Subgroup ($H$) information
| Description: | $C_7\times C_{28}$ |
| Order: | \(196\)\(\medspace = 2^{2} \cdot 7^{2} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Generators: |
$b^{14}c^{7}, c^{2}, b^{28}, b^{8}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 7$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_{14}^2.D_4$ |
| Order: | \(1568\)\(\medspace = 2^{5} \cdot 7^{2} \) |
| Exponent: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2\times C_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Outer Automorphisms: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_6\times C_7:C_3).C_2^6.C_2$ |
| $\operatorname{Aut}(H)$ | $C_2\times \GL(2,7)$, of order \(4032\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times C_6^2$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(224\)\(\medspace = 2^{5} \cdot 7 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $D_{14}:C_4$ |