Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(640\)\(\medspace = 2^{7} \cdot 5 \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(2,8,7)(3,6,9)\rangle$
|
| 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: | $C_2^4:S_5$ |
| Order: | \(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $1$ |
The ambient group is nonabelian, nonsolvable, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4:S_5$, of order \(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(S)$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_3\times D_4$ | ||||||||
| Normalizer: | $S_3\times D_4$ | ||||||||
| Normal closure: | $C_2^4:A_5$ | ||||||||
| Core: | $C_1$ | ||||||||
| Minimal over-subgroups: | $A_4$ | $A_4$ | $C_6$ | $C_6$ | $C_6$ | $S_3$ | $S_3$ | $S_3$ | $S_3$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of subgroups in this conjugacy class | $40$ |
| Möbius function | $8$ |
| Projective image | $C_2^4:S_5$ |