 31671377, 35456684, 4403503, 7751737], [25492769, 10417733, 35161508, 4403503, 7749385], [596891, 31671377, 5020679, 4403503, 32641735], [7706999, 31671377, 43885901, 32943967, 4705735], [38382884, 10417733, 5315855, 29593381, 32941615], [7711703, 31671377, 43969229, 4403503, 32646439], [743303, 31671377, 6627764, 4703383, 32939263], [25488065, 31671377, 42360968, 4403503, 7749385]], 'aut_group': None, 'aut_hash': 4497091926293446014, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 20736, 'aut_permdeg': 44, 'aut_perms': [1649097481739674138774890356162714568261959310904859058, 1170445641534250623987654218773733747089527309626347726, 2104781815748018619358662369500964740151598102696474265, 1299247415059279002814968609199487075610286533659245873, 2234059504695561892477724071384375485961809084018062225, 1589128904095310272093216601505988072313390564861667508, 588769029700997255207879372285809359573057848844427866], 'aut_phi_ratio': 36.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 3, 1, 2], [2, 6, 1, 2], [2, 18, 1, 1], [2, 18, 2, 1], [2, 36, 1, 1], [2, 108, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [3, 8, 3, 1], [3, 16, 3, 1], [4, 6, 1, 1], [4, 18, 1, 1], [4, 18, 2, 1], [4, 36, 1, 1], [4, 108, 1, 3], [6, 2, 1, 2], [6, 2, 2, 1], [6, 4, 1, 2], [6, 4, 2, 1], [6, 6, 1, 4], [6, 6, 2, 1], [6, 8, 3, 1], [6, 12, 1, 4], [6, 12, 2, 1], [6, 16, 3, 2], [6, 16, 6, 1], [6, 36, 1, 1], [6, 36, 2, 2], [6, 48, 3, 1], [12, 12, 1, 1], [12, 36, 1, 1], [12, 36, 2, 2], [12, 48, 3, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_6^2.C_6^2.C_2^4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '8.5', 'autcent_hash': 5, 'autcent_nilpotent': True, 'autcent_order': 8, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^3', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '2592.jb', 'autcentquo_hash': 6898096242996517164, 'autcentquo_nilpotent': False, 'autcentquo_order': 2592, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'S_3\\times C_6^2:D_6', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 3, 2], [2, 6, 2], [2, 18, 3], [2, 36, 1], [2, 108, 1], [3, 2, 2], [3, 4, 1], [3, 8, 3], [3, 16, 3], [4, 6, 1], [4, 18, 3], [4, 36, 1], [4, 108, 3], [6, 2, 4], [6, 4, 4], [6, 6, 6], [6, 8, 3], [6, 12, 6], [6, 16, 12], [6, 36, 5], [6, 48, 3], [12, 12, 1], [12, 36, 5], [12, 48, 3]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '864.4692', 'commutator_count': 1, 'commutator_label': '216.173', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '2.1', '2.1', '3.1', '3.1', '3.1'], 'composition_length': 9, 'conjugacy_classes_known': True, 'counter': 46903, 'cyclic': False, 'derived_length': 3, 'dihedral': False, 'direct_factorization': [['24.8', 1], ['72.43', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 3, 1, 2], [2, 6, 1, 2], [2, 18, 1, 3], [2, 36, 1, 1], [2, 108, 1, 1], [3, 2, 1, 2], [3, 4, 1, 1], [3, 8, 1, 3], [3, 16, 1, 3], [4, 6, 1, 1], [4, 18, 1, 3], [4, 36, 1, 1], [4, 108, 1, 3], [6, 2, 1, 2], [6, 2, 2, 1], [6, 4, 1, 2], [6, 4, 2, 1], [6, 6, 1, 4], [6, 6, 2, 1], [6, 8, 1, 3], [6, 12, 1, 4], [6, 12, 2, 1], [6, 16, 1, 6], [6, 16, 2, 3], [6, 36, 1, 3], [6, 36, 2, 1], [6, 48, 1, 3], [12, 12, 1, 1], [12, 36, 1, 3], [12, 36, 2, 1], [12, 48, 1, 3]], 'element_repr_type': 'GLZN', 'elementary': 1, 'eulerian_function': 40320, 'exponent': 12, 'exponents_of_order': [6, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[12, 0, 2]], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '864.4692', 'hash': 46903, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 2, 6, 6, 6], 'inner_gens': [[594539, 31671377, 47610623, 4403503, 32646439], [594539, 31671377, 6925292, 4403503, 7751737], [38382884, 10417733, 35456684, 7754089, 4701031], [594539, 31671377, 10652366, 4403503, 7751737], [25492769, 31671377, 42360968, 4403503, 7751737]], 'inner_hash': 4692, 'inner_nilpotent': False, 'inner_order': 864, 'inner_split': True, 'inner_tex': 'S_3\\times C_6:S_4', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': 12, 'irrQ_degree': 24, 'irrQ_dim': 24, 'irrR_degree': 24, 'irrep_stats': [[1, 8], [2, 26], [3, 8], [4, 20], [6, 14], [12, 5]], 'label': '1728.46903', 'linC_count': 432, 'linC_degree': 7, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 9, 'linQ_degree_count': 1152, 'linQ_dim': 9, 'linQ_dim_count': 1152, 'linR_count': 1152, 