Subgroup ($H$) information
| Description: | $C_2^4:C_4\times F_5$ |
| Order: | \(1280\)\(\medspace = 2^{8} \cdot 5 \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$\langle(2,3)(4,5)(6,8,7,9)(10,14)(11,16)(12,15)(13,17), (10,12)(11,13)(14,16)(15,17) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $C_2\wr S_3\times F_5$ |
| Order: | \(7680\)\(\medspace = 2^{9} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^4.\OD_{16}$, of order \(92160\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2^{10}.A_4.C_2^3\times F_5$ |
| $W$ | $C_2^4\times F_5$, of order \(320\)\(\medspace = 2^{6} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^3:S_4\times F_5$ |