Subgroup ($H$) information
| Description: | $C_3^5$ |
| Order: | \(243\)\(\medspace = 3^{5} \) |
| Index: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(1,27,15)(2,25,13)(3,26,14)(4,28,16)(5,30,17)(6,29,18)(7,31,20)(8,32,21) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and a $p$-group (hence elementary and hyperelementary). Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_3^8.(S_3\times D_4)$ |
| Order: | \(314928\)\(\medspace = 2^{4} \cdot 3^{9} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2\times C_3^4:D_4$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_3^4.Q_8.C_6.C_2^4.C_2$ |
| Outer Automorphisms: | $D_4\times S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^8.Q_8.D_6^2.C_2$ |
| $\operatorname{Aut}(H)$ | $\GL(5,3)$, of order \(475566474240\)\(\medspace = 2^{10} \cdot 3^{10} \cdot 5 \cdot 11^{2} \cdot 13 \) |
| $W$ | $C_2\times C_3^4:D_4$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | $C_3^5$ |
| Normalizer: | $C_3^8.(S_3\times D_4)$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^8.(S_3\times D_4)$ |