Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(3\) |
| Generators: |
$a^{4}b^{24}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_{48}.C_{12}$ |
| Order: | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and metacyclic (hence solvable, supersolvable, monomial, and metabelian).
Quotient group ($Q$) structure
| Description: | $C_{48}.C_4$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Automorphism Group: | $D_6\times D_{16}:C_4$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Outer Automorphisms: | $C_2^3\times C_4$, of order \(32\)\(\medspace = 2^{5} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:(C_2^3\times D_8:C_4.C_2)$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{48}.C_{12}$ | ||
| Normalizer: | $C_{48}.C_{12}$ | ||
| Complements: | $C_{48}.C_4$ | ||
| Minimal over-subgroups: | $C_3^2$ | $C_6$ | $C_6$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{48}.C_4$ |