Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(1372\)\(\medspace = 2^{2} \cdot 7^{3} \) |
| Exponent: | \(3\) |
| Generators: |
$b^{14}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a direct factor, cyclic (hence elementary, hyperelementary, metacyclic, and a Z-group), a $3$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_{21}.D_7^2$ |
| Order: | \(4116\)\(\medspace = 2^{2} \cdot 3 \cdot 7^{3} \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
Quotient group ($Q$) structure
| Description: | $C_7^2:D_{14}$ |
| Order: | \(1372\)\(\medspace = 2^{2} \cdot 7^{3} \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Automorphism Group: | $\He_7:C_6\wr C_2$, of order \(24696\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7^{3} \) |
| Outer Automorphisms: | $C_3\times S_3$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian and supersolvable (hence solvable and monomial).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_7:(C_6^2:C_2^2)$, of order \(49392\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7^{3} \) |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(24696\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7^{3} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{21}.D_7^2$ | ||||||||
| Normalizer: | $C_{21}.D_7^2$ | ||||||||
| Complements: | $C_7^2:D_{14}$ | ||||||||
| Minimal over-subgroups: | $C_{21}$ | $C_{21}$ | $C_{21}$ | $C_{21}$ | $C_{21}$ | $C_{21}$ | $C_6$ | $C_6$ | $C_6$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_7^2:D_{14}$ |