Subgroup ($H$) information
| Description: | $C_{10}.C_4^2$ |
| Order: | \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$\langle(3,4)(5,8)(6,7)(9,14,12,10)(13,15), (1,2)(3,4)(5,6)(7,8), (9,11,12,14,10) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $D_{10}.D_4\times S_4$ |
| Order: | \(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_5\times A_4).C_2^6.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2^2\wr C_2\times F_5$, of order \(640\)\(\medspace = 2^{7} \cdot 5 \) |
| $\operatorname{res}(S)$ | $C_2^4\times F_5$, of order \(320\)\(\medspace = 2^{6} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(16\)\(\medspace = 2^{4} \) |
| $W$ | $C_2^3\times F_5$, of order \(160\)\(\medspace = 2^{5} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $24$ |
| Number of conjugacy classes in this autjugacy class | $8$ |
| Möbius function | not computed |
| Projective image | $C_2^2\times F_5\times S_4$ |