Subgroup ($H$) information
| Description: | $C_3:C_{32}$ |
| Order: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Generators: |
$b, c^{10}, b^{16}, b^{2}, b^{4}, b^{8}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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: | $C_{10}$ |
| Order: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5$), hyperelementary, metacyclic, metabelian, a Z-group, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:((C_2\times C_4\times C_8).C_2^5)$ |
| $\operatorname{Aut}(H)$ | $C_8\times D_6$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_8\times D_6$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(64\)\(\medspace = 2^{6} \) |
| $W$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_{80}$ | |
| Normalizer: | $C_{15}:\SD_{64}$ | |
| Complements: | $C_{10}$ | |
| Minimal over-subgroups: | $C_3:C_{160}$ | $C_3:\SD_{64}$ |
| Maximal under-subgroups: | $C_{48}$ | $C_{32}$ |
Other information
| Möbius function | $1$ |
| Projective image | $C_{15}:D_{16}$ |