'linR_degree': 9, 'maximal_subgroups_known': True, 'metabelian': False, 'metacyclic': False, 'monomial': True, 'name': 'C6^2:(C2*S4)', '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 52, 'number_characteristic_subgroups': 59, 'number_conjugacy_classes': 81, 'number_divisions': 72, 'number_normal_subgroups': 90, 'number_subgroup_autclasses': 920, 'number_subgroup_classes': 1310, 'number_subgroups': 17536, 'old_label': None, 'order': 1728, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 219], [3, 80], [4, 420], [6, 672], [12, 336]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 6, 'outer_gen_orders': [2, 2, 6], 'outer_gen_pows': [592705, 592705, 592705], 'outer_gens': [[594539, 31671377, 35456684, 32943967, 29598085], [594539, 31671377, 47610623, 4403503, 32641735], [7706999, 31671377, 35461388, 4403503, 7751737]], 'outer_group': '24.14', 'outer_hash': 14, 'outer_nilpotent': False, 'outer_order': 24, 'outer_permdeg': 7, 'outer_perms': [16, 127, 847], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_6', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 14, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 22], [3, 8], [4, 14], [6, 10], [8, 4], [12, 5], [24, 1]], 'representations': {'PC': {'code': 2362315213057807382612637065800565756284382605137202126419572189841676309, 'gens': [1, 2, 3, 6, 8], 'pres': [9, 2, 2, 2, 2, 3, 2, 3, 2, 3, 542, 389, 74, 1443, 102, 1444, 12983, 6188, 3605, 158, 3048, 107143, 9529, 11698, 1987, 214, 101096, 15578]}, 'GLZN': {'d': 2, 'p': 84, 'gens': [892585, 7072411, 596891, 595057, 31672385, 25782667, 1042546, 29300557, 7705165]}, 'Perm': {'d': 14, 'gens': [997960320, 745, 264, 1680, 518918400, 403200, 3, 6266937600, 13412044800]}}, 'schur_multiplier': [2, 2, 2, 6], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 11, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_6^2:(C_2\\times S_4)', 'transitive_degree': 72, 'wreath_data': None, 'wreath_product': False}
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gps_groups • Show schema
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{'Agroup': True, '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': 6, 'aut_gen_orders': [2, 6], 'aut_gens': [[1, 2], [1, 10], [3, 2]], 'aut_group': '12.4', 'aut_hash': 4, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 12, 'aut_permdeg': 5, 'aut_perms': [6, 49], 'aut_phi_ratio': 3.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [3, 2, 1, 1], [6, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_6', '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': 6, 'autcentquo_group': '6.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': False, 'autcentquo_order': 6, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'S_3', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [3, 2, 1], [6, 2, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', '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': 4, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [['2.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 2, 1, 1], [6, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '1.1', 'frattini_quotient': '12.4', 'hash': 4, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 3], 'inner_gens': [[1, 10], [5, 2]], 'inner_hash': 1, 'inner_nilpotent': False, 'inner_order': 6, 'inner_split': True, 'inner_tex': 'S_3', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 2]], 'label': '12.4', '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': 'D6', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [0, 0, 0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 6, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 10, 'number_subgroups': 16, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 7], [3, 2], [6, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [0], 'outer_gens': [[7, 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': 5, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2]], 'representations': {'PC': {'code': 43377, 'gens': [1, 2], 'pres': [3, -2, -2, -3, 61, 16, 74]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [17, 35]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [31, 55, 56]}, 'Perm': {'d': 5, 'gens': [6, 1, 30]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_6', 'transitive_degree': 6, 'wreath_data': None, 'wreath_product': False}
