Subgroup ($H$) information
| Description: | $C_{4632}$ |
| Order: | \(4632\)\(\medspace = 2^{3} \cdot 3 \cdot 193 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(4632\)\(\medspace = 2^{3} \cdot 3 \cdot 193 \) |
| Generators: |
$a^{48}, a^{96}, b, a^{64}, a^{24}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal) and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,193$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{193}:C_{192}$ |
| Order: | \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
| Exponent: | \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Quotient group ($Q$) structure
| Description: | $C_8$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| 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) 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $A_4\times D_{12}:D_4$, of order \(1185792\)\(\medspace = 2^{11} \cdot 3 \cdot 193 \) |
| $\operatorname{Aut}(H)$ | $C_2^3\times C_{192}$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $W$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | $C_{9264}$ | ||
| Normalizer: | $C_{193}:C_{192}$ | ||
| Minimal over-subgroups: | $C_{9264}$ | ||
| Maximal under-subgroups: | $C_{2316}$ | $C_{1544}$ | $C_{24}$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{193}:C_8$ |