Subgroup ($H$) information
| Description: | $\He_{11}:C_{20}$ |
| Order: | \(26620\)\(\medspace = 2^{2} \cdot 5 \cdot 11^{3} \) |
| Index: | \(2\) |
| Exponent: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rrrr}
7 & 0 & 1 & 3 \\
8 & 5 & 1 & 1 \\
8 & 9 & 8 & 0 \\
0 & 8 & 3 & 6
\end{array}\right), \left(\begin{array}{rrrr}
7 & 5 & 10 & 1 \\
2 & 7 & 7 & 10 \\
3 & 8 & 6 & 6 \\
1 & 3 & 9 & 6
\end{array}\right), \left(\begin{array}{rrrr}
8 & 6 & 8 & 10 \\
10 & 8 & 2 & 8 \\
9 & 3 & 5 & 5 \\
5 & 9 & 1 & 5
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
5 & 3 & 0 & 0 \\
10 & 0 & 3 & 0 \\
10 & 8 & 7 & 9
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
9 & 5 & 2 & 0 \\
9 & 9 & 6 & 0 \\
5 & 8 & 8 & 1
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
6 & 10 & 0 & 0 \\
1 & 0 & 10 & 0 \\
0 & 10 & 6 & 1
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is normal, maximal, nonabelian, and solvable.
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.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_{11}:(C_5\times \GL(2,3))$, of order \(319440\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11^{3} \) |
| $\operatorname{Aut}(H)$ | $\He_{11}:C_{120}:C_2$, of order \(319440\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11^{3} \) |
| $W$ | $\He_{11}:(C_5\times Q_8)$, of order \(53240\)\(\medspace = 2^{3} \cdot 5 \cdot 11^{3} \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $\He_{11}:(C_5\times Q_8)$ |