Subgroup ($H$) information
| Description: | $C_7:C_8$ |
| Order: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Index: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Exponent: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Generators: |
$a, a^{4}, b^{15}, a^{2}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Hall subgroup, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{105}:C_8$ |
| Order: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Exponent: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^3\times F_5\times F_7$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times F_7$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(S)$ | $C_2^2\times F_7$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $D_7$, of order \(14\)\(\medspace = 2 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_{12}$ | |
| Normalizer: | $C_7:C_{24}$ | |
| Normal closure: | $C_{35}:C_8$ | |
| Core: | $C_{28}$ | |
| Minimal over-subgroups: | $C_{35}:C_8$ | $C_7:C_{24}$ |
| Maximal under-subgroups: | $C_{28}$ | $C_8$ |
Other information
| Number of subgroups in this conjugacy class | $5$ |
| Möbius function | $1$ |
| Projective image | $C_3\times D_{35}$ |