Subgroup ($H$) information
| Description: | $C_2^2\times D_{50}$ |
| Order: | \(400\)\(\medspace = 2^{4} \cdot 5^{2} \) |
| Index: | \(3\) |
| Exponent: | \(50\)\(\medspace = 2 \cdot 5^{2} \) |
| Generators: |
$a, d^{30}, c, d^{75}, b, d^{96}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a direct factor, nonabelian, a Hall subgroup, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_{150}:C_2^3$ |
| Order: | \(1200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \) |
| Exponent: | \(150\)\(\medspace = 2 \cdot 3 \cdot 5^{2} \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times D_{25}.C_{10}\times C_2^3.\PSL(2,7)$ |
| $\operatorname{Aut}(H)$ | $D_{25}.C_{10}\times C_2^3.\PSL(2,7)$ |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(672000\)\(\medspace = 2^{8} \cdot 3 \cdot 5^{3} \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $D_{25}$, of order \(50\)\(\medspace = 2 \cdot 5^{2} \) |
Related subgroups
| Centralizer: | $C_2^2\times C_6$ | ||
| Normalizer: | $C_{150}:C_2^3$ | ||
| Complements: | $C_3$ | ||
| Minimal over-subgroups: | $C_{150}:C_2^3$ | ||
| Maximal under-subgroups: | $C_2\times D_{50}$ | $C_2^2\times C_{50}$ | $C_2^2\times D_{10}$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_3\times D_{25}$ |