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gps_subgroup_search • Show schema
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{'Agroup': False, 'Zgroup': False, 'abelian': False, 'ambient': '3456.cp', 'ambient_counter': 68, 'ambient_order': 3456, 'ambient_tex': 'C_6^2.(D_4\\times D_6)', 'central': False, 'central_factor': False, 'centralizer_order': 4, 'characteristic': False, 'core_order': 2, 'counter': 1121, 'cyclic': False, 'direct': None, 'hall': 0, 'label': '3456.cp.72.fz1', 'maximal': False, 'maximal_normal': False, 'metabelian': True, 'metacyclic': False, 'minimal': False, 'minimal_normal': False, 'nilpotent': False, 'normal': False, 'old_label': '72.fz1', 'outer_equivalence': True, 'perfect': False, 'proper': True, 'quotient': None, 'quotient_Agroup': None, 'quotient_abelian': None, 'quotient_cyclic': None, 'quotient_hash': None, 'quotient_metabelian': None, 'quotient_nilpotent': None, 'quotient_order': 72, 'quotient_simple': None, 'quotient_solvable': None, 'quotient_supersolvable': None, 'quotient_tex': None, 'simple': False, 'solvable': True, 'special_labels': [], 'split': None, 'standard_generators': False, 'stem': False, 'subgroup': '48.43', 'subgroup_hash': 43, 'subgroup_order': 48, 'subgroup_tex': 'C_6:D_4', 'supersolvable': True, 'sylow': 0}
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gps_subgroup_data • Show schema
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{'ambient': '3456.cp', 'aut_centralizer_order': 16, 'aut_label': '72.fz1', 'aut_quo_index': None, 'aut_stab_index': 36, 'aut_weyl_group': '48.51', 'aut_weyl_index': 576, 'centralizer': '864.g1', 'complements': None, 'conjugacy_class_count': 2, 'contained_in': ['24.gb1', '24.gc1', '36.v1', '36.ci1', '36.cn1'], 'contains': ['144.ct1', '144.cv1', '144.dc1', '144.go1', '144.gs1', '144.hf1', '144.hg1', '216.by1'], 'core': '1728.a1', 'coset_action_label': None, 'count': 36, 'diagramx': None, 'generators': [1, 192, 24, 1732, 288], 'label': '3456.cp.72.fz1', 'mobius_quo': None, 'mobius_sub': None, 'normal_closure': '4.d1', 'normal_contained_in': None, 'normal_contains': None, 'normalizer': '18.a1', 'old_label': '72.fz1', 'projective_image': '1728.46671', 'quotient_action_image': None, 'quotient_action_kernel': None, 'quotient_action_kernel_order': None, 'quotient_fusion': None, 'short_label': '72.fz1', 'subgroup_fusion': None, 'weyl_group': '48.51'}
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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': 2, 'aut_exponent': 12, 'aut_gen_orders': [2, 2, 2, 2, 12], 'aut_gens': [[1, 2, 8], [25, 26, 8], [1, 2, 12], [1, 2, 40], [5, 6, 8], [25, 18, 12]], 'aut_group': '192.1147', 'aut_hash': 1147, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 192, 'aut_permdeg': 11, 'aut_perms': [21144, 12501360, 8361601, 15301704, 19436403], 'aut_phi_ratio': 12.