Subgroup ($H$) information
| Description: | $C_3^2$ |
| Order: | \(9\)\(\medspace = 3^{2} \) |
| Index: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | \(3\) |
| Generators: |
$\left(\begin{array}{ll}\alpha^{189} & \alpha^{196} \\ \alpha^{175} & \alpha^{241} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{88} & \alpha^{141} \\ \alpha^{120} & \alpha^{10} \\ \end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
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: | $C_3^2:S_3$ |
| Order: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $C_3^3:\GL(3,3)$, of order \(303264\)\(\medspace = 2^{5} \cdot 3^{6} \cdot 13 \) |
| Outer Automorphisms: | $\SL(3,3)$, of order \(5616\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 13 \) |
| 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 \(115562653240320\)\(\medspace = 2^{10} \cdot 3^{15} \cdot 5 \cdot 11^{2} \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $\operatorname{res}(S)$ | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(1989715104\)\(\medspace = 2^{5} \cdot 3^{14} \cdot 13 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_3^5$ | |
| Normalizer: | $C_3^4:S_3$ | |
| Complements: | $C_3^2:S_3$ | |
| Minimal over-subgroups: | $C_3^3$ | $C_3:S_3$ |
| Maximal under-subgroups: | $C_3$ |
Other information
| Number of subgroups in this autjugacy class | $1210$ |
| Number of conjugacy classes in this autjugacy class | $1210$ |
| Möbius function | $729$ |
| Projective image | $C_3^4:S_3$ |