Subgroup ($H$) information
| Description: | $C_2^3\times F_5$ |
| Order: | \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$\langle(6,9)(7,8)(10,11), (2,5,3,4)(6,9)(7,8)(10,11), (2,3)(4,5)(6,9)(7,8), (6,9)(7,8), (7,8), (1,4,3,2,5)(6,9)(7,8)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_2\times D_4\times S_5$ |
| Order: | \(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, nonsolvable, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^5.C_2^3.S_5$ |
| $\operatorname{Aut}(H)$ | $F_5\times C_2^3:\GL(3,2)$, of order \(26880\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \cdot 7 \) |
| $\operatorname{res}(S)$ | $C_2^2\wr C_2\times F_5$, of order \(640\)\(\medspace = 2^{7} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_2\times F_5$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
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 | $C_2^2\times S_5$ |