Subgroup ($H$) information
| Description: | $D_5^2:S_3^2$ |
| Order: | \(3600\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
| Index: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$ac^{5}d^{26}f^{10}, c^{4}d^{20}, d^{12}f^{6}, d^{30}e^{2}f^{3}, f^{10}, f^{3}, b^{3}d^{48}e^{4}, c^{6}d^{30}f^{6}$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_5^3.C_{12}^2.S_3^2$ |
| Order: | \(648000\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5^{3} \) |
| Exponent: | \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^3.C_{12}^2.(C_{12}\times S_3^2)$ |
| $\operatorname{Aut}(H)$ | $C_{15}^2.C_2^3.C_2^4$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $90$ |
| Möbius function | not computed |
| Projective image | not computed |