Subgroup ($H$) information
| Description: | $C_2$ |
| Order: | \(2\) |
| Index: | \(525\)\(\medspace = 3 \cdot 5^{2} \cdot 7 \) |
| Exponent: | \(2\) |
| Generators: |
$a$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $2$-Sylow subgroup (hence a Hall subgroup), a $p$-group, simple, and rational.
Ambient group ($G$) information
| Description: | $C_{15}:D_{35}$ |
| Order: | \(1050\)\(\medspace = 2 \cdot 3 \cdot 5^{2} \cdot 7 \) |
| Exponent: | \(210\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_5^2:C_4.S_5\times F_7$ |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $\operatorname{res}(S)$ | $C_1$, of order $1$ |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(5760\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_6$ | |||||||
| Normalizer: | $C_6$ | |||||||
| Normal closure: | $C_5:D_{35}$ | |||||||
| Core: | $C_1$ | |||||||
| Minimal over-subgroups: | $D_7$ | $D_5$ | $D_5$ | $D_5$ | $D_5$ | $D_5$ | $D_5$ | $C_6$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of subgroups in this conjugacy class | $175$ |
| Möbius function | $5$ |
| Projective image | $C_{15}:D_{35}$ |