Space group U-site Ru-site Si-site No. 139 I 4 / mmm 2 a 4 / mmm (0 , 0 , 0) 4 d ¯ 4 m 2 (0 , 1 $_{2}$, 1 $_{4}$) , ( 1 $_{2}$, 0 , 1 $_{4}$) 4 e 4 mm (0 , 0 , z ) , (0 , 0 , ¯ z ) No. 123 P 4 / mmm 1 a 4 / mmm + ○ 1 d 4 / mmm 4 i 2 mm . (0 , 1 $_{2}$, z ) , ... 2 g 4 mm (0 , 0 , z ) , ... + ○ 2 h 4 mm ( 1 $_{2}$, 1 $_{2}$, z ′ ) , ... No. 126 P 4 / nnc 2 a 422 4 d ¯ 4 .. 4 e 4 .. No. 128 P 4 / mnc 2 a 4 / m 4 d 2 . 22 4 e 4 .. No. 129 P 4 / nmm 2 c 4 mm (0 , 1 $_{2}$, z ) , ... 2 a ¯ 4 m 2 (0 , 0 , 0) , ... + ○ 2 b ¯ 4 m 2 (0 , 0 , 1 $_{2}$) , ... 2 c 4 mm (0 , 1 $_{2}$, z ) , ... + ○ 2 c 4 mm (0 , 1 $_{2}$, z ′ ) , ... No. 131 P 4$_{2}$ / mmc 2 c mmm . (0 , 1 $_{2}$, 0) , ... 2 e ¯ 4 m 2 (0 , 0 , 1 $_{4}$) , ... + 2 f ¯ 4 m 2 ( 1 $_{2}$, 1 $_{2}$, 1 $_{4}$) , ... 4 i 2 mm . (0 , 1 $_{2}$, z ) , ... No. 134 P 4$_{2}$ / nnm 2 a ¯ 42 m 4 d 2 . 22 4 g 2 . mm No. 136 P 4$_{2}$ / mnm 2 a m . mm 4 d ¯ 4 .. 4 e 2 . mm No. 137 P 4$_{2}$ / nmc 2 a ¯ 4 m 2 4 d 2 mm . (0 , 1 $_{2}$, z ) , ... 4 c 2 mm . Table I. Multiplicity, Wycko ff letter, and site symmetry of each atom in URu$_{2}$Si$_{2}$. First row; the case of No. 139, the mother space group: Below the second row; the maximal non-isomorphic k subgroups of No. 139. All the subgroups belong to D$_{4}$$_{h}$ . The site coordinates are shown, when the notation changes. In No. 131, the origin of the lattice changes, but the lattice does not need to be distorted. Author Name L etter 2 J. Phys. Soc. Jpn. temperatures. Therefore, the possibility of No. 128 or No. 134 is eliminated. type) is No. 139 ( I 4 / mmm ; D 17 4 $_{h}$), which has 7 maximal non- isomorphic t subgroups ( translationengleiche ), 8 maximal non-isomorphic k subgroups ( klassengleiche ), and two max- imal isomorphic subgroups of lower index. 23) t subgroups (No. 69 ( Fmmm ; D 23 2 $_{h}$), No. 71 (Immm ; D 25 2 $_{h}$), No. 87 (I 4 / m ; C 5 4 $_{h}$), No. 97 (I 422; D 9 $_{4}$), No. 107 ( I 4 mm ; C 9 4 $_{v}$), No. 119 (I ¯ 4 m 2; D 9 2 $_{d}$), No. 121 (I ¯ 42 m ; D 11 2 $_{d}$)) change the point group, and the isomorphic subgroup needs a super structure (( c ′ = 3 c ) or ( a ′ = 3 a , b ′ = 3 b )), so they should be easily observed by ex- periments. As the crystal class is unchanged, the point group of 8 maximal non-isomorphic k subgroups must be consid- ered. In No. 126 the local inversion symmetry at U-site is lost, then the U site belongs to 422; D$_{4}$ , while in No. 136 the lo- cal C$_{4}$ ( π/ 2 rotation) at U-site is lost, then the U site belongs to mmm ; D$_{2}$$_{h}$ . In both cases, not the lattice distortion, but the charge distortion at the U-site, caused by large Coulomb inter- action between 5 f electrons, must be the origin of the symme- try lowering. The charge distortion without inversion symme- try, could not be happened, without moving atoms. However, the charge distortion without C$_{4}$ could be possible — it is as- cribed to quadrupole ordering, i.e., a second order phase tran- sition from No. 139 to No 136 could be compatible with O$_{xy}$ - type anti-ferro quadrupole ordering with Q = (0 , 0 , 1), which does not couple with any lattice distortion. The schematic charge density map of the HO phase is shown in Fig. 1. Q = (0 , 0 , 1) is equivalent to Q = (1 , 0 , 0) or (0 , 1 , 0) in the left panel of Fig. 1. Multiplicity, Wycko ff letter, and site symmetry of each atom in URu$_{2}$Si$_{2}$ are listed in Table I, with all the k subgroups. All the space groups have the point symmetry D$_{4}$$_{h}$ . (from No. 123 ( D 1 4 $_{h}$) to No. 142 (D 20 4 $_{h}$)) Let us now investigate