Subgroup ($H$) information
| Description: | $D_{25}\times C_{83}$ |
| Order: | \(4150\)\(\medspace = 2 \cdot 5^{2} \cdot 83 \) |
| Index: | \(3\) |
| Exponent: | \(4150\)\(\medspace = 2 \cdot 5^{2} \cdot 83 \) |
| Generators: |
$a, b^{75}, b^{2490}, b^{498}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a direct factor, nonabelian, a Hall subgroup, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $D_{25}\times C_{249}$ |
| Order: | \(12450\)\(\medspace = 2 \cdot 3 \cdot 5^{2} \cdot 83 \) |
| Exponent: | \(12450\)\(\medspace = 2 \cdot 3 \cdot 5^{2} \cdot 83 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Quotient group ($Q$) structure
| Description: | $C_3$ |
| Order: | \(3\) |
| Exponent: | \(3\) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| 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, and simple.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_{82}\times D_{25}.C_{10}$, of order \(82000\)\(\medspace = 2^{4} \cdot 5^{3} \cdot 41 \) |
| $\operatorname{Aut}(H)$ | $C_{82}\times D_{25}.C_{10}$, of order \(41000\)\(\medspace = 2^{3} \cdot 5^{3} \cdot 41 \) |
| $W$ | $D_{25}$, of order \(50\)\(\medspace = 2 \cdot 5^{2} \) |
Related subgroups
| Centralizer: | $C_{249}$ | ||
| Normalizer: | $D_{25}\times C_{249}$ | ||
| Complements: | $C_3$ | ||
| Minimal over-subgroups: | $D_{25}\times C_{249}$ | ||
| Maximal under-subgroups: | $C_{2075}$ | $D_5\times C_{83}$ | $D_{25}$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_3\times D_{25}$ |