Compound having a garnet-like crystal structure of the general chemical formula: Li n [A (3-a'-a") A' (a') A" (a")][B (2-b'-b") B' (b') B" (b")][C' (c') C" (c")]O 12 Where A, A' and A'' each represent a dodecahedral position of the garnet-like crystal structure and A is at least one element selected from the group consisting of La, Y, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm and Yb, A' is at least one element selected from the group consisting of Ca, Sr and Ba, A'' is at least one element selected from the group consisting of Na and K, 0 ≤ a' < 2 and 0 ≤ a" < 1, where B, B' and B" each represent an octahedral position of the garnet-like crystal structure and B is at least one element selected from the group consisting of Zr, Hf and Sn, B' is at least one element selected from the group consisting of Ta, Nb, Sb and Bi, B'' is at least one element selected from the group consisting of Te, W and Mo, 0 ≤ b' ≤ 2 and 0 ≤ b" ≤ 2, where C' and C'' each represent a tetrahedral position of the garnet-like crystal structure and C' is at least one element selected from the group consisting of Al and Ga, C'' is at least one element selected from the group consisting of Si and Ge, 0 ≤ c' ≤ 0.5 and 0 ≤ c'' ≤ 0.4, and where n = 7 + a' + 2 · a'' - b' - 2·b'' - 3·c' - 4·c" and 5.5 ≤ n ≤ 6.875, where, when b' = 2, 6.0 < n < 6.875 or 5.5 ≤ n ≤ 6.875 and c' + c'' > 0 and when B' is Nb, 6.0 < n < 6.4 or 5.5 ≤ n ≤ 6.875 and c' + c'' > 0 and/or a' + a'' > 0. Where A, A' and A'' each represent a dodecahedral position of the garnet-like crystal structure and A is at least one element selected from the group consisting of La, Y, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm and Yb, A' is at least one element selected from the group consisting of Ca, Sr and Ba, A'' is at least one element selected from the group consisting of Na and K, 0 ≤ a' < 2 and 0 ≤ a" < 1, A is at least one element selected from the group consisting of La, Y, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm and Yb, A' is at least one element selected from the group consisting of Ca, Sr and Ba, A'' is at least one element selected from the group consisting of Na and K, 0 ≤ a' < 2 and 0 ≤ a" < 1, where B, B' and B" each represent an octahedral position of the garnet-like crystal structure and B is at least one element selected from the group consisting of Zr, Hf and Sn, B' is at least one element selected from the group consisting of Ta, Nb, Sb and Bi, B'' is at least one element selected from the group consisting of Te, W and Mo, 0 ≤ b' ≤ 2 and 0 ≤ b" ≤ 2, B is at least one element selected from the group consisting of Zr, Hf and Sn, B' is at least one element selected from the group consisting of Ta, Nb, Sb and Bi, B'' is at least one element selected from the group consisting of Te, W and Mo, 0 ≤ b' ≤ 2 and 0 ≤ b" ≤ 2, where C' and C'' each represent a tetrahedral position of the garnet-like crystal structure and C' is at least one element selected from the group consisting of Al and Ga, C'' is at least one element selected from the group consisting of Si and Ge, 0 ≤ c' ≤ 0.5 and 0 ≤ c'' ≤ 0.4, and C' is at least one element selected from the group consisting of Al and Ga, C'' is at least one element selected from the group consisting of Si and Ge, 0 ≤ c' ≤ 0.5 and 0 ≤ c'' ≤ 0.4, and where n = 7 + a' + 2 · a'' - b' - 2·b'' - 3·c' - 4·c" and 5.5 ≤ n ≤ 6.875, where, when b' = 2, 6.0 < n < 6.875 or 5.5 ≤ n ≤ 6.875 and c' + c'' > 0 and when B' is Nb, 6.0 < n < 6.4 or 5.5 ≤ n ≤ 6.875 and c' + c'' > 0 and/or a' + a'' > 0.