Subgroup ($H$) information
| Description: | $C_{15}\wr S_3:C_2$ |
| Order: | \(40500\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 5^{3} \) |
| Index: | \(2\) |
| Exponent: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Generators: |
$e^{3}, d^{3}e^{3}, b^{6}, c^{3}, b^{4}d^{3}e^{12}, c^{10}, ace, e^{10}, c^{5}d^{10}$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, and solvable. Whether it is monomial has not been computed.
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: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| 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, simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{15}^2.(C_{12}\times S_3^2)\times F_5$ |
| $\operatorname{Aut}(H)$ | $(C_5\times C_{15}^2).C_6^2.C_2^4$ |
| $W$ | $C_{15}^2:(S_3\times F_5)$, of order \(27000\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5^{3} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_{15}\wr S_3:C_4$ |