| Name: | $C_3\times Q_8$ |
| Order: | $24$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{5} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{7} & 0 \\0 & 0 & 0 & 0 & 0& \zeta_{12}^{7} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & -1 &0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $J_n(A(1,4)_1)$, $J_n(A(3,2))$${}^{\times 2}$, $A(3,4)$ |
| Minimal supergroups: | $J(B(T,2))$, $J_s(B(3,4))$, $J_s(B(3,4;4))$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$45$ |
$0$ |
$950$ |
$0$ |
$24465$ |
$0$ |
$708498$ |
$0$ |
$21988890$ |
| $a_2$ |
$1$ |
$2$ |
$12$ |
$95$ |
$970$ |
$11237$ |
$140569$ |
$1846070$ |
$25028110$ |
$346592381$ |
$4869395107$ |
$69103695080$ |
$987797501645$ |
| $a_3$ |
$1$ |
$0$ |
$16$ |
$0$ |
$2644$ |
$0$ |
$713050$ |
$0$ |
$231277788$ |
$0$ |
$80947583556$ |
$0$ |
$29346093039010$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$12$ |
$6$ |
$21$ |
$45$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$16$ |
$95$ |
$54$ |
$197$ |
$114$ |
$429$ |
$950$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$146$ |
$970$ |
$550$ |
$317$ |
$2133$ |
$1213$ |
$4774$ |
$2700$ |
$10770$ |
$24465$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1588$ |
$11237$ |
$900$ |
$6292$ |
$3541$ |
$25445$ |
$14214$ |
$7974$ |
$58056$ |
$32326$ |
$133132$ |
$73892$ |
$306565$ |
$708498$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$2644$ |
$18942$ |
$140569$ |
$10578$ |
$77764$ |
$43147$ |
$323645$ |
$24006$ |
$178613$ |
$98818$ |
$748257$ |
$411964$ |
$227350$ |
$1735572$ |
| $$ |
$953498$ |
$4036767$ |
$2213400$ |
$9411570$ |
$21988890$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&2&4&0&1&0&4&0&0&8\\0&3&0&3&0&12&0&0&15&0&9&0&21&35&0\\1&0&9&0&8&0&21&25&0&18&0&47&0&0&98\\0&3&0&7&0&20&0&0&29&0&13&0&48&68&0\\1&0&8&0&15&0&28&33&0&26&0&64&0&0&136\\0&12&0&20&0&78&0&0&118&0&58&0&186&286&0\\2&0&21&0&28&0&83&92&0&72&0&178&0&0&420\\4&0&25&0&33&0&92&117&0&92&0&215&0&0&516\\0&15&0&29&0&118&0&0&203&0&90&0&317&496&0\\1&0&18&0&26&0&72&92&0&88&0&180&0&0&448\\0&9&0&13&0&58&0&0&90&0&50&0&144&230&0\\4&0&47&0&64&0&178&215&0&180&0&434&0&0&1024\\0&21&0&48&0&186&0&0&317&0&144&0&528&802&0\\0&35&0&68&0&286&0&0&496&0&230&0&802&1267&0\\8&0&98&0&136&0&420&516&0&448&0&1024&0&0&2574\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&9&7&15&78&83&117&203&88&50&434&528&1267&2574&1327&1354&3338&2964&877\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $7/12$ | $1/6$ | $1/12$ | $0$ | $1/3$ | $0$ |
|---|
| $a_1=0$ | $7/12$ | $7/12$ | $1/6$ | $1/12$ | $0$ | $1/3$ | $0$ |
|---|
| $a_3=0$ | $1/2$ | $1/2$ | $1/6$ | $0$ | $0$ | $1/3$ | $0$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/2$ | $1/6$ | $0$ | $0$ | $1/3$ | $0$ |
|---|
