Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| Exponent: | \(3\) |
| Generators: |
$\left(\begin{array}{ll}\alpha^{189} & \alpha^{196} \\ \alpha^{175} & \alpha^{241} \\ \end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a semidirect factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_3^4:S_3$ |
| Order: | \(486\)\(\medspace = 2 \cdot 3^{5} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, and rational.
Quotient group ($Q$) structure
| Description: | $C_3^3:S_3$ |
| Order: | \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $\AGL(4,3)$, of order \(1965150720\)\(\medspace = 2^{9} \cdot 3^{10} \cdot 5 \cdot 13 \) |
| Outer Automorphisms: | $\PGL(4,3)$, of order \(12130560\)\(\medspace = 2^{8} \cdot 3^{6} \cdot 5 \cdot 13 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, and rational.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\AGL(5,3)$, of order \(115562653240320\)\(\medspace = 2^{10} \cdot 3^{15} \cdot 5 \cdot 11^{2} \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(S)$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(477531624960\)\(\medspace = 2^{9} \cdot 3^{15} \cdot 5 \cdot 13 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_3^5$ | |
| Normalizer: | $C_3^4:S_3$ | |
| Complements: | $C_3^3:S_3$ | |
| Minimal over-subgroups: | $C_3^2$ | $S_3$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of subgroups in this autjugacy class | $121$ |
| Number of conjugacy classes in this autjugacy class | $121$ |
| Möbius function | $-59049$ |
| Projective image | $C_3^4:S_3$ |