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 1, 2, 1], [2, 2, 2, 1], [2, 6, 2, 1], [3, 2, 1, 1], [4, 6, 2, 1], [6, 2, 1, 1], [6, 2, 2, 1], [6, 2, 4, 1]], 'aut_supersolvable': True, 'aut_tex': 'C_2^4:D_6', 'autcent_abelian': False, 'autcent_cyclic': False, 'autcent_exponent': 4, 'autcent_group': '32.27', 'autcent_hash': 27, 'autcent_nilpotent': True, 'autcent_order': 32, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2\\wr 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, 3], [2, 2, 2], [2, 6, 2], [3, 2, 1], [4, 6, 2], [6, 2, 7]], 'center_label': '4.2', 'center_order': 4, 'central_product': True, 'central_quotient': '12.4', 'commutator_count': 1, 'commutator_label': '6.2', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1', '2.1', '3.1'], 'composition_length': 5, 'conjugacy_classes_known': True, 'counter': 43, 'cyclic': False, 'derived_length': 2, 'dihedral': False, 'direct_factorization': [['2.1', 1], ['24.8', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3], [2, 2, 1, 2], [2, 6, 1, 2], [3, 2, 1, 1], [4, 6, 1, 2], [6, 2, 1, 3], [6, 2, 2, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 168, 'exponent': 12, 'exponents_of_order': [4, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [], 'familial': False, 'frattini_label': '2.1', 'frattini_quotient': '24.14', 'hash': 43, 'hyperelementary': 2, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 6, 'inner_gen_orders': [2, 2, 3], 'inner_gens': [[1, 6, 8], [5, 2, 40], [1, 18, 8]], 'inner_hash': 4, 'inner_nilpotent': False, 'inner_order': 12, 'inner_split': False, 'inner_tex': 'D_6', 'inner_used': [1, 2, 3], 'irrC_degree': -1, 'irrQ_degree': -1, 'irrQ_dim': -1, 'irrR_degree': -1, 'irrep_stats': [[1, 8], [2, 10]], 'label': '48.43', 'linC_count': 16, 'linC_degree': 3, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 4, 'linQ_degree_count': 4, 'linQ_dim': 4, 'linQ_dim_count': 4, 'linR_count': 4, 'linR_degree': 4, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': False, 'monomial': True, 'name': 'C6:D4', 'ngens': 5, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': [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': 11, 'number_conjugacy_classes': 18, 'number_divisions': 16, 'number_normal_subgroups': 27, 'number_subgroup_autclasses': 32, 'number_subgroup_classes': 54, 'number_subgroups': 108, 'old_label': None, 'order': 48, 'order_factorization_type': 31, 'order_stats': [[1, 1], [2, 19], [3, 2], [4, 12], [6, 14]], 'outer_abelian': False, 'outer_cyclic': False, 'outer_equivalence': False, 'outer_exponent': 4, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[25, 26, 8], [1, 2, 12], [5, 26, 8], [5, 2, 8]], 'outer_group': '16.11', 'outer_hash': 11, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 6, 'outer_perms': [6, 289, 1, 126], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2\\times D_4', 'pc_rank': 3, 'perfect': False, 'permutation_degree': 9, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2], 'quasisimple': False, 'rank': 3, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 8], [2, 6], [4, 2]], 'representations': {'PC': {'code': 2906379043649, 'gens': [1, 2, 4], 'pres': [5, -2, -2, -2, -2, -3, 61, 26, 408, 58, 409]}, 'GLZ': {'b': 3, 'd': 4, 'gens': [16563488, 35813463, 7233663]}, 'GLFp': {'d': 3, 'p': 7, 'gens': [34720862, 6536214, 34603218, 16771677, 34409418]}, 'Perm': {'d': 9, 'gens': [5045, 120, 127, 16, 45360]}}, 'schur_multiplier': [2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2], 'solvability_type': 7, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_6:D_4', 'transitive_degree': 24, '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': 