Subgroup ($H$) information
| Description: | $C_3^7.C_3:S_3^3$ |
| Order: | \(1417176\)\(\medspace = 2^{3} \cdot 3^{11} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$\langle(7,21,32)(8,19,33)(9,20,31)(10,35,23)(11,36,24)(12,34,22)(13,15,14)(16,17,18) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), nonabelian, and supersolvable (hence solvable and monomial). Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_3^4.\He_3^2:D_4:D_6$ |
| Order: | \(5668704\)\(\medspace = 2^{5} \cdot 3^{11} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_3^5.C_2^3.C_6.C_2^2$, of order \(34012224\)\(\medspace = 2^{6} \cdot 3^{12} \) |
| $\operatorname{Aut}(H)$ | $C_3^8.C_3^4.C_6^3.C_2^3$, of order \(918330048\)\(\medspace = 2^{6} \cdot 3^{15} \) |
| $W$ | $C_3^4.\He_3^2:D_4:D_6$, of order \(5668704\)\(\medspace = 2^{5} \cdot 3^{11} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^4.\He_3^2:D_4:D_6$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^4.\He_3^2:D_4:D_6$ |