Subgroup ($H$) information
| Description: | $C_3^3:(C_5^3:S_4)$ |
| Order: | \(81000\)\(\medspace = 2^{3} \cdot 3^{4} \cdot 5^{3} \) |
| Index: | \(2\) |
| Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Generators: |
$ef^{9}, d^{6}e^{4}f^{6}, b^{2}c^{5}d^{3}e^{4}f^{13}, c^{3}, f^{10}, d^{15}f^{3}, ac^{4}e^{2}, c^{4}d^{20}, c^{2}f^{5}, f^{3}$
|
| Derived length: | $4$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^3:D_5\wr S_3$ |
| Order: | \(162000\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 5^{3} \) |
| Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $4$ |
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^3.C_6^2.(C_{12}\times S_3^2)$ |
| $\operatorname{Aut}(H)$ | $C_5^3.C_6^2.(C_{12}\times S_3^2)$ |
| $W$ | $C_3^3:D_5\wr S_3$, of order \(162000\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 5^{3} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^3:D_5\wr S_3$ |