Subgroup ($H$) information
| Description: | $C_6$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Index: | \(243\)\(\medspace = 3^{5} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$f^{3}, a$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and central.
Ambient group ($G$) information
| Description: | $C_3^5:C_6$ |
| Order: | \(1458\)\(\medspace = 2 \cdot 3^{6} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 3$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_3^4:C_3$ |
| Order: | \(243\)\(\medspace = 3^{5} \) |
| Exponent: | \(3\) |
| Automorphism Group: | $C_3^5.(S_3\times C_3^2:\GL(2,3))$, of order \(629856\)\(\medspace = 2^{5} \cdot 3^{9} \) |
| Outer Automorphisms: | $C_3^5:(C_2\times \GL(2,3))$, of order \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^7.C_3^4.C_3^2.Q_8.D_6.C_2$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(S)$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(17006112\)\(\medspace = 2^{5} \cdot 3^{12} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_3^5:C_6$ | ||
| Normalizer: | $C_3^5:C_6$ | ||
| Complements: | $C_3^4:C_3$ | ||
| Minimal over-subgroups: | $C_3\times C_6$ | $C_3\times C_6$ | $C_3\times C_6$ |
| Maximal under-subgroups: | $C_3$ | $C_2$ |
Other information
| Number of subgroups in this autjugacy class | $9$ |
| Number of conjugacy classes in this autjugacy class | $9$ |
| Möbius function | $0$ |
| Projective image | $C_3^4:C_3$ |