Subgroup ($H$) information
| Description: | $C_6\times C_{192}$ |
| Order: | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Index: | \(2\) |
| Exponent: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Generators: |
$a^{6}, b^{96}, b^{6}, b^{12}, b^{3}, b^{64}, b^{48}, b^{24}, a^{4}b^{96}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), maximal, abelian (hence metabelian and an A-group), and metacyclic.
Ambient group ($G$) information
| Description: | $C_{192}.C_{12}$ |
| Order: | \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| Exponent: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and metacyclic (hence solvable, supersolvable, monomial, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:(C_2^3\times C_{32}.C_8.C_2^3)$ |
| $\operatorname{Aut}(H)$ | $Q_8.(C_8\times S_3).C_2^4$ |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^4\times C_{16}$, of order \(256\)\(\medspace = 2^{8} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $D_{96}$ |