Name: | $D_4:C_2$ |
Order: | $16$ |
Abelian: | no |
Generators: | $\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{7} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0\\0 & 0 & 0 & 0 & \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{5} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 & 0 & 0 \\0 & i & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & -i \\0 & 0& 0 & 0 & -i & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$ |
Maximal subgroups: | $J(A(1,4)_2)$${}^{\times 3}$, $J_s(A(1,4)_2)$${}^{\times 3}$, $B(1,4;2)_2$ |
Minimal supergroups: | $J(B(1,8)_1)$, $J(B(T,1))$, $J(B(2,4))$${}^{\times 2}$, $J(B(1,12;2))$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$33$ |
$0$ |
$920$ |
$0$ |
$28735$ |
$0$ |
$930132$ |
$0$ |
$30691122$ |
$a_2$ |
$1$ |
$2$ |
$11$ |
$89$ |
$1026$ |
$13372$ |
$182084$ |
$2527387$ |
$35479502$ |
$502054424$ |
$7148206446$ |
$102280581427$ |
$1469476480767$ |
$a_3$ |
$1$ |
$0$ |
$12$ |
$0$ |
$2966$ |
$0$ |
$966180$ |
$0$ |
$335126414$ |
$0$ |
$119999282352$ |
$0$ |
$43841219257418$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$11$ |
$4$ |
$15$ |
$33$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$12$ |
$89$ |
$44$ |
$176$ |
$97$ |
$400$ |
$920$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$133$ |
$1026$ |
$546$ |
$302$ |
$2288$ |
$1254$ |
$5299$ |
$2895$ |
$12320$ |
$28735$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1714$ |
$13372$ |
$939$ |
$7246$ |
$3959$ |
$30918$ |
$16850$ |
$9196$ |
$72207$ |
$39307$ |
$168996$ |
$91889$ |
$396186$ |
$930132$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$2966$ |
$23057$ |
$182084$ |
$12575$ |
$98856$ |
$53794$ |
$425991$ |
$29290$ |
$231429$ |
$125804$ |
$999831$ |
$542733$ |
$294784$ |
$2349792$ |
$$ |
$1274595$ |
$5528649$ |
$2996868$ |
$13020546$ |
$30691122$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&1&5&0&0&0&2&0&0&10\\0&2&0&2&0&9&0&0&16&0&7&0&24&39&0\\1&0&8&0&4&0&19&29&0&20&0&48&0&0&125\\0&2&0&6&0&19&0&0&33&0&14&0&56&84&0\\0&0&4&0&14&0&31&30&0&37&0&74&0&0&180\\0&9&0&19&0&80&0&0&144&0&64&0&234&370&0\\1&0&19&0&31&0&95&107&0&98&0&225&0&0&568\\5&0&29&0&30&0&107&157&0&114&0&271&0&0&711\\0&16&0&33&0&144&0&0&264&0&118&0&427&681&0\\0&0&20&0&37&0&98&114&0&123&0&253&0&0&632\\0&7&0&14&0&64&0&0&118&0&54&0&191&306&0\\2&0&48&0&74&0&225&271&0&253&0&570&0&0&1440\\0&24&0&56&0&234&0&0&427&0&191&0&718&1122&0\\0&39&0&84&0&370&0&0&681&0&306&0&1122&1788&0\\10&0&125&0&180&0&568&711&0&632&0&1440&0&0&3696\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&8&6&14&80&95&157&264&123&54&570&718&1788&3696&1928&2005&4910&4385&1296\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
---|
$-$ | $1$ | $1/2$ | $1/8$ | $0$ | $0$ | $0$ | $3/8$ |
---|
$a_1=0$ | $1/2$ | $1/2$ | $1/8$ | $0$ | $0$ | $0$ | $3/8$ |
---|
$a_3=0$ | $1/2$ | $1/2$ | $1/8$ | $0$ | $0$ | $0$ | $3/8$ |
---|
$a_1=a_3=0$ | $1/2$ | $1/2$ | $1/8$ | $0$ | $0$ | $0$ | $3/8$ |
---|