Subgroup ($H$) information
| Description: | $D_{30}$ |
| Order: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Index: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(1,9,13,8,14)(10,12,11), (2,4)(5,7)(8,13)(9,14)(10,11), (2,4)(3,6)(5,7), (10,11,12)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Ambient group ($G$) information
| Description: | $A_4\times D_6\times A_5$ |
| Order: | \(8640\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, an A-group, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_6\times S_4\times S_5$, of order \(34560\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $D_6\times F_5$, of order \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| $W$ | $S_3\times D_5$, of order \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $18$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $S_3\times A_4\times A_5$ |