Subgroup ($H$) information
| Description: | $C_3\times C_3^5.C_3$ |
| Order: | \(2187\)\(\medspace = 3^{7} \) |
| Index: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Generators: |
$\langle(13,15,14)(16,17,18)(25,26,27)(28,30,29), (1,3,2)(4,6,5)(7,8,9)(10,12,11) \!\cdots\! \rangle$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian. Whether it is a direct factor, a semidirect factor, metacyclic, monomial, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_3^6.C_3\wr C_2^2$ |
| Order: | \(236196\)\(\medspace = 2^{2} \cdot 3^{10} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
Quotient group ($Q$) structure
| Description: | $C_3:S_3^2$ |
| Order: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_3\wr S_3$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Outer Automorphisms: | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^5.C_3^5.C_6^3.C_2$, of order \(25509168\)\(\medspace = 2^{4} \cdot 3^{13} \) |
| $\operatorname{Aut}(H)$ | Group of order \(24794911296\)\(\medspace = 2^{6} \cdot 3^{18} \) |
| $W$ | $\PSOPlus(4,5)$, of order \(7200\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5^{2} \) |
Related subgroups
| Centralizer: | $C_3^4$ |
| Normalizer: | $C_3^6.C_3\wr C_2^2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |