Subgroup ($H$) information
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(10000\)\(\medspace = 2^{4} \cdot 5^{4} \) |
| Exponent: | \(2\) |
| Generators: |
$a^{2}bc^{4}d^{8}ef^{2}, b^{2}cd^{6}e^{4}f^{3}$
|
| 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: | $C_5^4:(C_2^2\times \OD_{16})$ |
| Order: | \(40000\)\(\medspace = 2^{6} \cdot 5^{4} \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^4.C_2.C_2^5.C_2^5.C_2^2$ |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_2^3\times C_4$ | |||
| Normalizer: | $C_2^3\times C_4$ | |||
| Normal closure: | $C_2\times C_5^4:C_2^2$ | |||
| Core: | $C_1$ | |||
| Minimal over-subgroups: | $D_{10}$ | $D_{10}$ | $C_2^3$ | $C_2^3$ |
| Maximal under-subgroups: | $C_2$ | $C_2$ |
Other information
| Number of subgroups in this autjugacy class | $2500$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $C_5^4:(C_2^2\times \OD_{16})$ |