Subgroup ($H$) information
Description: | $C_2$ |
Order: | \(2\) |
Index: | \(148224\)\(\medspace = 2^{8} \cdot 3 \cdot 193 \) |
Exponent: | \(2\) |
Generators: |
$b^{9264}$
|
Nilpotency class: | $1$ |
Derived length: | $1$ |
The subgroup is characteristic (hence normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, a $p$-group, simple, and rational.
Ambient group ($G$) information
Description: | $C_{18528}.C_{16}$ |
Order: | \(296448\)\(\medspace = 2^{9} \cdot 3 \cdot 193 \) |
Exponent: | \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Quotient group ($Q$) structure
Order: | \(148224\)\(\medspace = 2^{8} \cdot 3 \cdot 193 \) |
Exponent: | not computed |
Automorphism Group: | not computed |
Outer Automorphisms: | not computed |
Nilpotency class: | not computed |
Derived length: | not computed |
Properties have not been computed
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 $W = G / Z_G(H)$.
$\operatorname{Aut}(G)$ | $C_{1544}.C_{24}.C_4^2.C_2^5$ |
$\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
$W$ | $C_1$, of order $1$ |
Related subgroups
Centralizer: | $C_{18528}.C_{16}$ | ||||
Normalizer: | $C_{18528}.C_{16}$ | ||||
Minimal over-subgroups: | $C_{386}$ | $C_6$ | $C_4$ | $C_2^2$ | $C_4$ |
Maximal under-subgroups: | $C_1$ |
Other information
Möbius function | $0$ |
Projective image | not computed |