Subgroup ($H$) information
| Description: | $C_{15}^2:C_2^2$ |
| Order: | \(900\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
| Index: | \(3\) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$ac^{9}d^{6}, d^{3}, c^{10}, c^{10}d^{10}, b^{3}, c^{3}d^{6}$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, supersolvable (hence solvable and monomial), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{15}^2.C_{12}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $D_6\times F_5\wr C_2$, of order \(9600\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{2} \) |
| $\operatorname{res}(S)$ | $S_3\times C_5^2:\OD_{16}$, of order \(2400\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3\) |
| $W$ | $C_{15}:D_{10}$, 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 | $-1$ |
| Projective image | $(C_5\times C_{15}):D_6$ |