Subgroup ($H$) information
| Description: | $\He_3:\GL(2,3)$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Index: | $1$ |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Generators: |
$\left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 2 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right), \left(\begin{array}{rrrr}
2 & 2 & 2 & 0 \\
1 & 0 & 2 & 0 \\
2 & 0 & 2 & 0 \\
2 & 1 & 1 & 1
\end{array}\right), \left(\begin{array}{rrrr}
2 & 0 & 2 & 0 \\
2 & 1 & 0 & 2 \\
1 & 0 & 0 & 0 \\
2 & 0 & 1 & 1
\end{array}\right), \left(\begin{array}{rrrr}
2 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 \\
1 & 0 & 1 & 1 \\
2 & 0 & 0 & 0
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
2 & 2 & 2 & 0 \\
1 & 0 & 0 & 2
\end{array}\right), \left(\begin{array}{rrrr}
2 & 2 & 0 & 1 \\
0 & 2 & 0 & 0 \\
1 & 2 & 2 & 2 \\
2 & 1 & 0 & 0
\end{array}\right), \left(\begin{array}{rrrr}
1 & 0 & 0 & 0 \\
2 & 1 & 2 & 1 \\
1 & 2 & 2 & 2 \\
0 & 0 & 0 & 1
\end{array}\right), \left(\begin{array}{rrrr}
1 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 \\
2 & 1 & 1 & 2 \\
0 & 2 & 0 & 1
\end{array}\right)$
|
| Derived length: | $6$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, and a Hall subgroup.
Ambient group ($G$) information
| Description: | $\He_3:\GL(2,3)$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $6$ |
The ambient group is nonabelian and solvable.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\SU(3,2).C_3.C_6$, of order \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $\SU(3,2).C_3.C_6$, of order \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| $W$ | $\He_3:\GL(2,3)$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | $C_1$ | |||
| Normalizer: | $\He_3:\GL(2,3)$ | |||
| Complements: | $C_1$ | |||
| Maximal under-subgroups: | $\Unitary(3,2)$ | $\He_3:\SD_{16}$ | $C_3^3:D_6$ | $C_3:\GL(2,3)$ |
Other information
| Möbius function | $1$ |
| Projective image | $\He_3:\GL(2,3)$ |