Subgroup ($H$) information
| Description: | $C_{198}$ |
| Order: | \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rr}
228 & 0 \\
0 & 218
\end{array}\right), \left(\begin{array}{rr}
314 & 0 \\
0 & 120
\end{array}\right), \left(\begin{array}{rr}
124 & 0 \\
0 & 381
\end{array}\right), \left(\begin{array}{rr}
290 & 0 \\
0 & 256
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal) and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,11$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $D_{44}:C_{36}$ |
| Order: | \(3168\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 11 \) |
| Exponent: | \(792\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2^2:C_4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2^2\wr C_2$, of order \(32\)\(\medspace = 2^{5} \) |
| Outer Automorphisms: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{22}.C_{30}.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{30}$, of order \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times C_{30}$, of order \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(352\)\(\medspace = 2^{5} \cdot 11 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $D_{22}:C_4$ |