Subgroup ($H$) information
| Description: | $C_{12}:S_5$ |
| Order: | \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(6,9,8)(10,13,12,11), (10,12)(11,13), (1,2)(6,9,8)(10,13)(11,12), (1,5,4)(2,3)(10,13)(11,12), (6,9,8)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian and nonsolvable.
Ambient group ($G$) information
| Description: | $D_4\times A_4\times S_5$ |
| Order: | \(11520\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$, of order \(46080\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2\times D_4\times S_5$, of order \(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \) |
| $W$ | $C_2^2\times S_5$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^2\times A_4\times S_5$ |