Subgroup ($H$) information
| Description: | $C_3\times C_{15}$ |
| Order: | \(45\)\(\medspace = 3^{2} \cdot 5 \) |
| Index: | \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| Exponent: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Generators: |
$\langle(1,2,3,5,4)(6,12,13)(8,10,11)(9,15,14), (1,2,3,5,4), (1,5,2,4,3)(6,10,9)(8,14,13)(11,15,12)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 3$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $S_6:C_{10}$ |
| Order: | \(7200\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5^{2} \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $1$ |
The ambient group is nonabelian 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4\times S_6:C_2$, of order \(5760\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_4\times \GL(2,3)$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $W$ | $\SD_{16}$, of order \(16\)\(\medspace = 2^{4} \) |
Related subgroups
| Centralizer: | $C_3\times C_{15}$ | |
| Normalizer: | $F_9:C_{10}$ | |
| Normal closure: | $C_5\times A_6$ | |
| Core: | $C_5$ | |
| Minimal over-subgroups: | $S_3\times C_{15}$ | $C_{15}:S_3$ |
| Maximal under-subgroups: | $C_{15}$ | $C_3^2$ |
Other information
| Number of subgroups in this conjugacy class | $10$ |
| Möbius function | $0$ |
| Projective image | $S_6:C_2$ |