Subgroup ($H$) information
| Description: | $C_9:C_{15}$ |
| Order: | \(135\)\(\medspace = 3^{3} \cdot 5 \) |
| Index: | \(3\) |
| Exponent: | \(45\)\(\medspace = 3^{2} \cdot 5 \) |
| Generators: |
$a, c^{9}, c^{30}, bc^{10}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is normal, maximal, a semidirect factor, nonabelian, elementary for $p = 3$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $C_{15}.C_3^3$ |
| Order: | \(405\)\(\medspace = 3^{4} \cdot 5 \) |
| Exponent: | \(45\)\(\medspace = 3^{2} \cdot 5 \) |
| 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: | $C_3$ |
| Order: | \(3\) |
| Exponent: | \(3\) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^2:\GL(2,3)\times C_{12}$, of order \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $C_4\times C_3^2:S_3$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| $\operatorname{res}(S)$ | $C_4\times C_3^2:S_3$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3\) |
| $W$ | $C_3^2$, of order \(9\)\(\medspace = 3^{2} \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_3^3$ |