Subgroup ($H$) information
| Description: | $C_3^8$ |
| Order: | \(6561\)\(\medspace = 3^{8} \) |
| Index: | \(256\)\(\medspace = 2^{8} \) |
| Exponent: | not computed |
| Generators: |
$\langle(28,33,35)(29,31,36)(30,32,34), (10,15,17)(11,13,18)(12,14,16)(28,29,30) \!\cdots\! \rangle$
|
| Nilpotency class: | not computed |
| Derived length: | not computed |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), the socle, a semidirect factor, abelian (hence metabelian and an A-group), a $3$-Sylow subgroup (hence a Hall subgroup), and a $p$-group (hence elementary and hyperelementary). Whether it is a direct factor, elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_3^8:(C_4.D_4^2)$ |
| Order: | \(1679616\)\(\medspace = 2^{8} \cdot 3^{8} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_4.D_4^2$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_2^6.C_2^6.C_2$ |
| Outer Automorphisms: | $C_2^4:D_4$, of order \(128\)\(\medspace = 2^{7} \) |
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^8.C_2^2.C_2^3.C_2^6$ |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | \(256\)\(\medspace = 2^{8} \) |
Related subgroups
| Centralizer: | $C_3^8$ |
| Normalizer: | $C_3^8:(C_4.D_4^2)$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |