Subgroup ($H$) information
| Description: | $D_4\times C_{17}$ |
| Order: | \(136\)\(\medspace = 2^{3} \cdot 17 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(68\)\(\medspace = 2^{2} \cdot 17 \) |
| Generators: |
$a, b^{12}c^{17}, b^{8}, c^{2}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $\OD_{32}:C_{34}$ |
| Order: | \(1088\)\(\medspace = 2^{6} \cdot 17 \) |
| Exponent: | \(272\)\(\medspace = 2^{4} \cdot 17 \) |
| 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: | $C_8$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^2\times C_8).C_2^5$ |
| $\operatorname{Aut}(H)$ | $D_4\times C_{16}$, of order \(128\)\(\medspace = 2^{7} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $D_4\times C_{16}$, of order \(128\)\(\medspace = 2^{7} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
| Centralizer: | $C_{136}$ | ||
| Normalizer: | $\OD_{32}:C_{34}$ | ||
| Minimal over-subgroups: | $D_4:C_{34}$ | ||
| Maximal under-subgroups: | $C_{68}$ | $C_2\times C_{34}$ | $D_4$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_2^2:C_8$ |