Subgroup ($H$) information
| Description: | $C_2^2\wr C_2$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\langle(1,3)(8,10), (1,3)(4,7)(5,9)(8,10), (4,7)(8,10), (1,3)(2,6), (1,3)(2,7)(4,6)(5,10)(8,9)(11,12)(13,14)\rangle$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, and rational.
Ambient group ($G$) information
| Description: | $C_2^6:D_{10}$ |
| Order: | \(1280\)\(\medspace = 2^{8} \cdot 5 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_{16}.C_2^4.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2\wr S_3$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $\operatorname{res}(S)$ | $A_4:C_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(32\)\(\medspace = 2^{5} \) |
| $W$ | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
| Centralizer: | $C_2^4$ | ||
| Normalizer: | $C_2^4:D_4$ | ||
| Normal closure: | $C_2\wr D_5$ | ||
| Core: | $C_2^4$ | ||
| Minimal over-subgroups: | $C_2^4:D_5$ | $C_2^3:D_4$ | $C_2^3:D_4$ |
| Maximal under-subgroups: | $C_2^4$ | $C_2\times D_4$ | $C_2^2:C_4$ |
Other information
| Number of subgroups in this autjugacy class | $20$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $C_2^6:D_{10}$ |