Subgroup ($H$) information
| Description: | $C_7:C_{24}$ |
| Order: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Index: | \(2\) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Generators: |
$b, b^{2}, b^{4}, c^{7}, c^{3}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{21}:Q_{16}$ |
| Order: | \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| 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: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times D_4\times F_7$, of order \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2^3\times F_7$, of order \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^3\times F_7$, of order \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $D_{14}$, of order \(28\)\(\medspace = 2^{2} \cdot 7 \) |
Related subgroups
| Centralizer: | $C_{12}$ | ||
| Normalizer: | $C_{21}:Q_{16}$ | ||
| Minimal over-subgroups: | $C_{21}:Q_{16}$ | ||
| Maximal under-subgroups: | $C_{84}$ | $C_7:C_8$ | $C_{24}$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_7:D_4$ |