Subgroup ($H$) information
| Description: | $C_2\times F_5$ |
| Order: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$ac, c^{2}, d^{6}, b$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Ambient group ($G$) information
| Description: | $C_2\times D_6\times F_5$ |
| Order: | \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4:S_3^2\times F_5$, of order \(11520\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2\times F_5$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| $\operatorname{res}(S)$ | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $W$ | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $48$ |
| Number of conjugacy classes in this autjugacy class | $16$ |
| Möbius function | $-2$ |
| Projective image | $C_2\times D_6\times F_5$ |