Subgroup ($H$) information
| Description: | $C_{36}\times D_{33}$ |
| Order: | \(2376\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 11 \) |
| Index: | \(2\) |
| Exponent: | \(396\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rr}
125 & 0 \\
0 & 149
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 362
\end{array}\right), \left(\begin{array}{rr}
228 & 0 \\
0 & 218
\end{array}\right), \left(\begin{array}{rr}
99 & 0 \\
0 & 393
\end{array}\right), \left(\begin{array}{rr}
124 & 0 \\
0 & 381
\end{array}\right), \left(\begin{array}{rr}
110 & 0 \\
0 & 314
\end{array}\right), \left(\begin{array}{rr}
0 & 1 \\
1 & 0
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is normal, maximal, a semidirect factor, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $D_{132}:C_{18}$ |
| Order: | \(4752\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 11 \) |
| Exponent: | \(396\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| 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, simple, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{66}.C_{30}.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_{33}.C_{30}.C_2^4$ |
| $\card{\operatorname{res}(S)}$ | \(15840\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $D_{66}$, of order \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) |
Related subgroups
Other information
| Möbius function | not computed |
| Projective image | $D_{66}$ |