| Name: | $D_8:C_2$ |
| Order: | $32$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0& 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\\end{bmatrix}, \begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{5} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{7} & 0 \\0 & 0 & 0 & 0 & 0& \zeta_{12}^{7} \\\end{bmatrix}, \begin{bmatrix}i & 0 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & \zeta_{8}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -i& 0 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{7} \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $J(A(2,4))$, $J_s(A(2,4))$, $B(2,4;4)$, $J(A(1,8)_2)$${}^{\times 2}$, $J_s(A(1,8)_2)$${}^{\times 2}$ |
| Minimal supergroups: | $J(B(4,4))$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$27$ |
$0$ |
$620$ |
$0$ |
$17115$ |
$0$ |
$514332$ |
$0$ |
$16248078$ |
| $a_2$ |
$1$ |
$2$ |
$10$ |
$68$ |
$674$ |
$7952$ |
$101713$ |
$1356560$ |
$18561306$ |
$258366452$ |
$3640107575$ |
$51736907840$ |
$740153108747$ |
| $a_3$ |
$1$ |
$0$ |
$10$ |
$0$ |
$1818$ |
$0$ |
$521800$ |
$0$ |
$172358158$ |
$0$ |
$60613039320$ |
$0$ |
$22000674064446$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$10$ |
$3$ |
$12$ |
$27$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$10$ |
$68$ |
$33$ |
$124$ |
$69$ |
$274$ |
$620$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$92$ |
$674$ |
$360$ |
$206$ |
$1440$ |
$808$ |
$3267$ |
$1820$ |
$7455$ |
$17115$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1074$ |
$7952$ |
$597$ |
$4360$ |
$2425$ |
$17982$ |
$9956$ |
$5536$ |
$41373$ |
$22860$ |
$95558$ |
$52661$ |
$221382$ |
$514332$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1818$ |
$13384$ |
$101713$ |
$7413$ |
$55791$ |
$30750$ |
$234904$ |
$16968$ |
$128977$ |
$70934$ |
$545721$ |
$299116$ |
$164246$ |
$1270983$ |
| $$ |
$695569$ |
$2966495$ |
$1621074$ |
$6936804$ |
$16248078$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&5&0&0&0&1&0&0&6\\0&2&0&1&0&7&0&0&11&0&6&0&13&25&0\\1&0&7&0&2&0&13&20&0&9&0&30&0&0&71\\0&1&0&6&0&13&0&0&18&0&7&0&38&47&0\\0&0&2&0&13&0&19&18&0&26&0&47&0&0&100\\0&7&0&13&0&52&0&0&84&0&40&0&134&208&0\\0&0&13&0&19&0&62&58&0&50&0&131&0&0&308\\5&0&20&0&18&0&58&100&0&63&0&148&0&0&381\\0&11&0&18&0&84&0&0&150&0&68&0&226&368&0\\0&0&9&0&26&0&50&63&0&79&0&136&0&0&332\\0&6&0&7&0&40&0&0&68&0&36&0&101&170&0\\1&0&30&0&47&0&131&148&0&136&0&320&0&0&760\\0&13&0&38&0&134&0&0&226&0&101&0&400&590&0\\0&25&0&47&0&208&0&0&368&0&170&0&590&944&0\\6&0&71&0&100&0&308&381&0&332&0&760&0&0&1916\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&7&6&13&52&62&100&150&79&36&320&400&944&1916&1004&1048&2524&2244&684\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $1/8$ | $0$ | $0$ | $0$ | $3/8$ |
|---|
| $a_1=0$ | $1/2$ | $1/2$ | $1/8$ | $0$ | $0$ | $0$ | $3/8$ |
|---|
| $a_3=0$ | $1/2$ | $1/2$ | $1/8$ | $0$ | $0$ | $0$ | $3/8$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/2$ | $1/8$ | $0$ | $0$ | $0$ | $3/8$ |
|---|
