Name: | $S_3\times S_4$ |
Order: | $144$ |
Abelian: | no |
Generators: | $\begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{6}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{3}^{2} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{5} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 1 & 0 & 0 & 0 \\1 & 0 & 0& 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\\end{bmatrix}, \begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$ |
Maximal subgroups: | $J(C(6,2))$, $J_s(C(6,2))$, $J(D(3,1))$, $J(B(6,2))$, $J(D(2,2))$, $D(6,2)$ |
Minimal supergroups: | $J(D(6,6))$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$1$ |
$0$ |
$9$ |
$0$ |
$160$ |
$0$ |
$3955$ |
$0$ |
$114156$ |
$0$ |
$3568026$ |
$a_2$ |
$1$ |
$1$ |
$4$ |
$21$ |
$173$ |
$1856$ |
$22793$ |
$299405$ |
$4080261$ |
$56823432$ |
$802135079$ |
$11424726545$ |
$163736734846$ |
$a_3$ |
$1$ |
$0$ |
$4$ |
$0$ |
$442$ |
$0$ |
$115820$ |
$0$ |
$37961602$ |
$0$ |
$13398088944$ |
$0$ |
$4877169085690$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$1$ |
$1$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$4$ |
$1$ |
$4$ |
$9$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$4$ |
$21$ |
$10$ |
$34$ |
$19$ |
$72$ |
$160$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$25$ |
$173$ |
$92$ |
$56$ |
$350$ |
$202$ |
$779$ |
$440$ |
$1745$ |
$3955$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$262$ |
$1856$ |
$147$ |
$1024$ |
$579$ |
$4114$ |
$2306$ |
$1308$ |
$9375$ |
$5236$ |
$21478$ |
$11921$ |
$49406$ |
$114156$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$442$ |
$3068$ |
$22793$ |
$1724$ |
$12563$ |
$6986$ |
$52278$ |
$3886$ |
$28865$ |
$15982$ |
$120971$ |
$66592$ |
$36792$ |
$280829$ |
$$ |
$154203$ |
$653765$ |
$358050$ |
$1525608$ |
$3568026$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&0&2&0&0&0&0&0&0&1\\0&1&0&0&0&2&0&0&3&0&2&0&2&6&0\\0&0&3&0&0&0&3&5&0&1&0&7&0&0&16\\0&0&0&3&0&3&0&0&3&0&1&0&12&9&0\\0&0&0&0&5&0&4&4&0&9&0&11&0&0&22\\0&2&0&3&0&13&0&0&19&0&9&0&30&45&0\\0&0&3&0&4&0&18&10&0&7&0&30&0&0&67\\2&0&5&0&4&0&10&27&0&16&0&31&0&0&83\\0&3&0&3&0&19&0&0&35&0&17&0&45&82&0\\0&0&1&0&9&0&7&16&0&28&0&30&0&0&72\\0&2&0&1&0&9&0&0&17&0&10&0&18&39&0\\0&0&7&0&11&0&30&31&0&30&0&71&0&0&166\\0&2&0&12&0&30&0&0&45&0&18&0&100&123&0\\0&6&0&9&0&45&0&0&82&0&39&0&123&208&0\\1&0&16&0&22&0&67&83&0&72&0&166&0&0&420\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&3&3&5&13&18&27&35&28&10&71&100&208&420&240&238&582&513&196\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
---|
$-$ | $1$ | $49/72$ | $1/24$ | $31/72$ | $0$ | $1/12$ | $1/8$ |
---|
$a_1=0$ | $49/72$ | $49/72$ | $1/24$ | $31/72$ | $0$ | $1/12$ | $1/8$ |
---|
$a_3=0$ | $1/2$ | $1/2$ | $1/24$ | $1/4$ | $0$ | $1/12$ | $1/8$ |
---|
$a_1=a_3=0$ | $1/2$ | $1/2$ | $1/24$ | $1/4$ | $0$ | $1/12$ | $1/8$ |
---|