Subgroup ($H$) information
| Description: | $D_{21}$ |
| Order: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Index: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$ab, c^{15}, b^{2}c^{70}$
|
| 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: | $C_{15}:D_{42}$ |
| Order: | \(1260\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4\times S_3^2\times F_7$ |
| $\operatorname{Aut}(H)$ | $S_3\times F_7$, of order \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{res}(S)$ | $S_3\times F_7$, of order \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $D_{21}$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_5$ | ||
| Normalizer: | $C_5\times D_{21}$ | ||
| Normal closure: | $C_3:D_{21}$ | ||
| Core: | $C_7$ | ||
| Minimal over-subgroups: | $C_5\times D_{21}$ | $C_3:D_{21}$ | |
| Maximal under-subgroups: | $C_{21}$ | $D_7$ | $S_3$ |
Other information
| Number of subgroups in this conjugacy class | $6$ |
| Möbius function | $0$ |
| Projective image | $C_{15}:D_{42}$ |