Subgroup ($H$) information
| Description: | $C_8\times D_{193}$ |
| Order: | \(3088\)\(\medspace = 2^{4} \cdot 193 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(1544\)\(\medspace = 2^{3} \cdot 193 \) |
| Generators: |
$b^{772}, b^{8}, b^{386}, b^{193}, a^{4}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Ambient group ($G$) information
| Description: | $C_{1544}.C_8$ |
| Order: | \(12352\)\(\medspace = 2^{6} \cdot 193 \) |
| Exponent: | \(3088\)\(\medspace = 2^{4} \cdot 193 \) |
| 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
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{772}.C_{96}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_3^5:D_6$, of order \(296448\)\(\medspace = 2^{9} \cdot 3 \cdot 193 \) |
| $W$ | $C_{193}:C_8$, of order \(1544\)\(\medspace = 2^{3} \cdot 193 \) |
Related subgroups
| Centralizer: | $C_8$ | |||
| Normalizer: | $C_{1544}.C_8$ | |||
| Minimal over-subgroups: | $C_{1544}:C_4$ | |||
| Maximal under-subgroups: | $C_4\times D_{193}$ | $C_{1544}$ | $C_{193}:C_8$ | $C_2\times C_8$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{193}:C_8$ |