Subgroup ($H$) information
| Description: | $(C_5\times C_{15}):D_6$ |
| Order: | \(900\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$ac^{3}e^{10}, e^{3}, c^{2}d^{10}, d^{10}, bc^{3}d^{12}e^{10}, d^{3}e^{12}$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, monomial (hence solvable), and an A-group.
Ambient group ($G$) information
| Description: | $C_{15}^2:(C_2\times D_6)$ |
| Order: | \(5400\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5^{2} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $S_3$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), hyperelementary for $p = 2$, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^2.\He_3.C_4.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_{15}^2.C_{12}.C_2^3$ |
| $W$ | $C_{15}^2:(C_2\times D_6)$, of order \(5400\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5^{2} \) |
Related subgroups
Other information
| Möbius function | $3$ |
| Projective image | $C_{15}^2:(C_2\times D_6)$ |