Subgroup ($H$) information
| Description: | $C_5^3:C_2$ |
| Order: | \(250\)\(\medspace = 2 \cdot 5^{3} \) |
| Index: | \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$b^{6}, d^{3}e^{3}, e^{3}, c^{3}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_{15}\wr S_3:C_4$ |
| Order: | \(81000\)\(\medspace = 2^{3} \cdot 3^{4} \cdot 5^{3} \) |
| Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_3^3:D_6$ |
| Order: | \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Automorphism Group: | $C_3^3.S_3^2$, of order \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| Outer Automorphisms: | $C_3$, of order \(3\) |
| Derived length: | $3$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{15}^2.(C_{12}\times S_3^2)\times F_5$ |
| $\operatorname{Aut}(H)$ | $\AGL(3,5)$, of order \(186000000\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{6} \cdot 31 \) |
| $W$ | $C_5^3:(C_4\times S_3)$, of order \(3000\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{3} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_{15}\wr S_3:C_4$ |