Subgroup ($H$) information
| Description: | $C_5^3:C_{18}$ |
| Order: | \(2250\)\(\medspace = 2 \cdot 3^{2} \cdot 5^{3} \) |
| Index: | \(3\) |
| Exponent: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Generators: |
$a^{3}d^{5}, d^{3}, d^{10}, b, a^{2}, c^{3}d^{3}$
|
| Derived length: | $2$ |
The subgroup is normal, maximal, a semidirect factor, nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $(C_5\times C_{15}^2).C_6$ |
| Order: | \(6750\)\(\medspace = 2 \cdot 3^{3} \cdot 5^{3} \) |
| Exponent: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_3$ |
| Order: | \(3\) |
| Exponent: | \(3\) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| 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, and simple.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_5\times C_{15}^2).C_{12}^2.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_5^3.C_{12}^2.C_2^2$ |
| $W$ | $C_3\times C_5^3:C_6$, of order \(2250\)\(\medspace = 2 \cdot 3^{2} \cdot 5^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $3$ |
| Möbius function | $-1$ |
| Projective image | $C_3\times C_5^3:C_6$ |