Subgroup ($H$) information
| Description: | $C_2^2\times C_{100}$ |
| Order: | \(400\)\(\medspace = 2^{4} \cdot 5^{2} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
| Generators: |
$a, d^{20}, cd^{75}, d^{44}, b, c^{2}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary).
Ambient group ($G$) information
| Description: | $C_2^2\times C_4:C_{100}$ |
| Order: | \(1600\)\(\medspace = 2^{6} \cdot 5^{2} \) |
| Exponent: | \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8.C_2^4.C_{60}.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_2^3.(C_{20}\times S_4)$ |
| $\card{\operatorname{res}(S)}$ | \(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(16\)\(\medspace = 2^{4} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $16$ |
| Number of conjugacy classes in this autjugacy class | $8$ |
| Möbius function | $0$ |
| Projective image | $D_4$ |