Subgroup ($H$) information
| Description: | $C_{10}:D_4$ |
| Order: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Index: | \(540\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 5 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$\langle(1,2)(3,4), (5,9,8,6,7), (1,3)(2,4), (11,15)(12,13), (1,2,3,4)(5,6)(8,9)(10,14)(11,12)(13,15)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $(C_5\times A_4):S_6$ |
| Order: | \(43200\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(F_5\times S_4).A_6.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_2^2\wr C_2\times F_5$, of order \(640\)\(\medspace = 2^{7} \cdot 5 \) |
| $W$ | $C_2\times D_{10}$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $135$ |
| Möbius function | $8$ |
| Projective image | $(C_5\times A_4):S_6$ |