Subgroup ($H$) information
| Description: | $C_{68}$ |
| Order: | \(68\)\(\medspace = 2^{2} \cdot 17 \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(68\)\(\medspace = 2^{2} \cdot 17 \) |
| Generators: |
$c^{102}, c^{8}, c^{68}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal) and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,17$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{136}.D_4$ |
| Order: | \(1088\)\(\medspace = 2^{6} \cdot 17 \) |
| Exponent: | \(136\)\(\medspace = 2^{3} \cdot 17 \) |
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2\times D_4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2\wr C_2^2$, of order \(64\)\(\medspace = 2^{6} \) |
| Outer Automorphisms: | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{16}\times C_4.C_2^5.C_2$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{16}$, of order \(32\)\(\medspace = 2^{5} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times C_{16}$, of order \(32\)\(\medspace = 2^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(128\)\(\medspace = 2^{7} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_4:D_4$ |