Subgroup ($H$) information
| Description: | $S_4$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Index: | \(13500\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 5^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$a, b^{4}d^{3}e^{14}f^{10}, cd^{15}e^{14}f^{13}, d^{15}e^{2}f^{10}$
|
| Derived length: | $3$ |
The subgroup is nonabelian, monomial (hence solvable), and rational.
Ambient group ($G$) information
| Description: | $C_{15}^3.(C_4\times S_4)$ |
| Order: | \(324000\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 5^{3} \) |
| Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_{15}\wr S_3.C_4$, of order \(648000\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $W$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_4$ | ||
| Normalizer: | $C_4\times S_4$ | ||
| Normal closure: | $C_{15}^3.S_4$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $C_5^3:S_4$ | $C_3^3:S_4$ | $C_2\times S_4$ |
| Maximal under-subgroups: | $A_4$ | $D_4$ | $S_3$ |
Other information
| Number of subgroups in this autjugacy class | $3375$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_{15}^3.(C_4\times S_4)$ |