Subgroup ($H$) information
| Description: | $C_2\times C_{24}$ |
| Order: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Index: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$a^{2}b^{24}, b^{96}, b^{48}, b^{64}, a^{4}b^{48}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), central (hence abelian, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_8\times C_{192}$ |
| Order: | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Exponent: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
Quotient group ($Q$) structure
| Description: | $C_2\times C_{16}$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Automorphism Group: | $D_4:C_2^2$, of order \(32\)\(\medspace = 2^{5} \) |
| Outer Automorphisms: | $D_4:C_2^2$, of order \(32\)\(\medspace = 2^{5} \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2.C_4^3.C_2^6.C_2$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^2\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(512\)\(\medspace = 2^{9} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_8\times C_{192}$ | ||
| Normalizer: | $C_8\times C_{192}$ | ||
| Minimal over-subgroups: | $C_4\times C_{24}$ | $C_2\times C_{48}$ | |
| Maximal under-subgroups: | $C_2\times C_{12}$ | $C_{24}$ | $C_2\times C_8$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_2\times C_{16}$ |