Subgroup ($H$) information
| Description: | $\OD_{16}\times C_{51}$ |
| Order: | \(816\)\(\medspace = 2^{4} \cdot 3 \cdot 17 \) |
| Index: | \(2\) |
| Exponent: | \(408\)\(\medspace = 2^{3} \cdot 3 \cdot 17 \) |
| Generators: |
$c^{24}, c^{102}, c^{204}, c^{272}, c^{51}, a$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), maximal, a semidirect factor, nonabelian, elementary for $p = 2$ (hence hyperelementary), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $D_{12}.C_{68}$ |
| Order: | \(1632\)\(\medspace = 2^{5} \cdot 3 \cdot 17 \) |
| Exponent: | \(408\)\(\medspace = 2^{3} \cdot 3 \cdot 17 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| 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), a $p$-group, simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:(C_2\times C_{16}\times C_2^2\times D_4)$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times C_{16}\times D_4$, of order \(512\)\(\medspace = 2^{9} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $(C_2\times C_8) . C_2^5$, of order \(512\)\(\medspace = 2^{9} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_2\times D_6$ |