Subgroup ($H$) information
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(272\)\(\medspace = 2^{4} \cdot 17 \) |
| Exponent: | \(2\) |
| Generators: |
$b^{8}, c^{17}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Ambient group ($G$) information
| Description: | $D_{68}.C_8$ |
| Order: | \(1088\)\(\medspace = 2^{6} \cdot 17 \) |
| Exponent: | \(272\)\(\medspace = 2^{4} \cdot 17 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_{17}:\OD_{16}$ |
| Order: | \(272\)\(\medspace = 2^{4} \cdot 17 \) |
| Exponent: | \(136\)\(\medspace = 2^{3} \cdot 17 \) |
| Automorphism Group: | $D_4\times F_{17}$, of order \(2176\)\(\medspace = 2^{7} \cdot 17 \) |
| Outer Automorphisms: | $C_2\times C_8$, of order \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{17}:((C_2^2\times C_8).C_2^4)$ |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4352\)\(\medspace = 2^{8} \cdot 17 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{34}:C_{16}$ | ||
| Normalizer: | $D_{68}.C_8$ | ||
| Minimal over-subgroups: | $C_2\times C_{34}$ | $C_2\times C_4$ | $D_4$ |
| Maximal under-subgroups: | $C_2$ | $C_2$ |
Other information
| Möbius function | $0$ |
| Projective image | $D_{34}:C_8$ |