Subgroup ($H$) information
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(400\)\(\medspace = 2^{4} \cdot 5^{2} \) |
| Exponent: | \(2\) |
| Generators: |
$c^{2}, d^{25}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Ambient group ($G$) information
| Description: | $(C_2^2\times C_{100}):C_4$ |
| Order: | \(1600\)\(\medspace = 2^{6} \cdot 5^{2} \) |
| Exponent: | \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{50}.(C_2^4\times C_{20}).C_2^2$ |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_1$, of order $1$ |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(32000\)\(\medspace = 2^{8} \cdot 5^{3} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_2^2.D_{100}$ | |||
| Normalizer: | $C_2^2.D_{100}$ | |||
| Normal closure: | $C_2^3$ | |||
| Core: | $C_2$ | |||
| Minimal over-subgroups: | $C_2\times C_{10}$ | $C_2^3$ | $C_2^3$ | $C_2\times C_4$ |
| Maximal under-subgroups: | $C_2$ | $C_2$ | $C_2$ |
Other information
| Number of subgroups in this conjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $D_{50}.D_4$ |