Subgroup ($H$) information
| Description: | $C_5^2:D_{15}$ |
| Order: | \(750\)\(\medspace = 2 \cdot 3 \cdot 5^{3} \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(2,14,5,11,7)(4,12,15,10,8), (1,3)(5,11)(6,9)(7,14)(8,12)(10,15)(17,18), (2,5,7,14,11), (16,18,17), (1,9,13,6,3)\rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_5^3:(S_3\times S_4)$ |
| Order: | \(18000\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{3} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $S_4$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $3$ |
The quotient is nonabelian, monomial (hence solvable), and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\times D_5^3.D_6$, of order \(72000\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $S_3\times C_5^2:D_5.C_2.\PSL(3,5)$ |
| $W$ | $C_5^3:(S_3\times S_4)$, of order \(18000\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{3} \) |
Related subgroups
Other information
| Möbius function | $-12$ |
| Projective image | $C_5^3:(S_3\times S_4)$ |