Subgroup ($H$) information
| Description: | $C_2^2\times C_4^2$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Index: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$a^{5}bcd^{3}, b^{2}c^{3}e^{3}, d^{2}e^{2}, e^{2}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary).
Ambient group ($G$) information
| Description: | $C_4^4:C_{10}$ |
| Order: | \(2560\)\(\medspace = 2^{9} \cdot 5 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), metabelian, and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8.F_{16}:C_4$, of order \(245760\)\(\medspace = 2^{14} \cdot 3 \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2^8.\POPlus(4,3)$, of order \(147456\)\(\medspace = 2^{14} \cdot 3^{2} \) |
| $\operatorname{res}(S)$ | $C_2^4$, of order \(16\)\(\medspace = 2^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(256\)\(\medspace = 2^{8} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $60$ |
| Number of conjugacy classes in this autjugacy class | $12$ |
| Möbius function | $0$ |
| Projective image | $C_4^4:C_{10}$ |