Subgroup ($H$) information
| Description: | $C_2\times C_{11}^3:C_{10}$ |
| Order: | \(26620\)\(\medspace = 2^{2} \cdot 5 \cdot 11^{3} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
3 & 3 & 10 & 0 \\
5 & 3 & 0 & 10
\end{array}\right), \left(\begin{array}{rrrr}
3 & 1 & 4 & 10 \\
6 & 0 & 4 & 4 \\
1 & 0 & 2 & 10 \\
3 & 1 & 5 & 10
\end{array}\right), \left(\begin{array}{rrrr}
9 & 0 & 0 & 0 \\
0 & 9 & 0 & 0 \\
0 & 0 & 9 & 0 \\
0 & 0 & 0 & 9
\end{array}\right), \left(\begin{array}{rrrr}
10 & 0 & 0 & 0 \\
10 & 9 & 8 & 0 \\
4 & 4 & 0 & 0 \\
4 & 4 & 1 & 10
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
4 & 5 & 1 & 0 \\
6 & 6 & 8 & 0 \\
6 & 6 & 7 & 1
\end{array}\right), \left(\begin{array}{rrrr}
1 & 4 & 8 & 4 \\
8 & 6 & 7 & 8 \\
1 & 8 & 7 & 7 \\
1 & 1 & 3 & 1
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_{11}^3:(C_{10}\times D_{12})$ |
| Order: | \(319440\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11^{3} \) |
| Exponent: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $D_6$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, an A-group, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(51110400\)\(\medspace = 2^{9} \cdot 3 \cdot 5^{2} \cdot 11^{3} \) |
| $\operatorname{Aut}(H)$ | Group of order \(22619290848000\)\(\medspace = 2^{8} \cdot 3 \cdot 5^{3} \cdot 7 \cdot 11^{6} \cdot 19 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |