Subgroup ($H$) information
| Description: | $C_3\times C_{33}$ |
| Order: | \(99\)\(\medspace = 3^{2} \cdot 11 \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(33\)\(\medspace = 3 \cdot 11 \) |
| Generators: |
$b, d^{3}, c$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the socle (hence characteristic and normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 3$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_3\times D_{11}\times \He_3$ |
| Order: | \(1782\)\(\medspace = 2 \cdot 3^{4} \cdot 11 \) |
| Exponent: | \(66\)\(\medspace = 2 \cdot 3 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_3\times C_6$ |
| Order: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Outer Automorphisms: | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 3$ (hence hyperelementary), and metacyclic.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.Q_8.C_{33}.C_{30}.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_{10}\times \GL(2,3)$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{10}\times D_6$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(21384\)\(\medspace = 2^{3} \cdot 3^{5} \cdot 11 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $\He_3\times C_{33}$ | ||
| Normalizer: | $C_3\times D_{11}\times \He_3$ | ||
| Minimal over-subgroups: | $C_3^2\times C_{33}$ | $C_3^2\times D_{11}$ | |
| Maximal under-subgroups: | $C_{33}$ | $C_{33}$ | $C_3^2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-3$ |
| Projective image | $C_3^2\times D_{11}$ |