Subgroup ($H$) information
| Description: | $D_{50}.C_2^3$ |
| Order: | \(800\)\(\medspace = 2^{5} \cdot 5^{2} \) |
| Index: | \(2\) |
| Exponent: | \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
| Generators: |
$a, d^{64}, c, d^{50}, b, d^{20}, c^{2}$
|
| Derived length: | $2$ |
The subgroup is normal, maximal, nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $D_{50}.C_2^4$ |
| 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.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4.C_2^4.C_{75}.C_{10}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $D_{25}.C_{10}\times C_2^3.\PSL(2,7)$ |
| $\card{\operatorname{res}(S)}$ | \(96000\)\(\medspace = 2^{8} \cdot 3 \cdot 5^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $C_{50}:C_4$, of order \(200\)\(\medspace = 2^{3} \cdot 5^{2} \) |
Related subgroups
| Centralizer: | $C_2^3$ | |||
| Normalizer: | $D_{50}.C_2^4$ | |||
| Minimal over-subgroups: | $D_{50}.C_2^4$ | |||
| Maximal under-subgroups: | $D_{50}:C_4$ | $C_2^2\times D_{50}$ | $D_{50}:C_4$ | $C_2^3\times F_5$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $-1$ |
| Projective image | $C_{50}:C_4$ |