Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(2720\)\(\medspace = 2^{5} \cdot 5 \cdot 17 \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(1,4,11)(2,14,15)(3,6,12)(5,8,9)(10,17,13)\rangle$
|
| 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 $3$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $\SOMinus(4,4)$ |
| Order: | \(8160\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 17 \) |
| Exponent: | \(1020\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 17 \) |
| Derived length: | $1$ |
The ambient group is nonabelian, almost simple, and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\SL(2,16).C_4$, of order \(16320\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \cdot 17 \) |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_3\times D_5$ | ||||
| Normalizer: | $S_3\times D_5$ | ||||
| Normal closure: | $\SL(2,16)$ | ||||
| Core: | $C_1$ | ||||
| Minimal over-subgroups: | $C_{15}$ | $A_4$ | $C_6$ | $S_3$ | $S_3$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of subgroups in this conjugacy class | $136$ |
| Möbius function | $10$ |
| Projective image | $\SOMinus(4,4)$ |