Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(12348\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7^{3} \) |
| Exponent: | \(3\) |
| Generators: |
$\left(\begin{array}{rrrr}
2 & 0 & 0 & 0 \\
3 & 1 & 2 & 0 \\
5 & 0 & 4 & 0 \\
6 & 6 & 6 & 2
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $\He_7:(C_3^2\times D_6)$ |
| Order: | \(37044\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 7^{3} \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_7.(C_6\times S_3^2).C_2$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{14}:C_3^3$ | ||||
| Normalizer: | $C_{14}:C_3^3$ | ||||
| Normal closure: | $\He_7:C_3^2$ | ||||
| Core: | $C_1$ | ||||
| Minimal over-subgroups: | $C_{21}$ | $C_7:C_3$ | $C_3^2$ | $C_3^2$ | $C_6$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of subgroups in this autjugacy class | $98$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $\He_7:(C_3^2\times D_6)$ |