| Name: | $C_2^2$ |
| Order: | $4$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{5} & 0 & 0 &0 & 0 \\0 & 0 & \zeta_{6}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{1} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $J(A(1,1))$${}^{\times 2}$, $A(1,2)$ |
| Minimal supergroups: | $J(A(1,4)_1)$${}^{\times 2}$, $J_s(A(1,4)_2)$, $J(A(1,6)_1)$, $J(A(3,2))$, $J(A(1,4)_2)$${}^{\times 2}$, $J(A(1,6)_2)$, $J(B(3,1))$, $J_s(A(2,2))$, $J(A(2,2))$${}^{\times 6}$, $J_s(A(3,2))$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$5$ |
$0$ |
$123$ |
$0$ |
$3650$ |
$0$ |
$114835$ |
$0$ |
$3720150$ |
$0$ |
$122763102$ |
| $a_2$ |
$1$ |
$4$ |
$30$ |
$319$ |
$3994$ |
$53169$ |
$727395$ |
$10106772$ |
$141909786$ |
$2008193305$ |
$28592753305$ |
$409122110082$ |
$5877905280943$ |
| $a_3$ |
$1$ |
$0$ |
$42$ |
$0$ |
$11798$ |
$0$ |
$3863850$ |
$0$ |
$1340493518$ |
$0$ |
$479996954952$ |
$0$ |
$175364874474218$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$4$ |
$5$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$30$ |
$13$ |
$54$ |
$123$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$42$ |
$319$ |
$167$ |
$689$ |
$376$ |
$1579$ |
$3650$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$517$ |
$3994$ |
$2160$ |
$1187$ |
$9113$ |
$4983$ |
$21142$ |
$11535$ |
$49205$ |
$114835$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$6814$ |
$53169$ |
$3720$ |
$28918$ |
$15779$ |
$123567$ |
$67310$ |
$36706$ |
$288684$ |
$157105$ |
$675786$ |
$367388$ |
$1584471$ |
$3720150$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$11798$ |
$92111$ |
$727395$ |
$50198$ |
$395241$ |
$215017$ |
$1703676$ |
$117022$ |
$925467$ |
$503000$ |
$3998931$ |
$2170593$ |
$1178842$ |
$9398631$ |
| $$ |
$5097918$ |
$22113861$ |
$11986842$ |
$52081176$ |
$122763102$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&3&0&1&0&5&13&0&6&0&14&0&0&40\\0&5&0&8&0&36&0&0&64&0&28&0&96&159&0\\3&0&23&0&23&0&82&109&0&81&0&195&0&0&500\\0&8&0&21&0&76&0&0&132&0&56&0&224&339&0\\1&0&23&0&45&0&117&137&0&138&0&290&0&0&720\\0&36&0&76&0&320&0&0&576&0&256&0&936&1480&0\\5&0&82&0&117&0&371&438&0&388&0&900&0&0&2272\\13&0&109&0&137&0&438&587&0&487&0&1105&0&0&2844\\0&64&0&132&0&576&0&0&1053&0&472&0&1708&2727&0\\6&0&81&0&138&0&388&487&0&462&0&994&0&0&2528\\0&28&0&56&0&256&0&0&472&0&216&0&764&1224&0\\14&0&195&0&290&0&900&1105&0&994&0&2262&0&0&5760\\0&96&0&224&0&936&0&0&1708&0&764&0&2872&4488&0\\0&159&0&339&0&1480&0&0&2727&0&1224&0&4488&7143&0\\40&0&500&0&720&0&2272&2844&0&2528&0&5760&0&0&14784\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&5&23&21&45&320&371&587&1053&462&216&2262&2872&7143&14784&7703&7930&19640&17474&5175\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
| $a_1=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
| $a_3=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
