Subgroup ($H$) information
| Description: | $C_3^5:D_6\wr C_2$ |
| Order: | \(69984\)\(\medspace = 2^{5} \cdot 3^{7} \) |
| Index: | \(3\) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,2,4,8,9,16)(3,5,10,15,6,13)(7,14)(11,12)(17,18)(19,20)(21,23)(22,24) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, and solvable. Whether it is a direct factor, a semidirect factor, or monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^6.D_6\wr C_2$ |
| Order: | \(209952\)\(\medspace = 2^{5} \cdot 3^{8} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_3$ |
| Order: | \(3\) |
| Exponent: | \(3\) |
| 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), a $p$-group, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_3^2.C_3^5.C_2^3.C_2^4$, of order \(559872\)\(\medspace = 2^{8} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | $C_3^2.C_3^5.C_2^3.C_2^4$, of order \(279936\)\(\medspace = 2^{7} \cdot 3^{7} \) |
| $W$ | $C_3^5:D_6\wr C_2$, of order \(69984\)\(\medspace = 2^{5} \cdot 3^{7} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^6.D_6\wr C_2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^6.D_6\wr C_2$ |