Subgroup ($H$) information
| Description: | $C_4.\GL(2,\mathbb{Z}/4)$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Index: | \(5\) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(6,8)(7,11)(9,12)(10,13), (3,5,4)(6,8)(9,10)(12,13)(14,15), (7,11)(9,13) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, a Hall subgroup, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_2\times A_5:\SD_{16}$ |
| Order: | \(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^5:D_6$, of order \(15360\)\(\medspace = 2^{10} \cdot 3 \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $A_4.C_2^5.C_2^3$ |
| $\card{W}$ | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $5$ |
| Möbius function | not computed |
| Projective image | not computed |