Subgroup ($H$) information
| Description: | not computed |
| Order: | \(12960\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 5 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | not computed |
| Generators: |
$e^{2}, de^{2}f, a^{2}e^{3}, b^{16}, b^{20}, cf^{2}, b^{5}, b^{10}, b^{40}, f$
|
| Derived length: | not computed |
The subgroup is normal, a semidirect factor, nonabelian, and solvable. Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_2\times F_{81}:C_4$ |
| Order: | \(51840\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5 \) |
| Exponent: | \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times C_3^4.C_{80}:C_2.C_2$, of order \(103680\)\(\medspace = 2^{8} \cdot 3^{4} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $F_{81}:C_4$, of order \(25920\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5 \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_2\times F_{81}:C_4$ |