the correspondence of atom positions between the mother group and the k subgroups. No. 139 ( I 4 / mmm ; D 17 4 $_{h}$) belongs to the body centered tetragonal lattice, then 16 point symmetry op- erations and ( 1 $_{2}$, 1 $_{2}$, 1 $_{2}$) translation in the conventional lattice. Totally 32 symmetry operations transfer an atom to the crys- tallographically equivalent position, or rotate / reflect an atom at the same position. For example, U at (0, 0, 0) in No. 139, is transferred by ( 1 $_{2}$, 1 $_{2}$, 1 $_{2}$), but not by the 16 point symmetry op- erations, so that the order of the site symmetry (4 / mmm ; D$_{4}$$_{h}$ ) is 16. The multiplicity of the site (2) times the order of the site symmetry (16) is 32. In the case of Ru and Si, the multiplicity of the site is 4, and the order of the site symmetry is 8 (( ¯ 4 m 2; D$_{2}$$_{d}$ ) or (4 mm ; C$_{4}$$_{v}$ )), as listed in Table I. It is a naive consequence that the space group in the HO phase is No. 136. To conclude it, it is not necessary to claim the 5 f 2 local picture for URu$_{2}$Si$_{2}$. However, we take the local picture in the following discussion, because it is more easy to discuss the origin of the transition. As mentioned in the introduction, the low temperature phase should have a lower energy. Now, we propose the phase transition which does not require any movement of atoms. Therefore, the mechanism to reach the lower energy must be based on large Coulomb interactions. However, the anti-ferro quadrupole ordering itself does not lower the system energy, but raises it due to the local electronic charge distortion. We need a strong coupling constant for the anti-ferro quadrupole interaction. In the maximal subgroups, 16 symmetry operations remain. The lost operation brings the splitting of the site into two in- equivalent sites, or the lowering of the site symmetry. When the site is split as U and Si in No. 123, Ru and Si in No. 129, or Ru in No. 131, conventional experimental techniques can detect it. Even when the site is not split, the internal parameter arises in 4 i of No. 123, 2 c of No. 129, and 4 d of No. 137, re- sulting in atoms moving. However, any kind of lattice distor- tion is not expected, when the lower space group is No. 126, No. 128, No. 134, or No. 136. One may expect nesting character in the Fermi surface of the non- f reference compound ThRu$_{2}$Si$_{2}$, as in LaRu$_{4}$P$_{12}$ for PrRu$_{4}$P$_{12}$. In the local 4 f 2 picture of PrRu$_{4}$P$_{12}$, corresponding to LaRu$_{4}$P$_{12}$, the conduction band with strong nesting charac- ter provides the large coupling constant for the higher multi- pole interaction. 4) However, no clear indication of such nest- ing property observed in the band structure of ThRu$_{2}$Si$_{2}$, as shown later. In this letter, we examine which kind of orbitals can gain energy by introducing anti-ferro quadrupole ordering. For the simplicity, we take one dimensional chain along the z - direction with p$_{x}$ and p$_{y}$ orbitals at each site. The lattice constant is a , and each lattice point have 2 sites with a dis- tance a / 2. Therefore, the energy dispersion is expressed by NQR measurement is often a powerful tool to probe the phase transition. 24,25) Ru-site NQR measurements have been already preformed, 26) but never show any anomaly around at T$_{O}$ . If the Ru-site loses the 4-fold symmetry axis due to the phase transition, asymmetric parameter η becomes non zero to a ff ect the NQR frequency. The unchanged NQR frequency indicates that the 4-fold axis of Ru-site survives even in lower