Subgroup ($H$) information
| Description: | $(C_3^2\times S_3^2):C_4$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$ac^{3}, b^{2}c^{16}de^{2}, c^{8}, b^{3}, c^{12}d^{2}e, e, d, a^{2}c^{12}de$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
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 \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $\SOPlus(4,2)^2.C_2$, of order \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \) |
| $W$ | $C_3^4:(C_4\times D_4)$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $C_3^4:(C_4\times \SD_{16})$ |