Subgroup ($H$) information
| Description: | $C_{16}$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Generators: |
$b^{14}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Frattini subgroup (hence characteristic and normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and a $p$-group.
Ambient group ($G$) information
| Description: | $C_{15}:\SD_{64}$ |
| Order: | \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| Exponent: | \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $S_3\times C_{10}$ |
| Order: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Automorphism Group: | $C_4\times D_6$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Outer Automorphisms: | $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:((C_2\times C_4\times C_8).C_2^5)$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_3:C_{160}$ | ||||
| Normalizer: | $C_{15}:\SD_{64}$ | ||||
| Minimal over-subgroups: | $C_{80}$ | $C_{48}$ | $D_{16}$ | $Q_{32}$ | $C_{32}$ |
| Maximal under-subgroups: | $C_8$ |
Other information
| Möbius function | $6$ |
| Projective image | $C_{15}:D_{16}$ |