Subgroup ($H$) information
| Description: | $C_3:C_{40}$ |
| Order: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Index: | \(35\)\(\medspace = 5 \cdot 7 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Generators: |
$a^{5}, a^{10}, a^{20}, a^{8}, b^{70}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{105}:C_{40}$ |
| Order: | \(4200\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \) |
| Exponent: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times C_4\times F_5\times S_3\times F_7$ |
| $\operatorname{Aut}(H)$ | $C_{12}:C_2^3$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_{12}:C_2^3$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_{20}$ | ||
| Normalizer: | $C_3:C_{40}$ | ||
| Normal closure: | $C_{105}:C_{40}$ | ||
| Core: | $C_{60}$ | ||
| Minimal over-subgroups: | $C_{21}:C_{40}$ | $C_{15}:C_{40}$ | |
| Maximal under-subgroups: | $C_{60}$ | $C_{40}$ | $C_3:C_8$ |
Other information
| Number of subgroups in this conjugacy class | $35$ |
| Möbius function | $1$ |
| Projective image | $D_{105}$ |