Subgroup ($H$) information
| Description: | $D_{44}$ |
| Order: | \(88\)\(\medspace = 2^{3} \cdot 11 \) |
| Index: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Exponent: | \(44\)\(\medspace = 2^{2} \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rr}
124 & 0 \\
0 & 381
\end{array}\right), \left(\begin{array}{rr}
290 & 0 \\
0 & 256
\end{array}\right), \left(\begin{array}{rr}
282 & 0 \\
0 & 252
\end{array}\right), \left(\begin{array}{rr}
0 & 1 \\
1 & 0
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
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: | $C_{36}$ |
| Order: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Automorphism Group: | $C_2\times C_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Outer Automorphisms: | $C_2\times C_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{22}.C_{30}.C_2^5$ |
| $\operatorname{Aut}(H)$ | $D_4\times F_{11}$, of order \(880\)\(\medspace = 2^{4} \cdot 5 \cdot 11 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $D_4\times F_{11}$, of order \(880\)\(\medspace = 2^{4} \cdot 5 \cdot 11 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $W$ | $D_{44}$, of order \(88\)\(\medspace = 2^{3} \cdot 11 \) |
Related subgroups
| Centralizer: | $C_{36}$ | ||
| Normalizer: | $D_{44}:C_{36}$ | ||
| Complements: | $C_{36}$ $C_{36}$ | ||
| Minimal over-subgroups: | $C_3\times D_{44}$ | $D_{44}:C_2$ | |
| Maximal under-subgroups: | $C_{44}$ | $D_{22}$ | $D_4$ |
Other information
| Möbius function | $0$ |
| Projective image | $D_{22}:C_{36}$ |