Subgroup ($H$) information
| Description: | $C_5^2:D_{15}$ |
| Order: | \(750\)\(\medspace = 2 \cdot 3 \cdot 5^{3} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$a^{6}b^{9}, b^{2}, de^{2}, cd^{3}, e$
|
| 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: | $C_5^3:(S_3\times C_{12})$ |
| Order: | \(9000\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5^{3} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, monomial (hence solvable), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{12}$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_5^3.(C_4^2\times S_3)$ |
| $\operatorname{Aut}(H)$ | $C_5^3:(S_3\times C_4^2)$, of order \(12000\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{3} \) |
| $W$ | $C_5^3:(C_4\times S_3)$, of order \(3000\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{3} \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_5^3:(S_3\times C_{12})$ |