Subgroup ($H$) information
| Description: | $C_3^2$ |
| Order: | \(9\)\(\medspace = 3^{2} \) |
| Index: | \(1215\)\(\medspace = 3^{5} \cdot 5 \) |
| Exponent: | \(3\) |
| Generators: |
$a^{10}b^{4}c^{2}d^{2}f^{2}, b^{3}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), 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: | $C_3^6.C_{15}$ |
| Order: | \(10935\)\(\medspace = 3^{7} \cdot 5 \) |
| Exponent: | \(45\)\(\medspace = 3^{2} \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and metabelian (hence solvable). Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_3\wr C_5$ |
| Order: | \(1215\)\(\medspace = 3^{5} \cdot 5 \) |
| Exponent: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Automorphism Group: | $C_2\times F_{81}:C_4$, of order \(51840\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5 \) |
| Outer Automorphisms: | $\OD_{32}:C_4$, of order \(128\)\(\medspace = 2^{7} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, monomial (hence solvable), metabelian, and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^4:C_2.C_2^4$, of order \(1399680\)\(\medspace = 2^{7} \cdot 3^{7} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $W$ | $C_3$, of order \(3\) |
Related subgroups
| Centralizer: | $C_3^5:C_{15}$ | |||
| Normalizer: | $C_3^6.C_{15}$ | |||
| Minimal over-subgroups: | $C_3\times C_{15}$ | $C_9:C_3$ | $C_3^3$ | $C_9:C_3$ |
| Maximal under-subgroups: | $C_3$ | $C_3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-81$ |
| Projective image | $C_3^5:C_{15}$ |