Subgroup ($H$) information
| Description: | $C_{44}.C_{10}$ |
| Order: | \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| Index: | \(121\)\(\medspace = 11^{2} \) |
| Exponent: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rrrr}
8 & 6 & 8 & 10 \\
10 & 8 & 2 & 8 \\
9 & 3 & 5 & 5 \\
5 & 9 & 1 & 5
\end{array}\right), \left(\begin{array}{rrrr}
10 & 0 & 0 & 0 \\
3 & 4 & 8 & 0 \\
10 & 10 & 7 & 0 \\
2 & 1 & 7 & 8
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
5 & 3 & 0 & 0 \\
10 & 0 & 3 & 0 \\
10 & 8 & 7 & 9
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
9 & 5 & 2 & 0 \\
9 & 9 & 6 & 0 \\
5 & 8 & 8 & 1
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
6 & 10 & 0 & 0 \\
1 & 0 & 10 & 0 \\
0 & 10 & 6 & 1
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, and metacyclic (hence solvable, supersolvable, monomial, and metabelian).
Ambient group ($G$) information
| Description: | $\He_{11}:(C_5\times Q_8)$ |
| Order: | \(53240\)\(\medspace = 2^{3} \cdot 5 \cdot 11^{3} \) |
| Exponent: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_{11}:(C_5\times \GL(2,3))$, of order \(319440\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11^{3} \) |
| $\operatorname{Aut}(H)$ | $S_4\times F_{11}$, of order \(2640\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11 \) |
| $W$ | $C_{22}:C_{10}$, of order \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $121$ |
| Möbius function | $-1$ |
| Projective image | $\He_{11}:(C_5\times Q_8)$ |