Subgroup ($H$) information
| Description: | $C_3\times C_5^2:S_3$ |
| Order: | \(450\)\(\medspace = 2 \cdot 3^{2} \cdot 5^{2} \) |
| Index: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$b^{5}, cd^{3}, de^{3}, a^{4}, e$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, monomial (hence solvable), and an A-group.
Ambient group ($G$) information
| Description: | $C_5^3:(S_3\times C_{12})$ |
| Order: | \(9000\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5^{3} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, monomial (hence solvable), and an A-group.
Quotient group ($Q$) structure
| Description: | $F_5$ |
| Order: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Automorphism Group: | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_5^3.(C_4^2\times S_3)$ |
| $\operatorname{Aut}(H)$ | $C_5^2:(C_4\times D_6)$, of order \(1200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \) |
| $W$ | $C_5^2:(C_4\times S_3)$, of order \(600\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_5^3:(C_4\times S_3)$ |