Subgroup ($H$) information
| Description: | $C_3^4.C_3^3$ |
| Order: | \(2187\)\(\medspace = 3^{7} \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(4,5,6)(7,8,9)(16,17,18)(19,20,21)(28,29,30)(31,32,33), (1,3,2)(10,11,12) \!\cdots\! \rangle$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), a semidirect factor, nonabelian, a $3$-Sylow subgroup (hence a Hall subgroup), a $p$-group (hence elementary and hyperelementary), and metabelian. Whether it is metacyclic, monomial, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_3^5:F_9:C_2$ |
| Order: | \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $\SD_{16}$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metacyclic (hence 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.C_3^4.Q_8.C_6.C_2^3$, of order \(839808\)\(\medspace = 2^{7} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | Group of order \(612220032\)\(\medspace = 2^{7} \cdot 3^{14} \) |
| $W$ | $C_3^4:\SD_{16}$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | $C_3^3$ | |
| Normalizer: | $C_3^5:F_9:C_2$ | |
| Complements: | $\SD_{16}$ | |
| Minimal over-subgroups: | $C_3^5.C_3.S_3$ | $C_3^4.C_3^3.C_2$ |
| Maximal under-subgroups: | $C_3^3:\He_3$ | $C_3^5:C_3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_3^5:F_9:C_2$ |