Subgroup ($H$) information
| Description: | $C_{10}$ |
| Order: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Index: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$d^{5}, d^{2}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and central.
Ambient group ($G$) information
| Description: | $C_2^2:C_{12}\times C_{20}$ |
| Order: | \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2\times C_4\times C_{12}$ |
| Order: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2^7:S_4$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| Outer Automorphisms: | $C_2^7:S_4$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^7.C_2^6.C_2$ |
| $\operatorname{Aut}(H)$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4096\)\(\medspace = 2^{12} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_2^2:C_{12}\times C_{20}$ | |||
| Normalizer: | $C_2^2:C_{12}\times C_{20}$ | |||
| Minimal over-subgroups: | $C_{30}$ | $C_2\times C_{10}$ | $C_2\times C_{10}$ | $C_2\times C_{10}$ |
| Maximal under-subgroups: | $C_5$ | $C_2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_2\times C_4\times C_{12}$ |