Subgroup ($H$) information
| Description: | $C_3\wr C_3:C_6$ |
| Order: | \(486\)\(\medspace = 2 \cdot 3^{5} \) |
| Index: | \(2\) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$\left(\begin{array}{rrrr}
1 & 2 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 2 & 1 & 0 \\
0 & 1 & 0 & 1
\end{array}\right), \left(\begin{array}{rrrr}
2 & 0 & 2 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
2 & 0 & 1 & 1
\end{array}\right), \left(\begin{array}{rrrr}
1 & 2 & 1 & 1 \\
0 & 1 & 0 & 0 \\
0 & 2 & 2 & 1 \\
0 & 1 & 2 & 0
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
2 & 1 & 1 & 2 \\
1 & 2 & 0 & 2
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0 & 2 & 2 & 2 \\
0 & 0 & 1 & 0 \\
0 & 1 & 2 & 0
\end{array}\right), \left(\begin{array}{rrrr}
1 & 2 & 1 & 1 \\
2 & 1 & 1 & 0 \\
0 & 0 & 1 & 0 \\
2 & 0 & 1 & 1
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, and supersolvable (hence solvable and monomial).
Ambient group ($G$) information
| Description: | $C_3^3.S_3^2$ |
| Order: | \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.S_3^2$, of order \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $C_3^4:S_3^2$, of order \(2916\)\(\medspace = 2^{2} \cdot 3^{6} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_3^3:D_6$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3\) |
| $W$ | $C_3^3:D_6$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | $C_3$ | ||||
| Normalizer: | $C_3^3.S_3^2$ | ||||
| Complements: | $C_2$ $C_2$ | ||||
| Minimal over-subgroups: | $C_3^3.S_3^2$ | ||||
| Maximal under-subgroups: | $C_3^3:C_3^2$ | $C_3^3:S_3$ | $C_3\wr S_3$ | $S_3\times \He_3$ | $C_3\wr S_3$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_3^3.S_3^2$ |