Subgroup ($H$) information
| Description: | $\SOPlus(4,2)$ |
| Order: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Index: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,2)(3,6)(4,8)(5,9)(7,10)(11,12)(13,14)(15,16), (3,7,9)(5,10,6), (4,11) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian, monomial (hence solvable), and rational.
Ambient group ($G$) information
| Description: | $C_3^4:C_4^2:C_2^2$ |
| Order: | \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.C_2^3.C_2^5.C_2^4$ |
| $\operatorname{Aut}(H)$ | $F_9:C_2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $W$ | $\SOPlus(4,2)$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_2^2$ | ||
| Normalizer: | $C_6^2:D_4$ | ||
| Normal closure: | $C_2\times C_3^4:D_4$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $C_3^4:D_4$ | $S_3^2:C_2^2$ | $S_3^2:C_2^2$ |
| Maximal under-subgroups: | $S_3^2$ | $C_3^2:C_4$ | $D_4$ |
Other information
| Number of subgroups in this autjugacy class | $288$ |
| Number of conjugacy classes in this autjugacy class | $16$ |
| Möbius function | $0$ |
| Projective image | $C_3^4:C_4^2:C_2^2$ |