Subgroup ($H$) information
| Description: | $C_3\times C_{33}$ |
| Order: | \(99\)\(\medspace = 3^{2} \cdot 11 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(33\)\(\medspace = 3 \cdot 11 \) |
| Generators: |
$b^{2}, c^{9}, c^{66}$
|
| 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: | $S_3\times D_{99}$ |
| Order: | \(1188\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 11 \) |
| Exponent: | \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $D_6$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, an A-group, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_3\times C_{99}).C_{30}.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_{10}\times \GL(2,3)$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^2\times C_{10}$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(891\)\(\medspace = 3^{4} \cdot 11 \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
Other information
| Möbius function | $-6$ |
| Projective image | $S_3\times D_{99}$ |