Subgroup ($H$) information
| Description: | $C_2$ |
| Order: | \(2\) |
| Index: | \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| Exponent: | \(2\) |
| Generators: |
$b^{2}c^{84}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), stem (hence central), a $p$-group, simple, and rational.
Ambient group ($G$) information
| Description: | $C_{168}:Q_8$ |
| 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.
Quotient group ($Q$) structure
| Description: | $C_{24}.D_{14}$ |
| Order: | \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Automorphism Group: | $C_{42}.(C_2^5\times C_6)$ |
| Outer Automorphisms: | $C_2^2\times C_6$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{42}.(C_2^5\times C_6).C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_1$, of order $1$ |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(64512\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 7 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{168}:Q_8$ | ||||
| Normalizer: | $C_{168}:Q_8$ | ||||
| Minimal over-subgroups: | $C_{14}$ | $C_6$ | $C_2^2$ | $C_4$ | $C_4$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{24}.D_{14}$ |