Subgroup ($H$) information
| Description: | $D_4:C_6$ |
| Order: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Index: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$b, d^{2}, d, e^{21}, e^{14}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_{84}.C_2^4$ |
| Order: | \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $D_{14}$ |
| Order: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Automorphism Group: | $C_2\times F_7$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(96768\)\(\medspace = 2^{9} \cdot 3^{3} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2^2\times S_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^2\times S_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \) |
| $W$ | $C_2^4$, of order \(16\)\(\medspace = 2^{4} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-14$ |
| Projective image | $C_{21}:C_2^5$ |