Subgroup ($H$) information
| Description: | $C_5^2:C_6$ |
| Order: | \(150\)\(\medspace = 2 \cdot 3 \cdot 5^{2} \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$b^{3}, c^{3}d^{3}, d^{3}, b^{2}c^{2}d^{12}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_{15}^2:D_6$ |
| Order: | \(2700\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 5^{2} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{15}^2.C_{12}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $F_{25}:C_2$, of order \(1200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \) |
| $\operatorname{res}(S)$ | $F_{25}:C_2$, of order \(1200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_5^2:D_6$, of order \(300\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $0$ |
| Projective image | $C_{15}^2:D_6$ |