12, 'aut_gen_orders': [12, 4, 6, 6, 2, 2, 6, 4], 'aut_gens': [[1, 2, 8, 48, 576], [3385, 1194, 296, 3216, 576], [2253, 218, 2152, 1792, 864], [1345, 1634, 328, 1392, 576], [1633, 3010, 8, 1488, 2880], [3197, 3198, 1736, 3152, 2880], [2785, 706, 40, 2832, 2880], [193, 306, 8, 336, 2880], [2021, 1762, 1864, 64, 3168]], 'aut_group': None, 'aut_hash': 2854184908137106426, 'aut_nilpotency_class': -1, 'aut_nilpotent': False, 'aut_order': 27648, 'aut_permdeg': 60, 'aut_perms': [898073812465090508261804222866337955529437797434612261306459903839973094203935698, 107519382830165913834474519894549954806690337742036874410772693333223987093532182, 7176166427783373644875276319678201858668425159338427279191057885625298875584634497, 4850783008570589621628149286405949396279465953473216973249711200749357671809879823, 7009213955156925731387292437674699531982230694686431897542475141734926174080791356, 5828767241085032992150700757481721598441711634675026642402040200366659491880405167, 7169119928095440787863323749961288826298570770620071168547432401437160029823526550, 8034370876822106038188584111338841608607256913608852470957478406561256714400892062], 'aut_phi_ratio': 24.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 1], [2, 4, 1, 1], [2, 12, 1, 1], [2, 12, 2, 1], [2, 24, 1, 1], [2, 27, 2, 1], [2, 36, 1, 1], [2, 36, 2, 1], [2, 54, 1, 1], [2, 72, 1, 1], [2, 108, 1, 1], [3, 2, 1, 1], [3, 4, 1, 2], [3, 8, 1, 2], [4, 6, 2, 1], [4, 12, 2, 2], [4, 18, 2, 1], [4, 36, 2, 2], [6, 2, 1, 1], [6, 4, 1, 5], [6, 8, 1, 8], [6, 8, 2, 1], [6, 16, 1, 3], [6, 16, 2, 1], [6, 24, 2, 3], [6, 48, 1, 3], [6, 48, 2, 2], [6, 72, 1, 1], [6, 72, 2, 1], [6, 144, 1, 1], [8, 72, 2, 1], [8, 216, 2, 1], [12, 24, 2, 6], [12, 36, 2, 1], [12, 48, 2, 2], [12, 72, 2, 2], [24, 144, 2, 1]], 'aut_supersolvable': False, 'aut_tex': 'C_3^3.C_2^6.C_2^4', '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': False, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^4', 'autcentquo_abelian': False, 'autcentquo_cyclic': False, 'autcentquo_exponent': 12, 'autcentquo_group': '1728.46671', 'autcentquo_hash': 46671, 'autcentquo_nilpotent': False, 'autcentquo_order': 1728, 'autcentquo_solvable': True, 'autcentquo_supersolvable': False, 'autcentquo_tex': 'D_6^2:D_6', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 1], [2, 4, 1], [2, 12, 3], [2, 24, 1], [2, 27, 2], [2, 36, 3], [2, 54, 1], [2, 72, 1], [2, 108, 1], [3, 2, 1], [3, 4, 2], [3, 8, 2], [4, 6, 2], [4, 12, 4], [4, 18, 2], [4, 36, 4], [6, 2, 1], [6, 4, 5], [6, 8, 10], [6, 16, 5], [6, 24, 6], [6, 48, 7], [6, 72, 3], [6, 144, 1], [8, 72, 2], [8, 216, 2], [12, 24, 12], [12, 36, 2], [12, 48, 4], [12, 72, 4], [24, 144, 2]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '1728.46671', 'commutator_count': 1, 'commutator_label': '216.143', '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'], 'composition_length': 10, 'conjugacy_classes_known': True, 'counter': 68, 'cyclic': False, 'derived_length': 3, '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, 12, 1, 3], [2, 24, 1, 1], [2, 27, 1, 2], [2, 36, 1, 3], [2, 54, 1, 1], [2, 72, 1, 1], [2, 108, 1, 1], [3, 2, 1, 1], [3, 4, 1, 2], [3, 8, 1, 2], [4, 6, 1, 2], [4, 12, 1, 4], [4, 18, 1, 2], [4, 36, 1, 4], [6, 2, 1, 1], [6, 