Subgroup ($H$) information
| Description: | $C_6\times \OD_{16}$ |
| Order: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$ab, c^{6}, a^{2}c^{3}, b^{10}c^{3}, c^{9}, c^{4}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_{60}.C_4^2$ |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5:(C_2^4.C_2^6.C_2)$ |
| $\operatorname{Aut}(H)$ | $C_2^5:D_4$, of order \(256\)\(\medspace = 2^{8} \) |
| $\operatorname{res}(S)$ | $D_4:C_2^4$, of order \(128\)\(\medspace = 2^{7} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(16\)\(\medspace = 2^{4} \) |
| $W$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $5$ |
| Möbius function | $1$ |
| Projective image | $D_{20}$ |