Subgroup ($H$) information
| Description: | $C_3^3:C_6$ |
| Order: | \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
89 & 54 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
73 & 54 \\
18 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 45 \\
45 & 46
\end{array}\right), \left(\begin{array}{rr}
31 & 20 \\
30 & 31
\end{array}\right), \left(\begin{array}{rr}
1 & 60 \\
0 & 1
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_3^3:S_3^2$ |
| Order: | \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_6$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4:(D_4\times D_6)$, of order \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_3^3:\GL(3,3)$, of order \(606528\)\(\medspace = 2^{6} \cdot 3^{6} \cdot 13 \) |
| $\operatorname{res}(S)$ | $C_3^2:D_6^2$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3\) |
| $W$ | $C_3^2:S_3^2$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
Related subgroups
Other information
| Möbius function | $1$ |
| Projective image | $C_3^3:S_3^2$ |