Name: | $C_2$ |
Order: | $2$ |
Abelian: | yes |
Generators: | $\begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 &0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0\\\end{bmatrix}$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$1$ |
$0$ |
$6$ |
$0$ |
$60$ |
$0$ |
$805$ |
$0$ |
$12978$ |
$0$ |
$237006$ |
$a_2$ |
$1$ |
$1$ |
$3$ |
$11$ |
$58$ |
$382$ |
$2922$ |
$24642$ |
$222728$ |
$2122232$ |
$21092807$ |
$217061737$ |
$2300105315$ |
$a_3$ |
$1$ |
$0$ |
$3$ |
$0$ |
$116$ |
$0$ |
$10630$ |
$0$ |
$1422428$ |
$0$ |
$236405736$ |
$0$ |
$45382171296$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$1$ |
$1$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$3$ |
$1$ |
$3$ |
$6$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$3$ |
$11$ |
$6$ |
$16$ |
$10$ |
$30$ |
$60$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$12$ |
$58$ |
$34$ |
$23$ |
$101$ |
$65$ |
$198$ |
$125$ |
$395$ |
$805$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$77$ |
$382$ |
$48$ |
$233$ |
$147$ |
$735$ |
$460$ |
$292$ |
$1482$ |
$925$ |
$3026$ |
$1876$ |
$6237$ |
$12978$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$116$ |
$554$ |
$2922$ |
$348$ |
$1785$ |
$1107$ |
$5908$ |
$687$ |
$3631$ |
$2241$ |
$12189$ |
$7464$ |
$4597$ |
$25356$ |
$$ |
$15477$ |
$53109$ |
$32298$ |
$111888$ |
$237006$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&0&1&0&0&0&0&0&0&0\\0&1&0&0&0&1&0&0&1&0&1&0&0&1&0\\0&0&2&0&0&0&1&1&0&0&0&2&0&0&2\\0&0&0&2&0&1&0&0&0&0&0&0&3&1&0\\0&0&0&0&3&0&1&1&0&2&0&2&0&0&2\\0&1&0&1&0&4&0&0&3&0&2&0&4&5&0\\0&0&1&0&1&0&5&1&0&0&0&4&0&0&6\\1&0&1&0&1&0&1&6&0&2&0&3&0&0&6\\0&1&0&0&0&3&0&0&6&0&2&0&3&6&0\\0&0&0&0&2&0&0&2&0&6&0&2&0&0&4\\0&1&0&0&0&2&0&0&2&0&3&0&1&4&0\\0&0&2&0&2&0&4&3&0&2&0&9&0&0&10\\0&0&0&3&0&4&0&0&3&0&1&0&11&7&0\\0&1&0&1&0&5&0&0&6&0&4&0&7&13&0\\0&0&2&0&2&0&6&6&0&4&0&10&0&0&20\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&2&2&3&4&5&6&6&6&3&9&11&13&20&12&11&20&16&10\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
---|
$-$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_1=0$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_3=0$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_1=a_3=0$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|