Subgroup ($H$) information
| Description: | $C_2\times C_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$b^{3}c^{7}, c^{84}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_6.(C_4\times D_{28})$ |
| Order: | \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{42}.(C_2^5\times C_6).C_2^3$ |
| $\operatorname{Aut}(H)$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $\operatorname{res}(S)$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(5376\)\(\medspace = 2^{8} \cdot 3 \cdot 7 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{28}:Q_8$ | ||||
| Normalizer: | $C_4^2.D_{14}$ | ||||
| Normal closure: | $C_6:C_4$ | ||||
| Core: | $C_2^2$ | ||||
| Minimal over-subgroups: | $C_2\times C_{28}$ | $C_6:C_4$ | $C_4^2$ | $C_4^2$ | $C_4:C_4$ |
| Maximal under-subgroups: | $C_2^2$ | $C_4$ |
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $0$ |
| Projective image | $S_3\times D_{28}$ |