Subgroup ($H$) information
| Description: | $C_3^3:S_3^2$ |
| Order: | \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| Index: | $1$ |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$\left(\begin{array}{rr}
1 & 0 \\
0 & 17
\end{array}\right), \left(\begin{array}{rr}
7 & 6 \\
0 & 13
\end{array}\right), \left(\begin{array}{rr}
10 & 9 \\
9 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 9 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 12 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 12 \\
6 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 10 \\
0 & 1
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, and supersolvable (hence 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: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.S_3^3$, of order \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $C_3^3.S_3^3$, of order \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
| $W$ | $C_3^3:S_3^2$, of order \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
Related subgroups
| Centralizer: | $C_1$ | |||||||
| Normalizer: | $C_3^3:S_3^2$ | |||||||
| Complements: | $C_1$ | |||||||
| Maximal under-subgroups: | $C_3^4:C_6$ | $C_3^4:S_3$ | $C_3^4:S_3$ | $C_3^3:D_6$ | $C_3^2:S_3^2$ | $C_3^2:S_3^2$ | $C_3^2.S_3^2$ | $C_3^2.S_3^2$ |
Other information
| Möbius function | $1$ |
| Projective image | $C_3^3:S_3^2$ |