Subgroup ($H$) information
| Description: | $F_5$ |
| Order: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Index: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$\langle(2,3,5,4)(6,11)(7,10)(14,15), (2,5)(3,4), (1,3,5,2,4)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_2\times \GL(2,\mathbb{Z}/4):F_5$ |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_5\times A_4).C_2^4.C_2^6$ |
| $\operatorname{Aut}(H)$ | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $\operatorname{res}(S)$ | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(256\)\(\medspace = 2^{8} \) |
| $W$ | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $12$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $C_2\times \GL(2,\mathbb{Z}/4):F_5$ |