Subgroup ($H$) information
| Description: | $C_{11}^3$ |
| Order: | \(1331\)\(\medspace = 11^{3} \) |
| Index: | \(600\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \) |
| Exponent: | \(11\) |
| Generators: |
$\left(\begin{array}{rrrr}
8 & 0 & 7 & 4 \\
3 & 0 & 1 & 7 \\
4 & 4 & 2 & 0 \\
0 & 4 & 8 & 5
\end{array}\right), \left(\begin{array}{rrrr}
7 & 9 & 2 & 3 \\
8 & 0 & 6 & 2 \\
10 & 1 & 2 & 2 \\
5 & 10 & 3 & 6
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
5 & 6 & 4 & 0 \\
2 & 2 & 7 & 0 \\
2 & 2 & 6 & 1
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $11$-Sylow subgroup (hence a Hall subgroup), and a $p$-group (hence elementary and hyperelementary). Whether it is a direct factor has not been computed.
Ambient group ($G$) information
| Description: | $C_{11}^3:(S_3\times C_{10}^2)$ |
| Order: | \(798600\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \cdot 11^{3} \) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, solvable, and an A-group. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $S_3\times C_{10}^2$ |
| Order: | \(600\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Automorphism Group: | $\GL(2,5)\times S_4\times S_3$ |
| Outer Automorphisms: | $S_4\times \GL(2,5)$, of order \(11520\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), metabelian, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(63888000\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{3} \cdot 11^{3} \) |
| $\operatorname{Aut}(H)$ | $\GL(3,11)$, of order \(2124276000\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{3} \cdot 7 \cdot 11^{3} \cdot 19 \) |
| $\card{W}$ | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
Related subgroups
| Centralizer: | $C_{11}^2\times C_{110}$ |
| Normalizer: | $C_{11}^3:(S_3\times C_{10}^2)$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |