Subgroup ($H$) information
| Description: | $C_3^2:C_4\times \SD_{16}$ |
| Order: | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$ad^{2}e^{2}, c^{18}d^{2}e^{2}, c^{12}e, b^{3}d^{2}e^{2}, a^{2}d^{2}, c^{8}, b^{2}c^{16}de^{2}, c^{3}de$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, monomial (hence solvable), and metabelian.
Ambient group ($G$) information
| Description: | $C_3^4:(C_4\times \SD_{16})$ |
| Order: | \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.C_8^2.C_2^2$, of order \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $(C_2\times F_9).C_2^5$, of order \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \) |
| $W$ | $C_3^2:C_4\times D_4$, of order \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $9$ |
| Möbius function | $-1$ |
| Projective image | $C_3^4:(C_4\times \SD_{16})$ |