Subgroup ($H$) information
| Description: | $C_6$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Index: | \(512\)\(\medspace = 2^{9} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$c^{8}, b^{4}c^{8}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), the socle, and cyclic (hence elementary ($p = 2,3$), hyperelementary, metacyclic, and a Z-group).
Ambient group ($G$) information
| Description: | $C_3\times Q_{16}^2:C_2^2$ |
| Order: | \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Nilpotency class: | $7$ |
| Derived length: | $3$ |
The ambient group is nonabelian and elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary).
Quotient group ($Q$) structure
| Description: | $D_8\wr C_2$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Automorphism Group: | $(C_4\times \OD_{16}).D_4^2$, of order \(4096\)\(\medspace = 2^{12} \) |
| Outer Automorphisms: | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $6$ |
| Derived length: | $3$ |
The quotient is nonabelian and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times C_4^2.C_2^4.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8192\)\(\medspace = 2^{13} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $D_8\wr C_2$ |