Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(135\)\(\medspace = 3^{3} \cdot 5 \) |
| Exponent: | \(3\) |
| Generators: |
$c^{30}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), stem (hence central), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_{15}.\He_3$ |
| Order: | \(405\)\(\medspace = 3^{4} \cdot 5 \) |
| Exponent: | \(45\)\(\medspace = 3^{2} \cdot 5 \) |
| Nilpotency class: | $3$ |
| 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: | $C_5\times \He_3$ |
| Order: | \(135\)\(\medspace = 3^{3} \cdot 5 \) |
| Exponent: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Automorphism Group: | $C_4\times C_3^2:\GL(2,3)$, of order \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Outer Automorphisms: | $C_4\times \GL(2,3)$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, elementary for $p = 3$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.(S_3\times C_{12})$, of order \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{15}.\He_3$ | |||||
| Normalizer: | $C_{15}.\He_3$ | |||||
| Minimal over-subgroups: | $C_{15}$ | $C_3^2$ | $C_9$ | $C_9$ | $C_9$ | $C_9$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_5\times \He_3$ |