Subgroup ($H$) information
| Description: | $C_3^4:C_6$ |
| Order: | \(486\)\(\medspace = 2 \cdot 3^{5} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
35 & 13 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 12 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 15 \\
27 & 10
\end{array}\right), \left(\begin{array}{rr}
25 & 24 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
25 & 0 \\
0 & 25
\end{array}\right), \left(\begin{array}{rr}
25 & 4 \\
24 & 1
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is normal, a direct factor, nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_2\times C_3^3:C_6^2$ |
| Order: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| 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: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.C_6^2.C_3^3.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_3^4:C_3.D_6^2$ |
| $\card{\operatorname{res}(S)}$ | \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_3^3:C_6$, of order \(162\)\(\medspace = 2 \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $2$ |
| Projective image | $C_2\times C_3^3:D_6$ |