Subgroup ($H$) information
| Description: | $C_{10}$ |
| Order: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Index: | \(1452\)\(\medspace = 2^{2} \cdot 3 \cdot 11^{2} \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rr}
47 & 35 \\
51 & 74
\end{array}\right), \left(\begin{array}{rr}
3 & 0 \\
0 & 3
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_5\times C_{11}^2:D_{12}$ |
| Order: | \(14520\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11^{2} \) |
| Exponent: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^2.C_6.C_{10}.C_4.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $66$ |
| Möbius function | $0$ |
| Projective image | $C_{11}^2:D_{12}$ |