Subgroup ($H$) information
Description: | $F_{193}$ |
Order: | \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
Index: | $1$ |
Exponent: | \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
Generators: |
$a^{24}, b, a^{96}, a^{48}, a^{12}, a^{3}, a^{64}, a^{6}$
|
Derived length: | $2$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, and a Z-group (hence supersolvable, monomial, metacyclic, metabelian, and an A-group).
Ambient group ($G$) information
Description: | $F_{193}$ |
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 and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group).
Quotient group ($Q$) structure
Description: | $C_1$ |
Order: | $1$ |
Exponent: | $1$ |
Automorphism Group: | $C_1$, of order $1$ |
Outer Automorphisms: | $C_1$, of order $1$ |
Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.
$\operatorname{Aut}(G)$ | $F_{193}$, of order \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
$\operatorname{Aut}(H)$ | $F_{193}$, of order \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
$W$ | $F_{193}$, of order \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
Related subgroups
Centralizer: | $C_1$ | ||
Normalizer: | $F_{193}$ | ||
Complements: | $C_1$ | ||
Maximal under-subgroups: | $C_{193}:C_{96}$ | $C_{193}:C_{64}$ | $C_{192}$ |
Other information
Möbius function | $1$ |
Projective image | $F_{193}$ |