Subgroup ($H$) information
| Description: | $C_2$ |
| Order: | \(2\) |
| Index: | \(544\)\(\medspace = 2^{5} \cdot 17 \) |
| Exponent: | \(2\) |
| Generators: |
$b^{4}c^{34}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, a $p$-group, simple, and rational.
Ambient group ($G$) information
| Description: | $Q_8\times C_{136}$ |
| Order: | \(1088\)\(\medspace = 2^{6} \cdot 17 \) |
| Exponent: | \(136\)\(\medspace = 2^{3} \cdot 17 \) |
| Nilpotency class: | $2$ |
| 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: | $\OD_{16}:C_{34}$ |
| Order: | \(544\)\(\medspace = 2^{5} \cdot 17 \) |
| Exponent: | \(136\)\(\medspace = 2^{3} \cdot 17 \) |
| Automorphism Group: | $C_2^2\times C_{16}\times S_4$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Outer Automorphisms: | $C_{48}:C_2^3$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_{16}\times C_2^4:C_3.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_1$, of order $1$ |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(12288\)\(\medspace = 2^{12} \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $Q_8\times C_{136}$ | ||||
| Normalizer: | $Q_8\times C_{136}$ | ||||
| Minimal over-subgroups: | $C_{34}$ | $C_2^2$ | $C_4$ | $C_4$ | $C_4$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $0$ |
| Projective image | $\OD_{16}:C_{34}$ |