4, 1, 5], [6, 8, 1, 10], [6, 16, 1, 5], [6, 24, 1, 6], [6, 48, 1, 7], [6, 72, 1, 3], [6, 144, 1, 1], [8, 72, 1, 2], [8, 216, 1, 2], [12, 24, 1, 12], [12, 36, 1, 2], [12, 48, 1, 4], [12, 72, 1, 4], [24, 144, 1, 2]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1526353920, 'exponent': 24, 'exponents_of_order': [7, 3], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[8, 1, 4], [16, 1, 3]], 'familial': False, 'frattini_label': '4.2', 'frattini_quotient': '864.4686', 'hash': 6075597529126993144, 'hyperelementary': 1, 'inner_abelian': False, 'inner_cyclic': False, 'inner_exponent': 12, 'inner_gen_orders': [2, 4, 6, 6, 6], 'inner_gens': [[1, 1734, 488, 240, 3168], [2021, 2, 1864, 64, 3168], [97, 2226, 8, 336, 576], [385, 34, 296, 48, 2880], [1441, 1442, 8, 1200, 576]], 'inner_hash': 46671, 'inner_nilpotent': False, 'inner_order': 1728, 'inner_split': True, 'inner_tex': 'D_6^2:D_6', 'inner_used': [1, 2, 3, 4, 5], 'irrC_degree': 8, 'irrQ_degree': 8, 'irrQ_dim': 8, 'irrR_degree': 8, 'irrep_stats': [[1, 16], [2, 20], [4, 34], [8, 24], [16, 5]], 'label': '3456.cp', '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': 'C6^2.(D4*D6)', 'ngens': 10, '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], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 68, 'number_characteristic_subgroups': 66, 'number_conjugacy_classes': 99, 'number_divisions': 99, 'number_normal_subgroups': 148, 'number_subgroup_autclasses': 2194, 'number_subgroup_classes': 3556, 'number_subgroups': 60844, 'old_label': None, 'order': 3456, 'order_factorization_type': 33, 'order_stats': [[1, 1], [2, 463], [3, 26], [4, 240], [6, 1022], [8, 576], [12, 840], [24, 288]], 'outer_abelian': True, 'outer_cyclic': False, 'outer_equivalence': True, 'outer_exponent': 2, 'outer_gen_orders': [2, 2, 2, 2], 'outer_gen_pows': [0, 0, 0, 0], 'outer_gens': [[289, 2, 8, 48, 2880], [1, 2, 328, 240, 2880], [1, 2, 8, 336, 576], [1753, 26, 40, 1968, 2880]], 'outer_group': '16.14', 'outer_hash': 14, 'outer_nilpotent': True, 'outer_order': 16, 'outer_permdeg': 8, 'outer_perms': [5040, 120, 6, 1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2^4', 'pc_rank': 5, 'perfect': False, 'permutation_degree': 17, 'pgroup': 0, 'primary_abelian_invariants': [2, 2, 2, 2], 'quasisimple': False, 'rank': 4, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 16], [2, 20], [4, 34], [8, 24], [16, 5]], 'representations': {'PC': {'code': '22194041636284845264910194822832557408417368692568034347174407172396453685847822148383364226097796808369451985441503269', 'gens': [1, 2, 4, 6, 9], 'pres': [10, 2, 2, 2, 2, 3, 2, 2, 3, 2, 3, 34681, 51, 2182, 19523, 37293, 3303, 113, 20004, 5614, 424, 14405, 1935, 3625, 2555, 175, 33606, 4496, 8426, 206, 30727, 8337, 7707, 285128, 142578, 5458, 268, 230409, 115219, 4859]}, 'Perm': {'d': 17, 'gens': [20197905895, 47096395360741, 67040185771441, 43172454124855, 90679393733357, 21197261433600]}}, 'schur_multiplier': [2, 2, 2, 2, 2, 2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2, 2, 2], 'solvability_type': 17, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': False, 'sylow_subgroups_known': True, 'tex_name': 'C_6^2.(D_4\\times D_6)', 'transitive_degree': 24, 'wreath_data': None, 'wreath_product': False}
