Subgroup ($H$) information
| Description: | $C_{12}.D_{15}$ |
| Order: | \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Generators: |
$a, b^{3}, a^{4}, b^{10}, a^{2}, d^{2}$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_{60}.S_4$ |
| Order: | \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times F_5\times C_3:S_3:S_4$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times \AGL(2,3)\times F_5$ |
| $\card{\operatorname{res}(S)}$ | \(8640\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $C_3:D_{15}$, of order \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | $-1$ |
| Projective image | $C_{15}:S_4$ |