| Name: | $D_8:C_2$ |
| Order: | $32$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0& 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\\end{bmatrix}, \begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{5} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{7} & 0 \\0 & 0 & 0 & 0 & 0& \zeta_{12}^{7} \\\end{bmatrix}, \begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0& -1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 &0 & 0 & -1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 &-i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\-i & 0 & 0 & 0 & 0 & 0 \\0 & -i & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $B(2,4)$, $J(A(2,4))$, $J_n(A(2,4))$, $J_s(B(1,4)_2)$${}^{\times 2}$, $J_s(B(1,4;2)_2)$${}^{\times 2}$ |
| Minimal supergroups: | $J(B(4,4))$, $J_s(B(T,2))$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$27$ |
$0$ |
$620$ |
$0$ |
$17115$ |
$0$ |
$514332$ |
$0$ |
$16248078$ |
| $a_2$ |
$1$ |
$2$ |
$9$ |
$65$ |
$664$ |
$7922$ |
$101622$ |
$1356287$ |
$18560486$ |
$258363992$ |
$3640100194$ |
$51736885697$ |
$740153042317$ |
| $a_3$ |
$1$ |
$0$ |
$10$ |
$0$ |
$1822$ |
$0$ |
$521890$ |
$0$ |
$172359726$ |
$0$ |
$60613064520$ |
$0$ |
$22000674456706$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$9$ |
$3$ |
$12$ |
$27$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$10$ |
$65$ |
$33$ |
$124$ |
$69$ |
$274$ |
$620$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$92$ |
$664$ |
$360$ |
$207$ |
$1440$ |
$809$ |
$3268$ |
$1820$ |
$7455$ |
$17115$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1074$ |
$7922$ |
$597$ |
$4360$ |
$2426$ |
$17982$ |
$9958$ |
$5542$ |
$41376$ |
$22866$ |
$95564$ |
$52661$ |
$221382$ |
$514332$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1822$ |
$13384$ |
$101622$ |
$7417$ |
$55791$ |
$30755$ |
$234904$ |
$16968$ |
$128984$ |
$70940$ |
$545731$ |
$299128$ |
$164274$ |
$1271001$ |
| $$ |
$695597$ |
$2966523$ |
$1621074$ |
$6936804$ |
$16248078$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&4&0&1&0&2&0&0&6\\0&2&0&1&0&7&0&0&11&0&6&0&13&25&0\\1&0&6&0&3&0&14&19&0&9&0&30&0&0&71\\0&1&0&6&0&13&0&0&18&0&7&0&39&46&0\\0&0&3&0&12&0&18&20&0&26&0&46&0&0&100\\0&7&0&13&0&52&0&0&84&0&40&0&134&208&0\\0&0&14&0&18&0&62&58&0&48&0&131&0&0&308\\4&0&19&0&20&0&58&96&0&68&0&151&0&0&381\\0&11&0&18&0&84&0&0&150&0&69&0&224&369&0\\1&0&9&0&26&0&48&68&0&80&0&133&0&0&332\\0&6&0&7&0&40&0&0&69&0&36&0&99&171&0\\2&0&30&0&46&0&131&151&0&133&0&318&0&0&760\\0&13&0&39&0&134&0&0&224&0&99&0&406&587&0\\0&25&0&46&0&208&0&0&369&0&171&0&587&946&0\\6&0&71&0&100&0&308&381&0&332&0&760&0&0&1916\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&6&6&12&52&62&96&150&80&36&318&406&946&1916&1018&1039&2542&2245&720\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $0$ | $0$ | $1/4$ | $0$ | $1/4$ |
|---|
| $a_1=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $1/4$ | $0$ | $1/4$ |
|---|
| $a_3=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $1/4$ | $0$ | $1/4$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $1/4$ | $0$ | $1/4$ |
|---|
