Subgroup ($H$) information
| Description: | $\PSL(3,4):C_2$ |
| Order: | \(40320\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Index: | $1$ |
| Exponent: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Generators: |
$\langle(1,39)(2,22)(3,34)(4,29)(5,41)(6,40)(7,38)(8,28)(9,24)(10,42)(11,23)(12,26) \!\cdots\! \rangle$
|
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a direct factor, nonabelian, a $40320$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), almost simple, nonsolvable, and rational.
Ambient group ($G$) information
| Description: | $\PSL(3,4):C_2$ |
| Order: | \(40320\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Exponent: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian, almost simple, nonsolvable, and rational.
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 $p$), and perfect.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PSL(3,4):D_6$, of order \(241920\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $\PSL(3,4):D_6$, of order \(241920\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5 \cdot 7 \) |
| $W$ | $\PSL(3,4):C_2$, of order \(40320\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_1$ | |||||||||
| Normalizer: | $\PSL(3,4):C_2$ | |||||||||
| Complements: | $C_1$ | |||||||||
| Maximal under-subgroups: | $\PSL(3,4)$ | $A_6.C_2$ | $A_6.C_2$ | $A_6.C_2$ | $C_4^2.S_4$ | $\PGL(2,7)$ | $\PGL(2,7)$ | $\PGL(2,7)$ | $C_2\times \PSU(3,2)$ | $S_5$ |
Other information
| Möbius function | $1$ |
| Projective image | $\PSL(3,4):C_2$ |