Subgroup ($H$) information
| Description: | $C_8.D_{44}$ |
| Order: | \(704\)\(\medspace = 2^{6} \cdot 11 \) |
| Index: | $1$ |
| Exponent: | \(176\)\(\medspace = 2^{4} \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rr}
97 & 0 \\
0 & 222
\end{array}\right), \left(\begin{array}{rr}
81 & 0 \\
0 & 91
\end{array}\right), \left(\begin{array}{rr}
9 & 0 \\
0 & 306
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 352
\end{array}\right), \left(\begin{array}{rr}
34 & 0 \\
0 & 135
\end{array}\right), \left(\begin{array}{rr}
231 & 0 \\
0 & 217
\end{array}\right), \left(\begin{array}{rr}
0 & 1 \\
1 & 0
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, supersolvable (hence monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $C_8.D_{44}$ |
| Order: | \(704\)\(\medspace = 2^{6} \cdot 11 \) |
| Exponent: | \(176\)\(\medspace = 2^{4} \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{22}.(C_2^4\times C_{20})$ |
| $\operatorname{Aut}(H)$ | $C_{22}.(C_2^4\times C_{20})$ |
| $W$ | $D_{44}$, of order \(88\)\(\medspace = 2^{3} \cdot 11 \) |
Related subgroups
| Centralizer: | $C_8$ | |||
| Normalizer: | $C_8.D_{44}$ | |||
| Complements: | $C_1$ | |||
| Maximal under-subgroups: | $C_8.D_{22}$ | $C_2\times C_{176}$ | $C_{11}:\OD_{32}$ | $\OD_{32}:C_2$ |
Other information
| Möbius function | $1$ |
| Projective image | $D_{44}$ |