Subgroup ($H$) information
| Description: | $C_4^2.D_{10}$ |
| Order: | \(320\)\(\medspace = 2^{6} \cdot 5 \) |
| Index: | \(3\) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Generators: |
$a, b^{4}, c^{15}, b^{2}, b, c^{12}, c^{30}$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a Hall subgroup, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $C_{12}.(C_2\times D_{20})$ |
| Order: | \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^7\times S_3\times F_5$, of order \(15360\)\(\medspace = 2^{10} \cdot 3 \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_5:(C_2^5.C_2^6)$, of order \(10240\)\(\medspace = 2^{11} \cdot 5 \) |
| $\card{\operatorname{res}(S)}$ | \(2560\)\(\medspace = 2^{9} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_2\times D_{10}$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $-1$ |
| Projective image | $S_3\times D_{10}$ |